(ebook-pdf) Mathematics - Handbook of Mathematical Functions

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Unformatted text preview: Preface’ The present volume is an outgrowth of a Conference on Mathematical Tables held at Cambridge, Mass., on September 15-16, 1954, under the auspices of the National Science Foundation and the Massachusetts Institute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale computing machines. It was the consensus of opinion that in spite of the increasing use of the new machines the basic need for tables would continue to exist. Numerical tables of mathematical functions are in continual demand by scientists and engineers. A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers. In the latter connection, the tables serve mainly forpreliminarysurveys of problems before programming for machine operation. For those without easy access to machines, such tables are, of course, indispensable. Consequently, the Conference recognized that there was a pressing need for a modernized version of the classical tables of functions of Jahnke-Emde. To implement the project, the National Science Foundation requested the National Bureau of Standards to prepare such a volume and established an Ad Hoc Advisory Committee, with Professor Philip M. Morse of the Massachusetts Institute of Technology as chairman, to advise the staff of the National Bureau of Standards during the ~course of its preparation. In addition to the Chairman, the Committee consisted of A. Erdelyi, M. C. Gray, N. Metropolis, J. B. Rosser, H. C. Thacher, Jr., John Todd, C. B. Tompkins, and J. W. Tukey. The primary aim has been to include a maximum of useful information within the limits of a moderately large volume, with particular attention to the needs of scientists in all fields. An attempt has been made to cover the entire field of special functions. To carry out the goal set forth by tbe Ad Hoc Committee, it has been necessary to supplement the tables by including the mathematical properties that are important in computation work, as well as by providing numerical methods which demonstrate the use and extension of the tables. The Handbook was prepared under the direction of the late Milton Abramowitz, Its success has depended greatly upon the cooperation of and Irene A. Stegun. Their efforts together with the cooperation of the Ad HOC many mathematicians. Committee are greatly appreciated. The particular contributions of these and other individuals are acknowledged at appropriate places in the text. The sponsorship of the National Science Foundation for the preparation of the material is gratefully recognized. It is hoped that this volume will not only meet the needs of all table users but will in many cases acquaint its users with new functions. ALLEN V. ASTIN, L?imctor. Washington, D.C. Preface to the Ninth Printing The enthusiastic reception accorded the “Handbook of Mathematical Functions” is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614. Only four and one-half years after the first copy came from the press in 1964, Myron Tribus, the Assistant Secretary of Commerce for Science and Technology, presented the 100,OOOth copy of the Handbook to Lee A. DuBridge, then Science Advisor to the President. Today, total distribution is approaching the 150,000 mark at a scarcely diminished rate. The successof the Handbook has not ended our interest in the subject. On the contrary, we continue our close watch over the growing and changing world of computation and to discuss with outside experts and among ourselves the various proposals for possible extension or supplementation of the formulas, methods and tables that make up the Handbook. In keeping with previous policy, a number of errors discovered since the last printing have been corrected. Aside from this, the mathematical tables and accompanying text are unaltered. However, some noteworthy changes have been made in Chapter 2: Physical Constants and Conversion Factors, pp. 6-8. The table on page 7 has been revised to give the values of physical constants obtained in a recent reevaluation; and pages 6 and 8 have been modified to reflect changes in definition and nomenclature of physical units and in the values adopted for the acceleration due to gravity in the revised Potsdam system. The record of continuing acceptance of the Handbook, the praise that has come from all quarters, and the fact that it is one of the most-quoted scientific publications in recent years are evidence that the hope expressed by Dr. Astin in his Preface is being amply fulfilled. LEWIS M. BRANSCOMB, Director National Bureau of Standards November 1970 Foreword This volume is the result of the cooperative effort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had under consideration, for at least IO years, the production of a compendium like the present one. During a Conference on Tables, called by the NBS Applied Mathematics Division on May 15, 19.52, Dr. Abramowitz of t,hat Division mentioned preliminary plans for such an undertaking, but indicated the need for technical advice and financial support. The Mathematics Division of the National Research Council has also had an active interest in tables; since 1943 it has published the quarterly journal, “Mathematical Tables and Aids to Computation” (MTAC),, editorial supervision being exercised by a Committee of the Division. Subsequent to the NBS Conference on Tables in 1952 the attention of the National Science Foundation was drawn to the desirability of financing activity in table production. With its support a z-day Conference on Tables was called at the Massachusetts Institute of Technology on September 15-16, 1954, to discuss the needs for tables of various kinds. Twenty-eight persons attended, representing scientists and engineers using tables as well as table producers. This conference reached consensus on several cpnclusions and recomlmendations, which were set forth in tbe published Report of the Conference. There was general agreement, for example, “that the advent of high-speed cornputting equipment changed the task of table making but definitely did not remove the need for tables”. It was also agreed that “an outstanding need is for a Handbook of Tables for the Occasional Computer, with tables of usually encountered functions and a set of formulas and tables for interpolation and other techniques useful to the occasional computer”. The Report suggested that the NBS undertake the production of such a Handbook and that the NSF contribute financial assistance. The Conference elected, from its participants, the following Committee: P. M. Morse (Chairman), M. Abramowitz, J. H. Curtiss, R. W. Hamming, D. H. Lehmer, C. B. Tompkins, J. W. Tukey, to help implement these and other recommendations. The Bureau of Standards undertook to produce the recommended tables and the To provide technical guidance National Science Foundation made funds available. to the Mathematics Division of the Bureau, which carried out the work, and to provide the NSF with independent judgments on grants ffor the work, the Conference Committee was reconstituted as the Committee on Revision of Mathematical Tables of the Mathematics Division of the National Research Council. This, after some changes of membership, became the Committee which is signing this Foreword. The present volume is evidence that Conferences can sometimes reach conclusions and that their recommendations sometimes get acted on. V ,/” VI FOREWORD Active work was started at the Bureau in 1956. The overall plan, the selection of authors for the various chapters, and the enthusiasm required to begin the task were contributions of Dr. Abramowitz. Since his untimely death, the effort has continued under the general direction of Irene A. Stegun. The workers at the Bureau and the members of the Committee have had many discussions about content, style and layout. Though many details have had t’o be argued out as they came up, the basic specifications of the volume have remained the same as were outlined by the Massachusetts Institute of Technology Conference of 1954. The Committee wishes here to register its commendation of the magnitude and quality of the task carried out by the staff of the NBS Computing Section and their expert collaborators in planning, collecting and editing these Tables, and its appreciation of the willingness with which its various suggestions were incorporated into the plans. We hope this resulting volume will be judged by its users to be a worthy memorial to the vision and industry of its chief architect, Milton Abramowitz. We regret he did not live to see its publication. P. M. MORSE, Chairman. A. ERD~LYI M. C. GRAY N. C. METROPOLIS J. B. ROSSER H. C. THACHER. Jr. JOHN TODD ‘C. B. TOMPKINS J. W. TUKEY. Handbook of Mathematical Functions with Formulas, Edited Graphs, by Milton 1. and Mathematical Abramowitz and Irene A. Stegun Introduction The present Handbook has been designed to provide scientific investigators with a comprehensive and self-contained summary of the mathematical functions that arise in physical and engineering problems. The well-known Tables of Funct.ions by E. Jahnke and F. Emde has been invaluable to workers in these fields in its many editions’ during the past half-century. The present volume ext,ends the work of these authors by giving more extensive and more accurate numerical tables, and by giving larger collections of mathematical properties of the tabulated functions. The number of functions covered has also been increased. The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. C. P. Miller, and L. Rosenhead. In general, the chapters contain numerical tables, graphs, polynomial or rational approximations for automatic computers, and statements of the principal mathematical properties of the tabulated functions, particularly those of computa- 2. Tables Accuracy The number of significant figures given in each table has depended to some extent on the number available in existing tabulations. There has been no attempt to make it uniform throughout the Handbook, which would have been a costly and laborious undertaking. In most tables at least five significant figures have been provided, and the tabular’ intervals have generally been chosen to ensure that linear interpolation will yield. fouror five-figure accuracy, which suffices in most physical applications. Users requiring higher 1 The most recent, the sixth, with F. Loesch added as cc-author, was published in 1960 by McGraw-Hill, U.S.A., and Teubner, Germany. 2 The second edition, with L. J. Comrie added as co-author, was published in two volumes in 1962 by Addison-Wesley, U.S.A., and Scientific Computing Service Ltd., Great Britain. tional importance. Many numerical examples are given to illustrate the use of the tables and also the computation of function values which lie outside their range. At the end of the text in each chapter there is a short bibliography giving books and papers in which proofs of the mathematical properties stated in the chapter may be found. Also listed in the bibliographies are the more important numerical tables. Comprehensive lists of tables are given in the Index mentioned above, and current information on new tables is to be found in the National Research Council quarterly Mathematics of Computation (formerly Mathematical Tables and Other Aids to Computation). The ma.thematical notations used in this Handbook are those commonly adopted in standard texts, particularly Higher Transcendental Functions, Volumes 1-3, by A. ErdBlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi (McGrawHill, 1953-55). Some alternative notations have also been listed. The introduction of new symbols has been kept to a minimum, and an effort has been made to avoid the use of conflicting notation. of the Tables precision in their interpolates may obtain them by use of higher-order interpolation procedures, described below. In certain tables many-figured function values are given at irregular intervals in the argument. An example is provided by Table 9.4. The purpose of these tables is to furnish “key values” for the checking of programs for automatic computers; no question of interpolation arises. The maximum end-figure error, or “tolerance” in the tables in this Handbook is 6/& of 1 unit everywhere in the case of the elementary functions, and 1 unit in the case of the higher functions except in a few cases where it has been permitted to rise to 2 units. IX /- . INTRODUCTION X 3. Auxiliary Functions One of the objects of this Handbook is to provide tables or computing methods which enable the user to evaluate the tabulated functions over complete ranges of real values of their parameters. In order to achieve this object, frequent use has been made of auxiliary functions to remove the infinite part of the original functions at their singularities, and auxiliary arguments to co e with infinite ranges. An example will make t fi e procedure clear. The exponential integral of positive argument is given by 4. and Arguments recludes direct interThe logarithmic singularity polation near x=0. The Punctions Ei(x)-In x and x-liEi(ln x-r], however, are wellbehaved and readily interpolable in this region. Either will do as an auxiliary function; the latter was in fact selected as it yields slightly higher accuracy when Ei(x) is recovered. The function x-‘[Ei(x)-ln x-r] has been tabulated to nine decimals for the range 05x<+. For +<x12, Ei(x) is sufficiently well-behaved to admit direct tabulation, but for larger values of x, its exponential character predominates. A smoother and more readily interpolable function for large x is xe-“Ei(x); this has been tabulated for 2 <x510. Finally, the range 10 <x_<m is covered by use of the inverse argument x-l. Twenty-one entries of xe-“Ei(x), corresponding to x-l = .l(- .005)0, suffice to produce an interpolable table. Interpolation The tables in this Handbook are not provided with differences or other aids to interpolation, because it was felt that the space they require could be better employed by the tabulation of additional functions. Admittedly aids could have been given without consuming extra space by increasing the intervals of tabulation, but this would have conflicted with the requirement that linear interpolation is accurate to four or five figures. For applications in which linear interpolation is insufficiently accurate it is intended that Lagrange’s formula or Aitken’s method of iterative linear interpolation3 be used. To help the user, there is a statement at the foot of most tables of the maximum error in a linear interpolate, and the number of function values needed in Lagrange’s formula or Aitken’s method to interpolate to full tabular accuracy. As an example, consider the following extract from Table 5.1. Let us suppose that we wish to compute the value of xeZ&(x) for x=7.9527 from this table. We describe in turn the application of the methods of linear interpolation, Lagrange and Aitken, and of alternative methods based on differences and Taylor’s series. (1) Linear interpolation. The formula for this process is given by jp= (1 -P)joSPfi where jO, ji are consecutive tabular values of the function, corresponding to arguments x0, x1, respectively; p is the given fraction of the argument interval p= (x--x0>/(x1-~0> 775 ;:; E : 89608 89717 4302 8737 d0 g. I ze*El . 89823 .89927 90029 (z) 7113 7306 7888 8: ix 4 .90227 : 90129 4695 60”3 [1 ‘453 The numbers in the square brackets mean that the maximum error in a linear interpolate is 3X10m6, and that to interpolate to the full tabular accuracy five points must be used in Lagrange’s and Aitken’s methods. 8 A. C. Aitken On inte elation b iteration of roportional out the use of diherences, ‘Brot Edin i: urgh Math. 8 oc. 3.6676 parts, with. (1932). interpolate. jo=.89717 zez El (2) . 89268 7854 : 89384 6312 89497 9666 and jP the required instance, we have ji=.89823 4302 In the present 7113 p=.527 The most convenient way to evaluate the formula on a desk calculating machine is.to set o and ji in turn on the keyboard, and carry out t d e multiplications by l-p and p cumulatively; a partial check is then provided by the multiplier dial reading unity. We obtain j.6z,E.‘;9;72;&39717 4302)+.527(.89823 7113) Since it is known that there is a possible error of 3 X 10 -6 in the linear formula, we round off this result to .89773. The maximum possible error in this answer is composed of the error committed INTRODUCTION by the last roundingJ that is, .4403X 10m5, plus 3 X lo-‘, and so certainly cannot exceed .8X lo-‘. (2) Lagrange’s formula. In this example, the relevant formula is the 5-point one, given by The numbers in the third and fourth columns are the first and second differences of the values of xezEl(x) (see below) ; the smallness of the second difference provides a check on the three interpolations. The required value is now obtained by linear interpolation : f=A-,(p)f_z+A-,(p)f-1+Ao(p>fo+A,(p)fi +A&)fa Tables of the coefficients An(p) are given in chapter 25 for the range p=O(.Ol)l. We evaluate the formula for p=.52, .53 and .54 in turn. Again, in each evaluation we accumulate the An(p) in the multiplier register since their sum is unity. We now have the following subtable. x m=&(x) 7.952 .89772 .89774 0379 fn=.3(.89772 -2 10620 .; .89775 & 1 2 3 4 5 7.9 8.1 7.8 8.2 7.7 0999 Yn=ze”G@) : : . . 89823 89717 89927 89608 90029 89497 7113 4302 7888 8737 7306 9666 Yo. I 89773 :89774 Yo, 1.2. I Yo. 1, (I 44034 48264 2 90220 4 98773 2 35221 0379) In cases where the correct order of the Lagrange polynomial is not known, one of the prelimina interpolations may have to be performed witT polynomials of two or more different orders as a check on their adequacy. (3) Aitken’s method of iterative linear interpolation. The scheme for carrying out this process in the present example is as follows: 10622 7.954 9757)+.7(.89774 = 239773 7192. 9757 7.953 XI X,-X Yo.1.a.s.n .0473 0527 . 1473 -. 1527 . 2473 -. 2527 -. .89773 71499 2394 1216 2706 . 89773 71938 89773 ii 71930 30 Here 20-x 1 Yo x,-x x.--20 Yn x,-x 1 Yo.1 Yo.1 ,n=x,-x G--z1 l/O.” S2fl yo,n=- Yo. 1. .. ., m--l.m.n-- 1 ~n-%n safz wa l/0.1. . . ., n-1.98 Yo.1. . . -, m-1.n x,-x x,-x 1 If the quantities Z.-X and x~--5 are used as multipliers when forming the cross-product on a desk machine, their accumulation (~~-2) -(x,-x) in the multiplier register is the divisor to be used at that stage. An extra decimal place is usually carried in the intermediate interpolates to safeguard against accumulation of rounding errors. The order in which the tabular values are used is immaterial to some extent, but to achieve the maximum rate of convergence and at the same time minimize accumulation of rounding errors, we begin, as in this example, with the tabular argument nearest to the given argument, then take the nearest of the remaining tabular arguments, and so on. The number of tabular values required to achieve a given precision emerges naturally in the course of the iterations. Thus in the present example six values were used, even though it was known in advance that five would suffice. The extra row confirms the convergence and provides a valuable check. (4) Difference formulas. We use the central difference notation (chapter 25), Here Sf1l2=f1-f0, 8f3/a=fz-f1, . . . ,, a2/1=sf3ia-afiia=fa-2fi+fo ~af3~~=~aja-~aj~=fa-3j2+3fi-k 8'fa=~aj~fsla-6~3~2=f4-~f~+~ja-4f~+fo and so on. In the present example the relevant part of the difference table is as follows, the differences being lace of the written in units of the last decimal function, as is customary. The sma Bness of the high differences provides a check on the function values 7:9 8.0 Applying, formula xe=El(x) .89717 4302 . 89823 7113 for example, j~=(l-P)fo+E2(P)~*jo+E4(P)~4jo+ -2 -2 SY 2754 2036 Everett’s ... +Pfl+F2(P)~afl+F4(P)~4fl+ S4f -34 -39 interpolation . .** and takin the numerical values of the interpolation toe flf cients Es(p), E4 ), F,(p) and F,(p) from Table 25.1, we find t!l at ,,/ INTRODUCTION XII 10Qf.6,= .473(89717 4302) + .061196(2 2754) - .012(34) + .527(89823 7113) + .063439(2 2036) - .012(39) = 89773 7193. We may notice in passing that Everett’s formula shows that the error in a linear interpolate is approximately mPwfo+ F2(P)wl= m(P) can be used. We first compute as many of the derivatives ftn) (~0) as are significant, and then evaluate the series for the given value of 2. An advisable check on the computed values of the derivatives is to reproduce the adjacent tabular values by evaluating the series for z=zl and x1. In the present + ~2(P)lk?f0+wJ f’(z)=(l+Z-‘)f(Z)-1 f”(2)=(1+2-‘)f’(Z)--Z-Qf(2) f”‘(X) = (1 -i-z-y’(2) Inverse With linear interpolation there is no difference in principle between direct and inverse interpolation. In cases where the linear formula rovides an insufficiently accurate answer, two met fl ods are available. We may interpolate directly, for example, by Lagrange’s formula to prepare a new table at a fine interval in the neighborhood of the approximate value, and then apply accurate inverse linear interpolation to the subtabulated values. Alternatively, we may use Aitken’s method or even possibly the Taylor’s series method, with the roles of function and argument interchanged. It is important to realize that the accuracy of an inverse interpolate may be very different from that of a direct interpolate. This is particularly true in regions where the function is slowly varying, for example, near a maximum or minimum. The maximum precision attainable in an inverse interpolate can be estimated with the aid of the formula AxmAj/df dx in which Aj is the maximum possible error in the function values. Example. Given xe”Ei(z) = .9, find 2 from the table on page X. (i) Inverse linear interpolation. The formula for v is In the present ‘=.90029 example, we have .9 - .89927 7888 72 2112 7306.89927 7888=101=‘708357’ p:‘(xo)/k! i +22-y(2). (x--so) y’k’(x0)/k! .89717 4302 .89717 4302 - .01074 .00113 .00012 ; 3 ... 5. -22~Qf’(5) With x0=7.9 and x-x0= .0527 our computations are as follows: an extra decimal has been retained in the values of the terms in the series to safeguard against accumulation of rounding errors. ~(x,=~(xo)+(x-x,,~~+(x-xo,~~~ +(~-x,)q$+ we have f(x) =xeZEt(x) Since the maximum value of IEz(p)+Fz(p)I in the range O<p<l is fd, the maximum error in a linear interpolate is approximately (5) Taylor’s series. In cases where the successive derivatives of the tabulated function can be computed fairly easily, Taylor’s expansion example, 7621 0669 1987 -.ooooo .00056 .ooooo .a9773 3159 6033 7194 0017 5 3 9 Interpolation The desired z is therefore z=zQ+p(z,--2,,)=8.1+.708357(.1)=8.17083 57 To estimate the possible error in this answer, we recall that the maximum error of direct linear interpolation in this table is Aj=3X lOwe. An approximate value for dj/dx is the ratio of the first difference to the argument interval (chapter 25), in this case .OlO. Hence the maximum error in x is approximately 3XlO-e/(.OlO), that is, .0003. (ii) Subtabulation method. To improve the ap roximate value of x just obtained, we interpo P ate directly for p=.70, .7l and .72 with the aid of Lagrange’s 5-point formula, xe=El (x) X 8. 170 . 8.171 . 90000 6 QQ 89999 -.-_ 1 0151 3834 -2 1 0149 8. 172 Inverse gives 90001 linear Hencex=8.17062 An estimate 3983 interpolation in the new table 23. of the maximum error in this result is df~1x10-8_1x10-7 Ajl z .OlO (iii) Aitken’s method. This is carried out in the same manner as in direct interpolation. INTRODUCTION n 0 yn=xeeZE1(x) . 90029 7306 2, 8. 2 4 3 : 89927 90129 . 89823 7888 6033 7113 8. 3 1 8. 0 8. 17083 17023 8. 17113 5712 1505 8043 % : 90227 89717 4302 4695 7. 9 8. 4 8. 16992 17144 0382 9437 Z0.n QJ.98 8. 1706,l 8. 17062 2 8142 1 7335 6. Bivariate Bivariate interpolation is generally most simply performed as a sequence of univariate interpolations. We carry out the interpolation in one direction, by one of the methods already described, for several tabular values of the second argument in the neighborhood of its given value. The interpolates are differenced as a check, and Generation a.1 2.n 9521 2 5948 The estimate of the maximum error in this result is the same as in the subtabulation method. An indication of the error is also provided by the 7. XIII of Functions Many of the special mathematical functions which depend on a parameter, called their index, order or degree, satisfy a linear difference equation (or recurrence relation) with respect to this parameter. Examples are furnished by the Legendre function P,(z), the Bessel function Jn(z) and the exponential integral E,(x), for which we have the respective recurrence relations zo.l.2.3.74 231 415 8. 17062 discrepancy in the highest case xo .I ,2.3 A, and ZLI .2.8 .s. 6033 2112 2887 -. .00227 00282 2318 265 7306 -. . 00072 00129 -. 00176 2244 5G98 4695 interpolates, in this I Interpolation interpolation is then carried out in the second direction. An alternative procedure in the case of functions of a complex variable is to use the Taylor’s series expansion, provided that successive derivatives of the function can be computed without much difficulty. from Recurrence Relations (iii) the direction in which the recurrence applied. Examples are as follows. Stability-increasing Pm(x), Qnb), Q:(x) y&9, n KG) J-n-&), (x<l) z-t44 (n<d Stability-decreasing Jn+*-~Jn+J.-l=O P”(X), nE,+,+xE,,=e-=. is being p:(2) &Cd Particularly for automatic work, recurrence relations provide an important and powerful computing tool. If the values of P&r) or Jn(z) are known for two consecutive values of n, or E',(z) is known for one value of n, then the function may be computed for other values of n by successive applications of the relation. Since generation is carried out perforce with rounded values, it is vital to know how errors may be propagated in the recurrence process. If the errors do not grow relative to the size of the wanted function, the process is said to be stable. If, however, the relative errors grow and will eventually overwhelm the wanted function, the process is unstable. It is important to realize that st,ability may depend on (i) the particular solution of the difference equation being computed; (ii) the values of x or other parameters in the difference equation; l/n-u .00029 P.,(z) 7t @<l) Qnh), Q:(x) J&4, Jn+Hcd Em(z) F,,(t, Z.@) , Zn+&) (n >r) p) (Coulomb wave function) Illustrations of the generation of functions from their recurrence relations are given in the pertinent chapters. It is also shown that even in cases where the recurrence process is unstable, it may still be used when the starting values are known to sufficient accuracy. Mention must also be made here of a refinement, due to J. C. P. Miller, which enables a recurrence process which is stable for decreasing n to be applied without any knowledge of starting values for large n. Miller’s algorithm, which is wellsuited to automatic work, is described in 19.28, Example 1. INTRODUCTION XIV 8. Acknowledgments The production of this volume has been the result of the unrelenting efforts of many persons, all of whose contributions have been instrumental in accomplishing the task. The Editor expresses his thanks to each and every one. The Ad Hoc Advisory Committee individually and together were instrumental in establishing the basic tenets that served as a guide in the formation of the entire work. In particular, special thanks are due to Professor Philip M. Morse for his continuous encouragement and support. Professors J. Todd and A. Erdelyi, panel members of the Conferences on Tables and members of the Advisory Committee have maintained an undiminished interest, offered many suggestions and carefully read all the chapters. Irene A. Stegun has served eff ectively as associate editor, sharing in each stage of the planning of the volume. Without her untiring efforts, completion would never have been possible. Appreciation is expressed for the generous cooperation of publishers and authors in granting permission for the use of their source material. Acknowledgments for tabular material taken wholly or in part from published works are iven on the first page of each table. Myrtle R. Ke Yi lington corresponded with authors and publishers to obtain formal permission for including their material, maintained uniformity throughout the bibliographic references and assisted in preparing the introductory material. Valuable assistance in the preparation, checkin and editing of the tabular material was receive IFi from Ruth E. Capuano, Elizabeth F. Godefroy, David S. Liepman, Kermit Nelson, Bertha H. Walter and Ruth Zucker. Equally important has been the untiring cooperation, assistance, and patience of the members of the NBS staff in handling the myriad of detail necessarily attending the publication of a volume of this magnitude. Especially appreciated have been the helpful discussions and services from the members of the Office of Technical Information in the areas of editorial format, graphic art layout, printing detail, preprinting reproduction needs, as well as attention to promotional detail and financial support. In addition, the clerical and typing stafI of the Applied Mathematics Division merit commendation for their efficient and patient production of manuscript copy involving complicated technical notation. Finally, the continued support of Dr. E. W. Cannon, chief of the Applied Mathematics Division, and the advice of Dr. F. L. Alt, assistant chief, as well as of the many mathematicians in the Division, is gratefully acknowledged. M. ABRAMOWITZ. 1. Mathematical DAVID Constants S. LIEPMAN ’ Contents Page Mathematical Table 1.1. +i,nprime Constants <lOO, 20s. ............... 2 .................. 2 Some roots of 2, 3, 5, 10, 100, 1000, e, 20s .......... 2 e*n, n=l(l)lO, .................... 2 20s .................... 2 25s e*tns, n=l(l)lO, eas e*‘, , 20s ....................... ln n, log,, n, n=2(1)10, In 7~,In&, na, n=1(1)9, a*“, n=l(l)lO, ........ 24D r(4), l/r($), 3 3 25s .................... of T, powers and roots involving 26s 3 T, 25s ....... .................. 3 ....................... 3 15D ..................... Bureau of Standards. 3 3 24D. ................. r(2),l/r(z),lnr(2),2~3,a,g,q,~,g,g,~, 2 3 25s ..................... lo, l’, 1” in radians, 1 National 26, 25s 25s ................... 1 radian in degrees, ~,lny, primes <lOO, logI, ?r, log,, e, 25s ............... n In 10, n=1(1)9, Fractions 2 3 15D. ........ 3 MATHEMATICAL TABLE 1.1. 1 4142 i7320 2.2360 2..6457 3.3166 3.6055 4.1231 4.3588 4.7958 5.3851 5.5677 6.0827 6.4031 6.5574 6.8556 7.2801 7.6811 7.8102 8.1853 8.4261 8.5440 8.8881 9.1104 9.4339 9.8488 13562 50807 67977 51311 24790 51275 05625 98943 31523 64807 64362 62530 24237 38524 54600 09889 45747 49675 52771 49773 03745 94417 33579 81132 57801 37;: 56887 49978 06459 35539 46398 61766 54067 31271 13450 83002 29821 43284 30200 40104 28051 86860 90665 87244 i7635 31753 31558 14429 05660 79610 1) lj 2) 2) 3) 3) 3) 4) 81828 56098 53692 15003 31591 87934 33158 57987 83927 46579 45:04 93065 31876 31442 02576 92735 42845 04172 57538 48067 52353 02272 67740 39078 60342 12260 85992 82747 40077 16516 1) 2) 4) 5) 6) 8j 9) LO) 12) 13) 2.3140 5.3549 1.2391 2.8675 6.6356 1.5355 3.5533 8.2226 1.9027 4.4031 69263 16555 64780 13131 23999 29353 21280 31558 73895 50586 enr 27792 24764 79166 36653 34113 95446 84704 55949 29216 06320 69006 73650 97482 29975 42333 69392 43597 95275 12917 29011 26224 72417 14792 99019 64190 79852 47180 12288 94361 37912 59469 10149 41541 24577 85092 95272 49357 13344 38979 94215 95829 87204 17912 72066 00115 In n 55994 66810 11989 43410 22805 05531 67983 33621 99404 79837 46153 05621 16644 92914 98647 48514 64422 70430 69356 MATHEMATICAL :E: 19221 96890 86865 06523 41249 82711 81758 43941 99700 86306 11679 88501 88819 38113 47217 2.7182 7.3890 2.0085 5.4598 1.4841 4.0342 1.0966 2.9809 8.1030 2.2026 0.6931 1.0986 1.3862 1.6094 1.7917 1.9459 2.0794 2.1972 2.3025 2.3978 2.5649 2.8332 2.9444 3.1354 3.3672 3.4339 3.6109 I: % CONSTANTS 50488 72935 96964 05905 98491 92931 05498 35522 CONSTANTS 1O’fi 3.1622 77660 16837 93320 1O’fl 10"' 2.1544 1.7782 1.5848 79410 34690 93192 03892 03188 46111 %X 34853 4.6415 2.5118 5.6234 3.9810 1.2599 1.4422 1.1892 1.3160 7.0710 5.7735 4.4721 88833 86431 13251 71705 21049 49570 07115 74012 67811 02691 35954 61277 50958 90349 53497 89487 30740 06275 95249 86547 89625 99957 FKd 08040 25077 31648 83823 20667 24608 52440 76451 93928 4.8104 2.1932 11 2.0787 li 4.5593 1.6487 1) 6.0653 1.3956 1) 7.1653 -I ~- 77380 80050 95763 81277 21270 06597 12425 13105 96535 73801 50761 65996 70012 12633 08608 73789 16555 54566 90855 23677 81468 42360 95286 25043 1) 1) 2) 2j 3) 3) 4) 4) 4) 5) 3.6787 1.3533 4.9787 1.8315 6.7379 2.4787 9.1188 3.3546 1.2340 4.5399 94411e-“71442 52832 36612 06836 78639 63888 87341 46999 08546 52176 66635 19655 54516 26279 02511 98040 86679 92976 24848 32159 69189 42979 80293 70966 84230 20800 83882 54949 51535 - 2) - 3) - 5) - 6) - 7) - 9i -1oj -11) -13) -14j 4.3213 1.8674 8.0699 3.4873 1.5070 6. 5124 2.8142 1.2161 5.2554 2.2711 91826 42731 51757 42356 17275 12136 68457 55670 85176 01068 37722 70798 03045 20899 39006 07990 48555 94093 00644 32409 49774 88144 99239 54918 46107 07282 27211 08397 85552 38387 - 6.5988 5.6145 03584 94835 53125 66885 37077 16982 0102 7712 0205 9897 7815 4509 0308 5424 0000 0413 1139 2304 2787 3617 4623 4913 1. 5682 1.6127 1.6334 99956 12547 99913 00043 12503 80400 99869 25094 00000 92685 43352 48921 53600 27836 97997 61693 01724 83856 68455 101’5 1OOlfl loo”5 1000"' lOOO"5 2lB 3’B 2114 3114 2-m 3-m 5-‘fi (-- (-i. ((- 60287 30427 92853 11026 11156 83872 63720 43592 09997 95790 - 1) 1) lj ~--~ --~- ----- * * * ----- 55238 39995 34242 71802 36048 45167 31361 13891 76367 59152 e--nr 2) 1) log10 53094 96913 xz::: 50008 33051 59282 93827 56840 05440 67360 60802 04600 96908 40271 62459 44443 78038 24234 172321 952452 344642 007593 124774 053527 516964 904905 179915 619436 534874 495346 090274 067528 832720 291643 680957 667634 728425 *See page xx. 12 63981 l!id62 27962 36018 83643 14256 91943 39324 00000 15822 30683 37827 95282 01759 89895 83427 06699 71973 57958 19521 43729 39042 80478 63250 83071 58564 87459 00000 50407 67692 39285 89615 %Ei 26796 49968 54945 65264 37389 50279 74778 62611 87668 22163 12167 ~:~:: 50200 ::%i X%f 32847 66704 08451 09412 05088 MATHEMATICAL TABLE 1.1. CONSTANTS MATHEMATICAL 3.8501 3. 9702 4.0775 4.1108 4. 2046 4.2626 4. 2904 4.3694 4.4188 4.4886 4. 5747 47601 91913 37443 73864 92619 79877 59441 47852 40607 36369 10978 In n 71005 55212 90.571 17331 39096 04131 14839 46702 79659 73213 50338 85868 18341 94506 12487 60596 54213 11290 14941 79234 98383 28221 209507 444691 160.504 513891 700720 294545 921089 729455 754722 178155 167216 (-1) 1. 1447 9. 1893 29885 85332 84940 04672 01741 74178 43427 03296 ( ( ( ( ( 1) 1) 1) 1) 1) 2.3025 4.6051 6.9077 9.2103 1. 1512 1. 3815 1. 6118 1. 8420 2. 0723 85092 70185 55278 40371 92546 51055 09565 68074 26583 nln 10 99404 98809 98213 97618 49702 79642 09583 39523 69464 56840 13680 70520 27360 28420 74104 19788 65472 11156 17991 35983 53974 71966 08996 10795 12594 14393 16192 1) 1) 2) 2j 3) 3) 4) 4) 3.1415 9.8696 3. 1006 9. 7409 3.0601 9.6138 3.0202 9.4885 2.9809 9.3648 92653 04401 27668 09103 96847 91935 93227 31016 09933 04747 58979 08935 02998 40024 85281 75304 77679 07057 34462 60830 32384 86188 20175 37236 45326 43703 20675 40071 11666 20973 62643 34491 47632 44033 27413 02194 14206 28576 50940 71669 1. 5707 1. 0471 7.8539 1. 7724 1.4645 1.3313 2. 1450 2.3597 5. 5683 2. 2459 2. 5066 1. 2533 2. 2214 96326 97551 81633 53850 91887 35363 29397 30492 27996 15771 28274 14137 41469 79489 19659 97448 90551 56152 80038 11102 41469 83170 83610 63100 31550 07918 66192 77461 30961 60272 32630 97127 56000 68875 78452 45473 05024 02512 31235 31322 54214 56608 98167 20143 97535 77444 78474 84818 42715 15765 07883 07940 57. 2957 0. 0174 79513 53292 08232 51994 08767 32957 98155’ 69237r CONSTANTS-Continued log10 loglog ( ( ( ( ( ( (-1) t: 1) In In In ln r(i/3) r(2/3) r(lj4) r(3/4) 93571 60078 64214 01076 70082 71907 12045 29044 37607 64491 f ?624 74644 90456 41902 70338 64341 52860 59010 14279 39038 27847 48517 14219 32992 60656 85749 49132 92829 74387 94821 32760 23543 84362 98726 44819 94133 03251 85435 82765 12683 11289 92653 85307 77960 37061 96326 55592 14857 74122 33388 5s”9”19 17958 76937 43591 79489 15387 51285 87183 23081 32384 64769 97153 72953 66192 59430 52669 45907 39146 62643 25287 87930 85057 31322 77586 23850 70115 16379 3. 1415 6. 2831 !a. 4247 1) 1. 2566 1) 1.5707 1) 1. 8849 1) 2. 1991 1)2.5132 1) 2.8274 n 0. 5772 I-, -, r (714) 97857 75869 52011 29835 74802 58348 22860 27091 78092 90006 71734 4.9714 4.3429 (-1) (-1) logl0e 7P ( ( C ( ( ( ( ( 15664 90153 1. 7724 2. 6789 1. 3541 3. 6256 1. 2254 0.8929 0. 9027 0. 9064 0. 9190 0. 9854 0.3031 1.2880 0.2032 53850 38534 17939 09908 16702 79511 45292 02477 62526 20646 50275 22524 80951 905516 707748 426400 221908 465178 569249 950934 055477 848883 927767 147523 698077 431296 28606 06512 n 1.6720 1.7242 1.7708 1.7853 1.8260 1.. 8512 1. 8633 I.. 8976 1. 9190 1. 9493 1. 9867 *-” x 7 i 10 s-ii ;h; J2 r-1 13 ?r-l/4 +f3 ,-%I4 *-3/Z r--c (2r)-'12 (a/,)"2 2'f2/?r 3. 1830 1.. 0132 3.2251 1. 0265 3.2677 1. 0401 3. 3109 1. 0539 3.3546 1. 0678 98861 11836 53443 98225 63643 61473 36801 03916 80357 27922 83790 42337 31994 46843 05338 29585 77566 53493 20886 68615 67153 77144 89184 35189 54726 22960 76432 66633 91287 33662 77675 38795 42205 15278 28250 89838 59528 17287 39854 04078 4.7123 4.1887 4.4428 (-1) 5. 6418 (- 1) 6. 8278 (-1) 7. 5112 (-1) 4. 6619 (- 1) 4. 2377 (-1) 1. 7958 ( -2) 4.. 4525 (-1) 3. 9894 (-1) 7. 9788 (-1) 4.5015 88980 90204 82938 95835 40632 55444 40770 72081 71221 26726 22804 45608 81580 38468 78639 15836 47756 55295 64942 35411 23757 25166 69229 01432 02865 78533 98576 09846 62470 28694 68146 48285 61438 59679 56168 06151 67793 35587 03477 93965 16858 15881 80795 70208 87030 19885 10077 90820 35273 99461 98921 75996 0. 0002 0. 0000 ; 3 4 -1) -1) 90888 04848 20866 13681 57215 10953 96154r 59936r 48223 37662 -2j -2) -3) -31 -4j -4) -5) -5) 1’ 1” In Y -0. l/~U/‘4 l/Ul/3) l/r(2/3) mm) mw m4/3) mw) mw uw4 In r(4/3) In r(5/3) In r(5/4) in r(7/4) . *See page II. 5495 0. 0. 0. 0. 0. :l. l. l. :l. -0. -0. -0. -0. 39312 98164 5641 3732 7384 2758 8160 1198 1077 1032 0880 1131 1023 0982 0844 89583 82173 88111 15662 48939 46521 32167 62651 65252 91641 14832 71836 01121 547756 907395 621648 830209 098263 722186 432472 320837 131017 740343 960640 421813 020486 * i 2. Physical Constants and Conversion A. G. MCNISH Factors 1 Contents Table Table Table Table Table Table 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. Common Units and Conversion Factors . . . . . . . . . Names and Conversion Factors for Electric and Magnetic Units . . . . . . . . . . . . . . . . . . . . . . . Adjusted Values of Constants . . . . . . . . . . . . . Miscellaneous Conversion Factors. . . . . . . . . . . . Conversion Factors for Customary U.S. Units to Metric Units . . . . . . . . . . . . . . . . . . . . . . . Geodetic Constants . . . . . . . . . , . . . . . . . . * National Bureau of Standards. Page 6 6 7 8 8 8 2. Physical Constants and Table Quantity and Conversion Common Factors for Electric and = SI emu name name I Magnetic = force Magnetomotive force Magnetic flu* Magnetic flux density Electric displacement ampere coulomb volt ohm henry farad amp. turns/ meter amp. turns weber tesla I tbampere 1tbcoulomb abvolt abohm centimeter Conversion statampere statcoulomb statvolt statohm 10-l LO-’ 108 100 100 10-g 4*x centimeter gil bert maxwell gauss --._-_-______ Units = SI unit/ esu unit - oersted - Example: If the value assigned to a current *Divide this number by 4?r if unrationalized 6 and SI unit/ emu unit esu name - Current Charge Potential Resistance Inductance Capacitance Magnetizing Units Factors The SI unit of electric current is the ampere defined by the equation 2r,,,Z1ZJ4~= F giving the force in vacua per unit length between two infinitely long parallel conductors of infinitesimal cross-section. If F is in newtons, and rrn has the numerical value 477 X lo-‘, then I1 and Zr are in amperes. The customary equations define the other electric and magnetic units of SI such as the volt, ohm, farad, henry, etc. The force between electric charges in a vacuum in this system is given by Q, Qn/4nrerg= F, re having the numerical value 10r/4nc2 where c is the speed of light in meters per second (r,= 8.854 x 10-12). The CGS unrationalized system is obtained by deleting 4n in the denominators in these equations and expressing F in dynes, and r in centimeters. Setting r,,, equal to unity defines the CGS unrationalized electromagnetic system (emu), re then taking the numerical value of 1/c2. Setting re equal to unity defines the CGS unrationalized electrostatic system (esu), r,,, then taking the numerical value of l/cz. Mass-the kilogram -fixed by the international kilogram at S&vres, France. Time-the second- fixed as l/31,556,925.9747 of the tropical year 1900 at 12” ephemeris time, or the duration of 9,19‘2,631,770 cycles of the hyperfine transition frequency of cesiurn 133. Temperature-the degree-fixed on a thermodynamic basis by taking the temperature for the triple point of natural water as 273.16 “K. (The Celsius scale is obtained by adding -273.15 to the Kelvin scale.) Other units are defined in terms of them by assigning the value unity to the proportionThe ality constant in each defining equation. entire system, including electricity units, is called the Systi.?me International d’unitds (SI). Taking the l/100 part of the meter as the unit of length and the l/1000 part of the kilogram as the unit of mass, similarly, gives = 2.1. often used in physics ~ (1 meter - 1650763.73h). 2.2. Names Factors rise to the CGS system, and chemistry. The tables in this chapter supply some of the more commonly needed physical constants and conversion factors. All scientific measurements in the fields of mechanics and heat are based upon four international arbitrarily adopted units, the magnitudes of which are fixed by four agreed on standards: Lengththe meter -fixed by the vacuum wavelength of radiation corresponding to the transition 2Plu-5Da of krypton 86 Table Conversion -3x 100 -3 x 109 -(1/3)X 10-Z -(1/9)X 10-u %(1/9)X 10-l’ -9x 10” -3 x loo* IO-3* 4rX lo-I* 108 10’ __---___----__-______-_____ -3/10** -(1/3)X 10-z -(1/3)X 10-B -3x 105* 10-J* I_-..____..____ - - is 100 amperes its value in abamperes is 100X10-‘=lO. system is involved; other numbers are unchanged. 3. Elementary Analytical MILTON Methods ABRAMOWITZ l Contents Elementary 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.3. 3.9. 3.10. Analytical Methods Page 10 ................. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means ............... Inequalities ...................... Rules for Differentiation and Integration ......... Limits, Maxima and Minima .............. Absolute and Relative Errors .............. Infinite Series ..................... Complex Numbers and Functions ............ Algebraic Equations .................. Successive Approximation Methods ........... Theorems on Continued Fractions ............ Numerical 3.11. Methods Table 3.1. 19 19 23 ............................ Powers and Roots 13 14 14 16 17 18 19 19 ....................... Use and Extension of the Tables ............ 3.12. Computing Techniques. ................ References 10 10 11 .................. 24 n’“, k=l(l)lO, 24, l/2, l/3, l/4, l/5 n=2(1)999, Exact or 10s The author acknowledges the assistance of Peter J. U’Hara the preparation and checking of the table of powers and roots. 1 National Bureau of Standards. (Deceased.) and Kermit C. Nelson in 3. Elementary Analytical 3.1. Binomial Theorem and Binomial Coefficients; Arithmetic and Geometric Progressions; Arithmetic, Geometric, Harmonic and Generalized Means Binomial Methods 3.1.9 Sum a”-w+ . . . +b” Coefficients of Geometric (see chapter integer) s,=a+ar+a?+ n =nc7t=n(n-l) 24) . .k., (n--k+l)= Arithmetic Mean Geometric 3.1.12 A . - . +a. Mean of Mean n Quantities G (at>O,k=1,2,. G= (a,&. . . a,,)l’” Harmonic . .,n) H of n Quantities 3.1.13 3.1.5 .+$) 3.1.6 1+c)+@+. (ah>O,k=1,2,. 1-G)+@- 3.1.7 of Binomial Coefficients 0 M(t)=O(t<O, lim M(t) =max. t+m 3.1.17 t&M(t)=min. 3.1.16 1 1 1 1 1 7---8---9---lo_--_ ll---12---- 2 3 4 5 6 17 18 19 1 10 1 11 1 12 3.1.19 ; 1 6 4’ 1 10 10 5 15 20 15 1 6 10 II. (a,,%,. * * *, a,) =mtLx. a . .,a,)=min.a M(-l)=H 1 3.2. Inequalities 1 8 36 1 9 45 Relation 1 10 55 For a more extensive table see chapter 24. page a2, M(l)=A 3.1.20 21 35 35 21 7 28 56 70 56 28 36 84126126 84 45120210252210120 55165330462462330165 66220495792924792495220 @I, liiM(t)=G 3.1.18 3---4---5..--6-e-m some ak zero) z 3.1.8 2---I Mean M(t) ==(i g a:yf 3.1.14 . . . +(--I)“@=0 3.1.15 Table . .,n) . . +c)=2” General&d *See (--l<r<l) n (k-;-l) 3.1.4 a(l-P) l--P n Quantities of &h+az+ 3.1.11 @=(n:k)=(-l)k n Terms to lim s.=a/(l-r) n-t- n! (n-k)!k! 0k 3.1.3 Progreamion . . . +a+‘=--- 3.1.2 * (a+z)) 3.1.10 (n a positive Binomial 7+-l)&; last term in series=Z=a+(n-1)d an-%2 Sum +C) n Terms to . . . +(a++-114 =na+; Theorem a”-lb+@ Progression a-t-b+d)+b+24+ 3.1.1 (a+b)“=a”+c) of Arithmetic Between and Arithmetic, Generalized Geometric, Means Harmonic 3.2.1 A> G>H, 3.2.2 equality if and only if al=az= min. a<M(t)<mnx. a . . . =a,, ELEMENTARY 3.2.3 min. a<G<max. equality 3.2.5 Minkowski’s If p>l Inequalities Inequality for Sums and &, bk>O for all x:, 3.2.12 if t<s unless all ak are equal, or s<O and an an is zero. Triangle 11 METHODS a holds if all ak are equal, or t<O and an an is zero M(t)<&!(s) 3.2.4 ANALYTICAL (& (ak+bk)p~‘p<(& a;)“‘+($ holds if and only t?qUdity i-f bi$“j (c=con- bk=C& stant>O). Iall-la21_<lal+~2111all+la21 Minkowski’s 3.2,6 Inequality 5or Integrals If P>l, Chehyshev’s Inequality If alla2>a,z b,>b,>b,> 3.2.13 . . . >a, . . . >bn 3.2.7 n5 akbk> -(kc, k=l Hiilder’s If ;++,p>1, 2 (Jb a ‘> (& Inequality for I?(~)+9(z)I~~s)llp~(~b a a lJ~(z)l%iz)l’p +(Jb “) a Iscd lqp equality holds if and only if g(z) =cf(x) stant>O). Sums (c=con- fj>l 3.3. Rules for Differentiation and Integration Derivatives equality holds if and only if jbkl=cIuEIP-’ stant>O). If p=q=2 we get Cauchy’s (c=con- $; (cu) =c $9 c constant 3.3.1 3.3.2 Inequality -g (u+v)++g 3.2.9 [& akb,]2<& a; & b: (equality for &=Cbk, & (uv) =u E+v 3.3.3 2 c constant). Hiilder’s 4 Inequality for Integrals 3.3.4 & (u/v)= vdu/dx-udv/dx v2 - , 1r;++=1,p>1, q>l .g u(v) =g 3.3.5 gi 3.2.10 & (u”) =uD e %+1.n U 2) 3.3.6 equality holds if and only (c=constant>O). If p=p=2 we get Schwa&s 3.2.11 Inequality if jgCs)I=clflr)Ip-’ Leibniz’s Theorem for Differentiation of an Integral 3.3.7 b(c) d & s a(c) f (2, wx b(c) 3 a(cj s b ,J(x,c)dx+f@, 4 $--f (a, 4 2 12 ELEMENTARY Leibniz’s Theorem for Differentiation ANALYTICAL The following of a Product formulas S P(x)dx n>l 3.3.8 s METHODS is an integer. where P(x) (ux”+ fJx+cy (t&g u+(y) g g +(g ds g are useful for evaluating is a polynomial and 3.3.16 +...+~)d~rg+...+cg dx 2 (ax2+br+c)=(4c&c-bz)~ S dX -&=1g 3.3.9 (b2--4uc<O) a 2az+b- 3.3.17 I 2azfbf #x -d2y dy -3 dy2=dr2 0 zc 3.3.10 $= -B g-3 Integration 3.3.12 (g--j ($)-" -2 S by Parts S 3.3.20 jidu=w+dti (a+ bx$c+dxj=k jkudx=(jidx) Integrals v-s(judx) of Rational (Integration Algebraic 2 dx dx 1 =- arctan E! u2+b2;C2 ub U Functions 3.3.22 const,ants are omitted) ( Sux+f,)"dx=(ux+6)"+1 S 4n+1> S S (n#-1) 3.3.24 3.3.15 $&In )ax+b) S t(u+bx) of Irrational dx =h2 (c+dx)11’2 3.3.27 =+ 3.3.28 =h2 3.3.29 3.3.30 (x2;1-“a2)2=& arctan ~+20~(xf+U2) 3.3.25 Integrals 3.3.26 c+dx bc In Ia+bx I dx (a+bx)“P(c+dx)=[d(bc~ud)]1~2 S =[d(ad&]1/2 Algebraic arctan arcsin (ad # bc) _ S __ ln b2x21 b2+ S S 3.3.21 3.3.23 3.3.14 (P-4ac=O) 3.3.19 S 3.8.13 (b2-4m>O) =2az+b 3.3.18 3.3.11 (b2--4uc)t (P-4acy Functions -d(a+ 1 W<O) bx) 1’2 C b(c+dx) 2bdx+ad+ bc-ad bc > ln J[bd(n+ bx)]1/2+ b(c+dx)1/21 arctan ~~~~-J” d(u+bx)1’2-[d(ad-bc)]1’2 In d(a+bx)1’2+[d(ud-bc)]“2 (b>O, d<O) (bd>O) (d(u&-bc)<O) I @b-J--4>O) ELEMENTARY If zn=un+ivn, 3.7.23 then ~~+l:=u,+,+iv~+~ u,+~=xu~-~v,; ANALYTICAL where 3.8. Algebraic Solution v,+,=xv,+yu, 9?z” and 92” are called harmonic 17 METHODS polynomials. of Quadratic 3.8.1 0 2”, j-k gf, p= b2-4ac, .zi+,za= -b/u, 3.7.25 If a>O, two real roots, p=O, two equal roots, a<O, pair of complex conjugate Roots 3.7.26 z*=&=rte+rs=r+ cos @+iri sin $0 Solution If --?r<~< ?r this is the principal root. The other root has the opposite sign. The principa: root is given by 3.7.27 d=[+(r+x)]+&-i[$(r--x)]*=ufiv where sign is taken tc 2uv=y and where the ambiguous be the same as the sign of y. 3 . 7 . 28 I l&l-I221 I Let sl=[r-+(q3+r2)q+, Catwhy-Riemann Equations au -=-, av au -=-- by by ax ;, a av a~ ; f$=-$ If z=Tefff, 3.7.31 sz=[T’-((p3+?3*]* then f(z)=f(x+iy)=u(x,yy)+iv(x,y)whereu(x,y),v(z,y aA real, is unaly& at those points z=z+$ which 3.7.30 Equations one real root and a pair of complex c.onjugate roots, $+9=0, all roots real and at least two are equal, . p3+r2<0, all roots real (irreducible case). _<1z1~~2111211+I~21 Functions, roots. $+P>O, root if - ?r<0 5 7r) (k=l,2,3, a . ., n-1) Inequalities Complex If of Cubic Z~Z~=C/U Given Z3+a2z2+ulz+a0==0, let 3.8.2 Zl/n,Tl/nefe/n , (principal Other roots are Pet(B+2rn’ln 3.7.29 Equations Given az2+ bz+c=O, 21,2=- 3.7.24 Equations If zl, z2, z3 are the roots of the cubic equation g=; Z ~+Z~+Z~=-CIC~ Laplace’s ~~~2+~$,+&~,:=~~ Equation The functions U(X, y) and v(x, y) are callec harmonic functions and satisfy Laplace’s equation Cartesian Polar 3.7.33 r; (r g)+$=r of Quartic Equations Given 24+~323+a3~~+~1l~+ag=O, real root u1 of the cubic equation 3.8.3 Coordinates r i!?+$E~2+g2=o 3.7.32 Solution U3 - a2U2 i- (~23 - 4ao)U - (a; j- Uoa$ - kW3) find the = and determine the four roots of the quartic solutions of the two quadratic equations Coordinates ; (r g+g=o 0 as 18 ELEMENTARY ANALYTICAL If all roots of the cubic equation arc real, USC the value of U, which gives real coefficients in the *quadratic equation and select signs so that if METHODS Method of Iteration then -a3, 4142==0. Convergence of an fcxk) xk 2, . . .). Approximution Process (n=l, bk) will converge quadratically to x=5: (if instead of the condition (2) above), (1) Monotonic convergence, f(zO)r’(zo) >0 and f’(s), j”(z) do not change sign in the interval (Q, t), or (2) Osdato y conwgence, f(xJf” (x0)<0 and f’(s), f”(z) do not change sign in the interval (x0, x1), xo<E<xl. Newton’s 3.9.6 x=N’l” 3.9.2 Let zl, z2, x3, . . . be an infinite sequence of approximations to a number f. Then, if 1% n+~-~I<&n-tlk, Rule f’ Then, if f’(z)>0 and the constants cn are negative and bounded, the sequence x,, converges monotonically to the root [. If c,,=c=constant<O and f’(z)>O, then the process converges but not necessarily monotonically. of Approximations Methods 3.9.1 Let z=zl be an approximation to x=[ where f(t) =0 and both x1 and [ are in the interval a$r<b. We define Degree of Successive xk+l= (n=l, b. If z=zk is an approximation to the solution z= I of f(z) =0 then the sequence Comments GI+1=G+C&n) for aLzSb, Newton’s Approximation General Method will 3.9.5 -&, z1z2z3z4=ao. Czj2,=u2, 3.9. Successive Newton’s z2 j2,2t= IF’(s)J<q<l (2) a<xo &‘F(~)~xo’~ - If zl, z2, z3, z4 are the roots, z2 j= Substitution) 3.9.4 The iteration scheme Q+~=F(z~) converge to a zero of z=F(z) if (1) pl+p2=a2,plp2+pl+q2=a2,p~q2+p2q~=al, (Successive Method Applied to Real nth Roots Given x”=N, if zk is an approximation then the sequence xk+l=- ; [$i+(n-l)xk] 2, . . .) will converge quadratically to a. where A and k are independent of n, the sequence is said to have convergence of at most the kth degree (or order or index) to [. If k=l and A<1 the convergence is linear; if k=2 the convergence is quadratic. Regula Falsi (False Position) Aitken’e 3.9.3 Given y=f(z) to find 5 such that f(.$)=o, choose ~0 and x1 such that f(rO) and f(zl) have opposite signs and compute x*=x, -Hi f,= f 1~o-JoX, jlVfO for Acceleration of Sequences 3.9.7 If 2k, &+I, zri+2 are three successive iterates in p, sequence converging with an error which is approximately in geometric progression, then * &=xk- Then continue with x2 and either of x0 or x1 for which f(;ro) or j(zl) is of opposite sign to f(zl). Regula falsi is equivalent to inverse linear interpolation, G-Process (5k--k+1)*=;tk~k+2-2:+1. A*& is an improved A*Xk estimate OGtk) then Z=s+O(P), ’ of x. Ix\<~. In fact, if zk”x+* ELEMENTARY ANALYTICAL METHODS * 3.10. Theorems on Continued Fractions A,B,_l-A,-lB,=(-l)n-’ (4) (5) Definitions kiI al; For every n>O, j ,=b, 1 claI ClC& c2c3a3 &I-lW% c,bl+ czbz+ caba+ ’ * * c,b,’ (6) l+b,+b,b,+ . . . +bzb3. . . b, bz =-- 1 l- b,+l- =b,,+&e&. .. . . . +;=-& d+$+ If the number of terms is finite, j is called a ternlinating continued fraction. If the number of ternls is infinite, j is called an infinite cont’inued fraction and the terminating fraction -1 -a0 x+A ...l _- b3 b,+l- --& I . . . $yu 1 2 t(-1,n----5 aoGa2 aof 1 =- aox n1 n _ . . . a, _- a12 ___ al-x+ uo+ b, * ’ ‘--b,+l I12-xf %-1X * . . +un-2 is called the nth convergent of j. A (2) If lim -A exists, the infinite continued fracIt-+- 88 If uf= 1 and the tion j is said to be convergent. bt are integers there is always convergence. Theorems (1) If and br are positive at then j2n<j2n+2, fin-1 >f*n+, . (2) If j.=+ n A,=b,A,-~+a,A,-2 Bn=bnBn-l+anBn-2 where A-1=l, A,,=bo, B-1=0, B,=l. 0 .2 .4 .6 .8 FIGURE 3.1 1i y:=xn* 2, 5. *n=0,,5t 29 1, Numerical 3.11. Example using Table Use and Extension 1. Computti 3.1. of the Tables xl9 and x4’ for x=29 Methods Linear I 3p=x9. x10 =6.10326 1248. 102’ = (1.25184 9008. 1036)2/29 in Table Repetition for fourth yields the same result. gives 3.1 roots with 1 4 ~7~3+3(5.507144)-]=5.50714 [. = (1.45071 4598. 1013)(4.207072333. 1014) x4’= (x*4)2/x interpolation (919.826)“4-5.507144. By Newton’s method N=919.826, 3845 Thus, ~“~=5.50714 3845/10$=1.74151. 1796, ~-~“=zt/x=.18983 05683. =5.40388 2547. lO6* 3.12. Example 2. (9.19826)“‘= Compute x-3’4 for x=9.19826. (919.826/100)1’4= (919.826)1’4/10t Computing Techniques Example 3. Solve the quadratic equation x2- 18.2x+.056 given the coeflicients as 18.2 f .l, *see page II. .. 20 ELEMENTARY .056f .OOI. ANALYTICAL < METHODS Example 5. Solve the cubic equation x3- 18.12 -34.8=0. To use Newton’s method we first form the table of f(z)=23-1S.1r-34.8 From 3.8.1 the solution is z=4(18.2f-[(18.2)2-4(.05B)]:) =3(18.2~[:J31]t)=3(18.2~18.~) = 18.1969, .OOJ 4” -43.2 f(x) The smaller root may be obtained more accurately from * .05fi/18.1969= .0031& .OOOl. Example 5 6 7 Compute (-3 + .0076i)i. 4. From 3.7.26, (-3+.0976i)~=u+iv Y r!y u=2G? I,-= ( .3 72.6 181.5 We obtain by linear inverse interpolation: where O-(-.3) 72.6-(-.3)=5’oo4’ x,=5+ *, j”= (t”+y’)t > Thus Using Newton’s method, f’(x) =3x2- r=[(-3)2+(.0076)2]~=(9.00005776)~=3.00000 1 Ij= 3.00000 9627- (-3) 2 f= 9627 21 =zo-f&J/f’ .73205 2196 We note that the principal square root has been computed. Example Solve the quartic equation 6. ~‘-2.37752 Into Quadratic (22 + p12 + qd w by Inverse 00526 ' ' Repetition yields x1=5.00526 5097. Dividing f(x) by x-5.00526 5097 gives x2+5.00526 5097x i-6.95267 869 the zeros of which are -2.50263 2549 f.83036 8OOi. We seek that value of y, for which y(nJ =O. Inverse interpolation in ~(a,) gives ~(a,) =O for pl -2.003. Then, Factors + p2x + 92) Interpolation Starting with the trial value pI = 1 we compute successively QI 42 p1= a’--am 42-pll pz=an-p1 9.. 053 4 526 -2. 543 - 1. 093 2. 2 4. 115 -3. - 1. 284 . 165 106 QI Inverse interpolation -2. 550 172 between qI=2.2 :md thus, - 2: E Qz P2 PI 4. 51706 4.,51684 4. 51661 7640 2260 6903 -2. -2. -2. 55259 55282 55306 257 851 447 17506 . 17530 . 17553 Y h) 765 358 955 .00078 . 00001 -. 00075 552 655 263 gives q,=2.00420 2152, and we get finally, _. 2. 00420 4. 520 Inverse interpolation 2.003 gives ql=2.0041, 5. 383 . 729 2. 0041 2. 0042 2. 0043 P2 Ykll) . 011 Y(Qd=ql-t92+p*P2 - a2 _____ :: 171 ~~---- 2.003 q2=; C--.07215 9936jz5 57.020048 I 4922x3+6.07350 5741.x’ -11.17938 023s+9.05265 5259=0. Resolution 18.1 we get (d =5.004- .0076 u=&=2(1.73205 21g6)=.00219 392926 PI - . 2152 Qz 4. 51683 - PI 7410 -2. 55283 Y (Ql) P2 358 17530 8659 -. 00000 0011 and pl= 4 ELEMENTARY Double Precision Multiplication Desk Calculator and Division ANALYTICAL on a Example 7. MultiplyM=20243 97459 71664 32102 by m=69732 82428 43662 95023 on a 10X10X20 desk calculating machine. Let MO=20243 97459, Ml=71664 32102, mO= 69732 82428, ml=43662 95023. Then Mm= M0m0102’+ (Mom,+Mlmo) 101o+M~ml. (1) Multiply ,W1m1=31290 75681 96300 28346 and record the digits 96300 28346 appearing in positions 1 to 10 of the product dial. (2) Transfer the digits 31290 75681 from positions 11 to 20 of the product dial to positions 1 to 10 of the product dial. (3) Multiply cumulatively M,mo+Mom,+31290 75681=58812 67160 12663 25894 and record the digits 12663 25894 in positions 1 to 10. (4) Transfer the digits 58812 67160 from positions 11 to 20 to positions 1 t,0 10. (5) Multiply cumulatively Mom,+58812 67160 =14116 69523 40138 17612. The results as obtained are shown below, 9630028346 1266325894 14116695234013817612 141166952340138176121266325894963~?28346 If the product Mm is wanted to 20 digits, only the result obtained in step 5 need be recorded. Further, if the allowable error in the 20th place is a unit’, the operation MImI may be omitted. When either of the factors M or m contains less than 20 digits it is convenient to position the numbers as if they both had 20 digits. This multiplication process may be extended to any higher accuracy desired. 8. Divide N=14116 69523 40138 17612 by d=20243 97459 71664 32102. Method (1)--linear interpolation. Method @)--If N and d are numbers each not more than 19 digits let N=N1+NolOQ, d=dI+ dolO where No and do contain 10 digits and N, and dl not more than 9 digits. Then N NolOQ+N, zs- 1 [.N-y] d=,lOQ+d, dolO Here N= 14116 69523 40138 1761, d=20243# 97459 71664 3210 No= 14116 69523, do=20243 97459, d,=71664 3210 (1) NodI= 10116 63378 42188 8830 (productdial). (2) (Nod,)/do=49973 55504 (quotient dial). (3) N- (N&/d,= 14116 69!;22 90164 62106 (product dial). (4) [N- (NodI)/do]/dolOQ= .69732 82428=first 10 digits of quotient in quotient dial. Remainder =r=O8839 11654, in positions 1 to 10 of product dial. (5) r/(d010Q)=.43662 9502.10-“O=next 9 digits of quotient. N/d=.69732 82428 43662 9502. This method may be modified to give the quotient of 20 digit numbers. Method (1) may be extended to quotients of numbers containing more than 20 digits by employing higher order interpolation. 9. Sum the series S= l-&+*-i to 5D using the Euler transform. + The sum of the first 8 terms is .634524 to 6D. If u,=ljn we get Example n 9 %7 . 111111 10 . 100000 Difference X.71664 32102=24685 64402&10-*O (note this is an 11 X 10 multiplication). Quotient= (69732 82430 90519 39054-246856 44028).10-20 =.69732 82428 43662 95028 There is an error of 3 units in the 20th place due to neglect of the contribution from second differences. A*u, Au, A3u, A%,, -11111 Example N/20243 97459.101’= .69732 82430 90519 39054 N/20243 97460.10”= .69732 82427 46057 26941 Difference=3 44462 12113. 21 METHODS 2020 -505 -9091 11 . 090909 -7576 12 156 1515 . 083333 -349 1166 -6410 13 . 076923 From 3.6.27 we then obtain SC 634524+.111111 -_ 2 (-.011111)+.002020 23 22 -(-- .000505) +.000156 24 26 = .634524+ .055556+ .002778+ .000253 -+ .000032+ .000005 = .693148 (S=ln 2=.6931472 to 7D). 22 ELEMENTARY Example 10. Evaluate s0 -F =& dx=g s,. $?jk-‘=glI k-‘+l& (k+10)-2 transform. (Icfl)= y s,’ sin;;;;t) METHODS m sin 2 dx s Cl J: the integral =- G to 4D using the Euler ANALYTICAL dx 1 +jY&p- dt+% Evaluating the integrals in numerical integration we get (-l)f g dt. where f(k) the last sum by $, k-2=1.54976 s = (k+10)-2. ... Thus, 7731+.1 - .005 + .00016 6667 - .OOOOO 0333 = 1.64493 4065, Ic as compared 1.85194 Example .43379 12. A A2 A3 A4 4067. Compute arctanx=$3+g7+ .25661 . 18260 with $=1.64493 to 5D for x= .2. Here n>l, &,=O, b,=2n-1, x2 4x2 9x2 al=x, A-l=l, ... an=(n-l)2x2 for Bdl=O, A,,=O, . 14180 -2587 .11593 799 - 1788 -321 478 .09805 -1310 - 168 .08495 A0 -= Bo ’ 153 310 - 1000 A -r,*g .07495 The sum to k=3 is 1.49216. Applying Euler transform to the remainder we obtain Bl the A=.197368 f (.14180)-h (-.02587)+& B2 (.00799) A3 -; (-.00321)+$ = .07090 + .00647 + .00100+ .00020 + .00005 = .07862 We obtain the value of the integral compared with 1.57080. Example 11. as 1.57018 as Sum the series $I kep==f P the Euler-Maclaurin summation formula. From 3.6.28 we have for n= a, B=.197396 (.00153) using 3 [II A4 3.032 Bq = 15.36 Note that in carrying out the recurrence method for computing continued fractions the numerators A, and the denominators B, must be used as originally computed. The numerators and denominators obtained by reducing An/B, to lower terms must not be used. ELEMENTARY ANALYTICAL 23 METHODS References Texts [3.1] R. A. Buckingham, Numerical methods (Pitman Publishing Corp., New York, N.Y., 1957). [3.2] T. Fort, Finite differences (Clarendon Press, Oxford, England, 1948). [3.3] L. Fox, The use and construction of mathematical tables, Mathematical Tables, vol. 1, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1956). [3.4] G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [3.5] D. R. Hartree, Numerical analysis (Clarendon Press, Oxford, England, 1952). [3.6] F. B. Hildebrand, Introduction to numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1956). [3.7] A. S. Householder, Principles of numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [3.8] L. V. Kantorowitsch and V. I. Krylow, Naherungsmethoden der Hoheren Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1956; translated from Russian, Moscow, U.S.S.R., 1952). [3.9] K. Knopp, Theory and application of infinite series (Blackie and Son, Ltd., London, England, 1951). [3.10] Z. Kopal, Numerical analysis (John Wiley & Sons, Inc., New York, N.Y., 1955). [3.11] G. Kowalewski, Interpolation und genaherte Quadratur (B. G. Teubner, Leipzig, Germany, 1932). [3.12] K. S. Kuns, Numerical analysis (McGraw-Hill Book Co., Inc., New York, N.Y., 1957). [3.13] C. Lanczos, Applied analysis (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1956). [3.14] I. M. Longman, Note on a method for computing infinite integrals of oscillatory functions, Proc. Cambridge Philos. Sot. 52, 764 (1956). [3.15] S. E. Mikeladze, Numerical methods of mathematical analysis (Russian) (Gos. Izdat. TehnTeor. Lit., Moscow, U.S.S.R., 1953). [3.16] W. E. Milne, Numerical calculus (Princeton Univ. Press, Princeton, N.J., 1949). [3.17] L. M. Milne-Thomson, The calculus of finite differences (Macmillan and Co., Ltd., London, England, 1951). [3.18] H. [3.19] (3.201 [3.21] [3.22] [3.23] [3.24] [3.25] Mineur, Techniques de calcul numerique (Librairie Polytechnique Ch. B&anger, Paris, France, 1952). National Physical Laboratory, Modern computing methods, Notes on Applied1 Science No. 16 (Her Majesty’s Stationery Office, London, England, 1957). J. B. Rosser, Transformations to speed the convergence of series, J. Research NBS 46, 56-64 (1951). J. B. Scarborough, Numerical mathematical analysis, 3d ed. (The Johns Hopkins Press, Baltimore, Md.; Oxford Univ. Press, London, England, 1955). J. F. Steffensen, Interpolation (Chelsea Publishing Co., New York, N.Y., 1950). H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Inc., New York, N.Y., 1948). E. T. Whittaker and G. Robinson, The calculus of observations, 4th ed. (Blackie and Son, Ltd., London, England, 1944). R. Zurmtihl, Praktische M.athematik (SpringerVerlag, Berlin, Germany, 1953). Mathematical Tables and Collections of Formulas [3.26] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The Smithsonian Institution, Wa,shington, D.C., 1957). [3.27] L. J. Comrie, Barlow’s tables of squares, cubes, square roots, cube roots a,nd reciprocals of all integers up to 12,500 (Chelmical Publishing Co., Inc., New York, N.Y., 1954). [3.28] H. B. Dwight, Tables of integrals and other mathematical data, 3d ed. (The Macmillan Co., New York, N.Y., 1957). [3.29] Gt. Britain H.M. Nautical Almanac Office, Interpolation and allied tables (Her Majesty’s Stationery Office, London, England, 1956). [3.30] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [3.31] G. Schulz, Formelsammlung zur praktischen Mathematik (de Gruyter and Co., Berlin, Germany, 1945). 24 ELEMENTARY ANALYTICAL POWERS Table 3.1 AND ROOTS nk k : 3 4 : 7 fz 10 24 nl= for use See Examples 1-5 METHODS T&L n3, n‘L 15 2= .L 16 n7== 128 of the table. Floating decimal notation: 910=3486784401 = (9)3.4867 84401 s 8 25% .lO= $4, 1024 167 77216 &L 1.4142 13562 nl/3= .&4= 7$1/5= l/3 l/4 l/5 1.2599 I.1892 1.1486 2 i 190 24 l/2 l/3 l/4 l/5 (16)5.9604 64478 2.2360 1.7099 1.4953 66176 1.8171 20593 1.5650 84580 1.9158 2.6457 1.9129 1.6265 12314 51311 31183 76562 1.4309 1.4757 73162 (18)4.7383 81338 2.4494 1.3797 29662 24 v3 l/4 v5 3.1622 2.1544 69081 1 17 194 2143 9)2.3579 ( (10)2.5937 93192 1.7782 79410 2:: 3375 50625 7 59375 113 90625 1708 59375 24 l/2 l/3 l/4 l/5 79 10)6.1917 (25)7.9496 36422 84720 3.4641 2.2894 1.8612 1.6437 01615 28485 09718 51830 14 -.241 41U3 9j6.9757 2% 4913 83521 19857 _ -37569 38673 57441 16 256 4096 65536 10 48576 167 77216 2684 914.2949 (lOj6;8719 (12)1.0995 35456 67296 47674 11628 (llj1.1858 (12)2.0159 93900 11220 (28)1.6834 (28)7.9228 16251 (29)3.3944 86713 3.8729 2.4662 83346 12074 1.7187 71928 1.9679 89671 00000 4.0000 2.5198 2.0000 1.7411 42100 00000 01127 I 4.1231 2.5712 2.0305 78765 05626 81591 43185 1.7623 40348 13562 07911 9 7% 6561 59049 5 31441 47 82969 430 46721 3874 20489 9)3.4867 84401 41824 2.8284 2.0000 1.6017 1.5157 ( 66483 (21)4.7223 (22)7.9766 27125 00000 92831 16567 44308 00000 3.0000 2.0800 1.7320 1.5518 83823 50808 45574 13 1’946 169 --_ 2197 28561 3 71293 48 26809 627 48517 8157 30721 85984 31808 81696 80352 03906 11)5.7665 1.4142 1.3195 1342 17728 -. __._ 358 4299 9)5.1597 24790 80091 60287 94266 I 49570 74013 30940 9)1.0737 1728 20736 2 48832 61051 71561 87171 58881 47691 32676 3.3166 2.2239 1.8211 1.6153 ( 12 1:: 1331 14641 42460 (24)9.8497 77660 34690 1.5848 w (20) 89743 1’0: 1000 10000 00000 00000 00000 00000 00000 00000 00000 23543 64801 53607 75249 49167 00000 1.5874 01052 2.0000 516; 4096 32168 2 62144 20 97152 167 77216 1 17649 8 57 403 2824 (14)2.8147 8 3:; 2401 16807 100 77696 61977 75947 48781 1 10 100 1000 ( 9 11.0000 (10 1.0000 (24) 1.0000 1.4422 1.3160 1.2457 7 2:: 1296 7716 46656 2 79936 16 79616 604 1024 4096 16384 65536 2 62144 10 48516 (11)2.8242 95365 1.7320 50808 21050 07115 98355 6 2: 125 625 3125 15625 78125 3 90625 19 53125 97 65625 : 3 4 2:; 729 2187 6561 19683 59049 2: nL ngss= l/2 27 10)1.0604 11)1.3785 2744 38416 5 37824 75 29536 1054 13504 49937 84918 (26)5.4280 07704 3.6055 2.3513 1.6702 (27)3.2141 51275 34688 71652 1.8988 28922 1 18 340 6122 4.2426 2.6207 2.0597 1.7826 3:: 5832 04976 89568 12224 20032 58845 1 24 470 8938 (30)4.8987 40687 41394 67144 02458 57381 42264 1.9343 36420 1.6952 18203 10)1.1019 96058 11 1.9835 92904 12 1 3.5704 67227 (30)1.3382 99700 3.7416 2.4101 6859 30321 76099 45881 71739 62931 4.3588 2.6684 2.0877 1.8019 98944 01649 97630 83127 23 1 42000 8000 60000 1 94481 32 640 9) 1.2800 10)2.5600 (11)5.1200 (13)1.0240 24 w l/3 l/4 l/5 00000 00000 00000 00000 00000 00000 ( 9)1.8010 (10)3.7822 (11)7.9428 (13)1.6679 (31)1.6777 21600 (31)5.4108 4.4721 2.7144 2.1147 1.8205 35955 17617 42527 64203 4:: 10648 2 34256 51 53632 1133 79904 441 9261 40 857 4.5825 2.7589 2.1406 84101 66121 88541 5:: 13824 3 31776 79 62624 1911 02976 85936 00466 88098 19838 75695 24176 95143 1.8384 16287 (32)1.6525 4.6904 2.8020 2.1657 10926 15760 39331 36771 1.8556 00736 (32)4.8025 4.7958 2.8438 2.1899 1.8721 07640 31523 66980 38703 71231 (33)1.3337 4.8989 2.0844 2.2133 1.8881 35777 79486 99141 63839 75023 ELEMENTARY ANALYTICAL POWERS 6:: (33)3.5527 13679 5.0000 00000 2.9240 17738 2.2360 67977 1.9036 53939 26 676 17576 4 56976 118 81376 3089 15776 9 8.0318 10176 11 2.0882 70646 12 5.4295 03679 14 I 1.4116 70957 33)9.1066 85770 5.0990 19514 2.9624 96068 2.2581 00864 1.9186 45192 9:: 27000 6 10000 243 00000 7290 00000 10 2.1870 00000 11 6.5610 00000 13 1.9683 00000 Ii 14 5.9049 00000 (35)2.8242 95365 5.4772 25575 3.1072 32506 2.3403‘47319 1.9743 50486 31 961 29791 23521 29151 03681 61411 10374 62216 82870 26610 64363 80652 11062 40755 15625 3 90625 97 65625 2441 40625 35 1225 42875 15 00625 525 21075 24 l/2 l/3 l/4 l/5 15)2.7585 47354 (37)1.1419 13124 5.9160 79783 3.2710 66310 2.4322 99279 2.0361 68005 40 1600 64000 25 60000 1024 00000 24 l/2 l/3 l/4 l/5 6 7 9” 10 24 l/2 l/3 l/4 l/5 (16)1.0485 (38)2.8147 6.3245 3.4199 2.5148 2.0912 76000 49767 55320 51893 66859 79105 45 2025 91125 41 00625 1845 28125 ( 9)8.3037 65625 (11)3.7366 94531 (13 1.6815 12539 (14 7.5668 06426 (I6 I 3.4050 62892 (39)4.7544 50505 6.7082 03932 3.5568 93304 2.5900 20064 2.1411 27368 9 286 0675 10 2.7512 11 8.5289 13 2.6439 14 8.1962 (35)6.2041 5.5677 3.1413 2.3596 1.9873 II 46656 16 79616 604 66176 15)3.6561 (37)2.2452 6.0000 3.3019 2.4494 2.0476 28 1158 9)4.7501 11)1.9475 12)7.9849 (38)5.0911 6.4031 3.4482 2.5304 2.1016 58440 25771 00000 27249 89743 72511 16G 68921 25761 56201 04241 42739 25229 10945 24237 17240 39534 32478 25 METHODS AND ROOT!3 nk Table 21 5 143 3674 10 1.0460 11 2.8242 12 7.6255 14 I 2.0589 34)2.2528 5.1961 3.0000 2.2795 1.9331 729 19683 31441 48907 20489 35320 95365 97405 11321 39954 52423 00000 07057 82045 32768 10 48576 335 54432 (36)1.3292 5.6560 3~1748 2.3704 2.0000 27996 54249 02104 14230 00000 37 1369 50653 18 74161 693 43957 (37)4.3335 6.0827 3.3322 2.4663 2.0589 25711 62530 21852 25715 24137 17:: 74088 31 11696 1306 91232 (38)9.0778 6.4807 3.4760 2.5457 2.1117 49315 40698 26645 29895 85765 1 48 2293 (10 1.0779 11 5.0662 13 2.3811 15 1.1191 ii16 5.2599 (40)1.3500 6.8556 3.6088 2.6183 2.1598 22d9 03823 79681 45007 21533 31205 28666 30473 13224 46075 54600 26080 30499 30012 6 172 4818 10 1.3492 11 3.7780 13 1.0578 II 14 2.9619 (34)5.3925 5.2915 3.0365 2.3003 1.9472 7:: 21952 14656 10368 90304 92851 19983 45595 61661 32264 02622 88972 26634 94361 10;; 35937 11 85921 391 35393 9 1.2914'67969 '10 4.2618 44298 '12 1.4064 08618 113I 4.6411 48440 15)1.5315 78985 (36)2.7818 55434 5.7445 62647 3.2075 34330 2.3967 El727 2.0123 46617 30 1444 54672 20 85136 792 35168 9 3.0109 36384 11 1.1441 55826 ~12 4.3477 92138 14 1.6521 61013 15 I 6.2782 11848 (37)8.2187 60383 6.1644 14003 3.3619 75407 2.4828 23796 2.0699 35054 34 1470 9 6.3213 11 I 2.7181 13 1.1688 14 1 5.0259 16)2.1611 39)1.5967 6.5574 3.5033 2.5607 2.1217 18:; 79507 18801 08443 63049 86111 20028 26119 48231 72093 38524 98060 49602 47461 1 53 2548 10)1.2230 11 5.8706 13 12.8179 1.3526 37322 03230 41186 48026 43542 8:: 24389 7 07281 2105 11149 5948 23321 (35)1.2518 5.3851 3.0723 2.32!05 1.9610 (36)5. 6950 5.8309 3.2396 2.41147 2.0;!43 03680 51895 11801 36403 97459 19 15% 59319 23 13441 902 24199 (38)1.5330 6.2449 3. 3912 2.4989 2.0807 29700 97998 11443 99399 16549 19;: 85184 37 48096 1649 16224 (39)2.7'724 6.6332 3.5303 2.5'755 2.1'315 53276 49581 48335 09577 25513 47 21;: 97336 44 77456 2059 62976 (39)8.0572 6.7823 3.5830 2.6042 2.1505 70802 29983 47871 90687 60013 _ _._.. 57 64801 (40)3.6‘703 7.0000 3.6!593 2.6457 2.1'779 The numbers in square brackets at the bottom of the page mean that the maximum error in a linear interpolate is a X 10-P (p in parentheses), and that to interpolate to the full tabular accuracy 11~ poJnts must be used in Lagrange's and Aitkens methods for the respective functions W'. *See page xx. 49008 64807 16826 95787 09057 34 1156 39304 13 36336 454 35424 9)1.5448 04416 23:: 10592 08416 03968 59046 83423 28043 05461 40)2.2376 6.9282 3.6342 2.6321 2.1689 3.1 36822 00000 05710 51311 06425 26 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND ROOTS k : : 2 7 t 10 24 l/2 l/3 ::: 24 l/2 l/3 l/4 l/5 1 62 3125 10 1.5625 11 7.8125 13 3.9062 15 1.9531 16 I 9.7656 (40)5.9604 1.0710 3.6840 2.6591 2.1867 25:: 25600 50000 00000 00000 00000 50000 25000 25000 64478 67812 31499 47948 24148 30:: 1 66375 91 50625 5032 84375 10 2.7680 64063 12 1.5224 35234 '13 8.3733 93789 '15 I 4.6053 66584 :17)2.5329 51621 (41)5.8708 98173 7.4161 98487 3.8029 52461 2.7232 69815 2.2288 07384 1 67 3450 10 I 1.7596 11 8.9741 13 4.5167 1.1904 40)9.5870 7.1414 3.1084 2.6723 2.1954 7176 24 l/2 l/3 l/4 l/5 24 l/2 l/3 l/4 l/5 (42)4.7383 7.7459 3.9148 2.7831 2.2679 45118 01897 67858 1.4833 14774 3.8258 2.7355 2.2368 62366 64800 53829 60 3600 2 16000 129 60000 10)4.6656 29769 1.7488 74104 (41i2.4133 7.2801 3.1562 2.6981 2.2123 53110 09889 24 l/2 l/3 l/4 l/5 (44)1.9158 8.3666 4.1212 2.8925 2.3389 12314 00265 85300 07608 42837 .1[(-,,9] 67876 56822 (41)3.7796 7.3484 3.7197 2.7108 2.2206 38253 69228 63150 06011 43035 57 (42)1.3835 7.5498 3.8485 2.1476 2.2447 55344 34435 01131 96205 86134 34:: 2 05379 121 17361 7149 24299 (42)2.1002 7.6157 3.8708 2.7596 2.2526 54121 73106 76641 69021 07878 (42)3.1655 7.6811 3.8929 2.7714 2.2603 43453 45748 96416 88002 22470 64 (42) 7.0455 68477 7.8102 49676 3.9364 97183 2.7946 2.2754 (43) 82393 43032 (43)4.6671 8.1240 4.0412 2.8502 2.3115 ; 43; 1.5281 75339 91610 66263 55056 2.8060 2.2828 7.9372 53933 3.9790 2.8173 2.2901 57208 13241 72049 44:; 3 00763 74736 78950 38405 40021 69883 79249 79722 07874 1.0408 7.8740 3.9578 66 4356 2 81496 189 39% 2 50047 157 52961 9974 ___ .._. 36547._ 38:: 2 38328 147 76336 9161 32832 201 (43)6.6956 8.1853 4.0615 2.8610 2.3185 88867 (43)9.5546 52172 8.2462 48100 05553 41963 4.081'6 2.8716 2.3254 254 ( 9 I 1.8042 (11 1.2810 I 14 9.0951 12 6.4515 (16 I 4.5848 (18 3.2552 (44)2.6927 8.4261 4.1408 2.9027 2.3455 11681 29351 02839 20158 35312 50072 43551 76876 49773 17749 83108 81669 ,i[(-;)4] 3 73248 268 13856 ( 9I (11 13 14 1.9349 1.3931 1.0030 1.2220 17632 40695 61300 41363 [ :86{ :: :I;: (44)3.7668 8.4852 4.1601 2.9129 2.3521 i:::: 63772 81374 67646 50630 58045 (43)2.2300 8.0000 4.0000 2.8284 2.2973 30685 11251 55102 21711 22030 (44)1.3563 8.3066 4.1015 2.8821 2.3322 70007 23863 65930 21417 21626 54:: 4 05224 299 86576 3 89017 283 98241 -71593 9 2.0730 11 1.5133 13 1.1047 14 I 8.0646 74520 00000 00000 27125 96710 4766; 3 28509 226 67121 73 5329 5184 50:: 3 57911 40% 2 62144 167 77216 462648 3 14432 213 81376 51121 72 49:: 3 43000 240 10000 85154 3244 1 85193 105 56001 6016 92057 00000 4226: 2 74625 178 50625 ( 9)1.1602 90625 10)7.5418 89063 12)4.9022 27891 3.1864 48129 I ('l8)1.3462 74334 (43)3.2353 44710 8.0622 51148 4.0207 25159 2.8394 11514 2.3045 31620 48342 02551 11157 49614 44515 54 2916 1 51464 85 03056 4591 65024 53 2809 48877 90481 95493 36113 11140 69041 63592 il17 (41)1.5278 7.2111 3.7325 2.6853 2.2039 37;: 2 26981 138 45841 8445 96301 00000 81338 66692 67641 57684 33155 1 78 4181 2.2164 1.1747 6.2259 3.2997 10 12 13 15 94451 24238 33090 28429 nk 52 2704 1 40608 73 11616 3802 04032 26;: 32651 65201 25251 28780 06779 56 3136 1 75616 98 34496 5507 31776 (41)9.0471 METHODS 42263 39852 00919 1814:976 5830 2 58671 2 16 5 8871 ;44;5.2450 ,i[(-55131 8.5440 4.1793 2.9230 2.3586 38047 03745 39196 12786 55818 (44)7.2704 8.6023 4.1983 2.9329 2.3650 “:[(-55)2] 49690 25261 36454 72088 82169 ELEMENTARY POWERS 75 5625 4 21875 316 l/2 l/3 l/4 l/5 (45)1.0033 8.6602 4.2171 2.9428 2.3714 AND ROOTS 76 5776 4 38976 333 62176 40625 91278 54038 63326 30956 40610 ANALYTICAL (45)1.3788 79182 8.7177 97887 4.2358 2.0525 2.3777 23584 91724 30992 nk Table 59:; 4 56533 351 53041 (45)1.8870 8.7749 4.2543 2.9622 2.3839 23915 64387 20865 6078: 4'74552 370 15056 55503 8.8317 4.:2726 2.'3718 2. 3901 67;: 5 51368 452 12176 474 56638 27 METHO:DS 60866 58682 27866 15677 3.1 62:; 4 93039 389 50081 (45)3.4918 8.8881 4;2908 2.9813 2.3962 06676 94417 40427 07501 12991 81 b56i 6480: 12000 60000 00000 40000 52000 21600 77280 41824 l/2 l/3 l/4 l/5 (45)4.7223 8.9442 4.3088 2.9906 2.4022 522 24 71747 (45) a.5414 66801 9.0553 85138 81486 6 36056 547 00816 00625 44457 29672 70277 53252 85441 00000 48711 00000 24685 86 7396 72;: 6 14125 (46)2.6789 9.:!736 4.41140 3.0452 2.4372 8190: 39031 18495 04962 61646 47818 29000 9 11 I 13 15 24 l/2 l/3 l/4 l/5 24 l/2 l/3 l/4 l/5 65: 5.9049 5.3144 4.7829 4.3046 46)7.9766 9.4868 4.4814 3.0800 2.4595 8 814 9 7.7378 11 I 7.3509 13 6.9633 15 I b.6342 17 6.3024 19 5.9873 (47)2.919.3 9.7467 4.5629 3.1219 2.4862 82:: 53571 10000 00000 74961 21451 10000 92520 10194 25276 98001 6118i 04400 92014 69000 72100 04890 84401 44308 (47) 32981 04747 70288 09486 1.0399 9.!;393 4.4979 3.0885 2.4649 41445 90619 50932 96 9216 8 04736 90;: 57375 50625 09375 18906 72961 849 34656 02635 85641 49570 (19)6.6483 (47)3. 7541 9.?979 4.!,788 3.I.301 2.4914 4-55151 70:: 5 92704 497 87136 4.3444 3.0092 2.4141 16698 41771 87 7569 6 58503 572 89761 9)4.9842 09207 (46)3.5355 9.3273 4.4310 3.0540 2.4428 91351 79053 47622 75810 89656 92 8464 7 78688 716 39296 (46)1.1425 9. 1104 4. 3620 3. ~0183 2A200 a5726 63047 57435 41015 44749 97 9P"9 .-, 9 12673 885 29281 26360 32467 58971 56970 69160 61879 4-y] (17)7.6023 (19)7.3742 (47)4.8141 9.8488 4.5947 3.1382 2.4966 10587 41269 72219 57802 00892 a8993 30932 1 ;[(-y] (46)1.5230 9.1651 4.3795 3.0274 2.4258 (46)4.6514 9.:3808 4.4479 3.13628 2.4484 04745 31520 60181 14314 79851 (46)6.1004 9.4339 4.4647 3.0714 2.4540 25945 81132 45096 78656 19455 94 8836 86:; a 04357 748 05201 9)6.9568 83693 11 6.4699 01834 13 6.0170 08706 17 5.2041 19 4.8398 47)1:7522 9.6436 4. 5306 3.1054 2.4756 10388 51390 19140 00104 04834 89 7921 7 04969 627 42241 ., 15 5.!5958 (47)1.3517 9.5916 4.5143 3.0970 2.4703 47375 33579 70671 49479 01407 88 -I"44 6 81472 599 69536 a 30584 780 74896 18097 10830 23072 28603 5Oi'bl 54896 22799 91866 9d48 9 41192 922 36816 9)9.0392 07868 (47)2.2650 9.6953 4.5468 3.1137 2.4809 960 01461 59715 35944 37258 93182 98:; 9 70299 59601 11 8.8584 13 1 8.6812 15)8.5076 8.3374 23E109 55932 30226 77621 I 11 I 9.5099 9 9.4148 I 13 9.2274 15 9.3206 00499 01494 46944 53479 (47)6.1578 9.8994 4.0104 3.1463 2.5017 03265 945'37 (47)7.8567 9.9498 4.6260 3.1543 2.5068 81408 74371 65009 42146 42442 I::{ z: 2383% ;: 04313 94097 69392 90243 94345 611:; 5 71787 58321 66546 9.0000 4.3267 3.0000 2.4082 97562 48868 9.2195 4.3968 3.0363 2.4315 l/3 l/4 l/5 (45)6.X626 71910 69380 (46)2.0232 l/2 (19)1.2157 66483 40: 5 31441 430 46721 36292 46284 57527 ,,a[,-;)91 28 ELEMENTARY Table ANALYTICAL POWERS 3.1 METHODS AND ROOTS nk k 101 10201 10 30301 1040 60401 : : 5 103 10609 102 10404 10 61200 1002 43216 10 92121 1125 50001 1nr _. _ 10816 11 24864 1169 85856 7" : 10 24 l/2 l/3 l/4 l/5 (48)1.0000 ( 1)1.0000 4.6415 3.1622 2.5110 00000 00000 00034 11660 66432 48) 1.2691 2.5168 105 11025 11 57625 1215 50625 I i10 1.2762 i 12 1.4071 14 1.3400 24 l/2 l/3 l/4 l/5 24 l/3 l/4 l/5 l/3 l/4 l/5 (48)9.8497 ( 1)1.0408 4.7914 3.2385 2.5602 (49)2.0625 ( 1)1.0723 4.0629 3.2147 2.5830 17 2073 10)2.4883 12)2.9059 24 w l/3 l/4 l/5 107 11449 12 25043 1310 79601 (4Q4.0489 ( 1)1.0295 4.1326 3.2086 2.5413 32676 08048 19857 31040 21376 (20)6.1917 (49)7.9496 ( 1)1.0954 4.9324 3.3097 2.6051 17619 80529 44131 22111 90170 34641 63014 23491 80436 30642 (48)5.0723 ( 1)1.0344 4.1414 3.2162 2.5461 111 12321 13 67631 1518 07041 (49)1.2239 ( 1) 1.0535 4.0058 3.2458 2.5640 15658 65375 95534 67180 65499 __ ._- 120 14400 20000 60000 20000 04000 lb 1873 (49)4.3297 ( 1)1.0016 4.8909 3.2008 2.5920 14641 17 71561 2143 50001 22:: 36422 04720 45115 24149 50920 71085 (49)9.7017 ( 1)1.1000 4.9460 23378 00000 07443 3.3166 2.6094 24790 90635 11”[(-45)3] ,@‘] 59396 21453 07613 b2893 f3;;: 3:2531 2.5694 15 60096 1810 63936 20)4.4114 35079 (49)3.5236 41704 ( 1)1.0770 32961 4.0769 90961 3.2810 18035 2.5075,66964 66953 00043 112 12544 14 04928 1573 51936 (49)1.5170 ( 1):';;;; (50)1.1820 ( 1y;; 3:3234 2.6137 53123 70314 56186 97660 03156 09795 49001 lb30 1)1.0190 4.7026 3.1934 2.5316 109 11801 12 95029 1411 58161 (48)7.9110 ( 1)1.0440 4.7768 3.2311 2.5555 (49)1.8708 ( 1)1.0630 4.8345 3.2603 2.5740 09051 14581 00127 90439 42354 83175 30651 56181 46315 55391 114 12996 14 01544 41361 01613 88721 122 14884 15048 33456 00163 03959 10830 07213 02000 31415 50242 ;;a;; 108 11664 59712 40896 28077 74323 24269 30210 04627 24997 80737 30405 04165 03903 69375 36868 67508 40)2.5633 113 12769 14 42097 __-_, 20675 65303 73246 68168 12982 94106 89157 40148 32501 80003 12 1360 10 1.4693 12 1.5868 14 1.7138 16 1.8509 18 I 1.9990 20)2.1589 4Q6.3411 1)1.0392 4.lb22 3.2237 2.5500 94627 99944 95077 93980 85873 17482 115 13225 l/2 20720 71028 54546 (48)2.0327 1)1.0140 4.6075 3.1857 2.5267 55444 28216 15 20075 1749 00625 24 37249 4.6123 3.1719 2.5218 106 11236 11 91016 1262 41696 110 12100 13 31000 1464 10000 w 90229 (48)1.6084 95641 00423 81563 16 1.4774 18 1.5513 20 1.6280 (48)3.2250 ( 1)1.0246 4.7116 3.2010 2.5365 34649 lb 1930 (49)5.3109 ( l)l.O862 4.9040 3.2950 2.5964 118 13924 43032 71716 00621 70049 60131 73252 20703 123 15129 18 60061 2280 06641 I I (50)1.4370 ( 1)1.1090 4.9731 3.3302 2.6100 ,4-;V3] 80104 53651 89033 45713 68602 46q6)5] (49)2.3212 ( 1)1.0677 4.0480 3.2675 2.5705 lb 2005 (49)6.5031 ( 1)1.0900 4.9106 3.3028 2.6000 20685 07825 07586 79071 02140 119 14161 85159 33921 99444 71211 64734 33952 14507 124 15376 19 2364 10 2.9316 12 3.6352 14 4.5076 16 I 5.5095 10)6.9309 (20)8.5944 (50)1.7463 ( 1)1.1135 4.9066 3.3369 2.6223 06624 21376 25062 15077 66696 06703 80312 25506 06393 52073 30952 93965 11047 ELEMENTARY ANALYTICAL POWERS 125 15625 19 53125 2441 40625 10 3.0517 12 3.8146 14 4.7683 16 I 5.9604 18) 7.4505 57813 AND ROOTS 126 15876 20 Of3376 2520 47376 (10 1 3.1757 (12 4.0015 96938 04141 (14 I 5.0418 20 2601 10)3.3038 12 4.1958 14 5.3287 16 6.7675 18 8.5947 21 1.0915 50)3.0994 95218 20)9.3132 91266 71582 64478 80597 25746 24 50)2.1175 82368 (50)2.5638 52774 l/2 1)1.1180 33989 ( 1)1.1224 97216 5.0000 3.3431 2.6265 l/3 l/4 l/5 (16 8.0045 68619 5.0132 3.3503 2. 6307 07704 (50)6. ( 1)1.1401 75425 ( 1)1.1445 l/3 l/4 l/5 5.0657 3.3766 2.6472 97019 48375 24 1.3427 ( 1)1.1618 5.1299 3.4086 2.6672 l/2 l/3 l/4 l/5 1 1'0 24 l/2 l/3 l/4 l/5 2.0661 ( 1)1.1832 5.1924 3.4397 2.6867 2 3 4 5 6 7 8 28 3952 10 5.5730 12 7.8580 15 1.1079 17 1.5622 36000 35040 (21 I 2.8925 (51)3.2141 90379 5.1425 3.4149 2.6712 40000 (19 01028 1)1.1661 140 19600 27 44000 3841 60000 (12)7.5295 (15)1.0541 (17) 1.4757 02016 87418 18888 25688 37894 59535 56968 51)1.6030 27840 58099 68608 (10)5.3782 89056 19 2.2027 21)3.1059 04678 46550 99700 51)3.8129 15957 94102 90628 39790 ( 1) 1.1874 145 21025 30 48625 4420 50625 0 6.4097 34063 2 1 9.2941 14391 15)1.3476 46587 17)1.9540 87551 19)2.8334 26948 21)4.1084 69075 51)7.4616 01544 53078 23282 71840 3421 10 4.6525 12 1 6.3275 14 8.6054 17 1.1703 19 1.5916 21 I 2.1646 97252 95004 (51) l/3 l/4 l/5 24 ( 1)1.1489 136 18496 25 15456 135 18225 24 60375 3321 50625 l/2 5239 5.0787 3.3831 2.6512 11681 5.2048 3.4459 2.6905 5.0916 3.3895 2.6553 (51)1.9111 ( 1)1.1704 5.1551 3.4212 2.6751 63181 52970 OR461 141 19881 03221 54161 83670 41975 84764 1)1.2041 5.2535 3.4701 2.7056 59458 87872 00082 62363 13625 ( 0,1597 31428 67602 84058 1)1.2083 5.2656 3.4760 2.7093 Table 128 16384 20 97152 2684 35456 44882 69991 36735 13222 25206 19 2.3474 21 3.3333 (51) ( 129 16641 21. 46689 276’) 22881 91621 (50)3.7414 44192 (50)4.5097 56022 ( 1)1.1313 70850 ( 1)1.135;' 81669 5.0396 3.3635 2.6390 23 3129 10)4.1615 12)5.5349 14 7.3614 84200 85661 15822 5.0527 3.37O:L 2.6431. 133 17689 52637 00721 79589 00854 18136 134 17956 24 3224 I 21)1.7318 19)1.3021 16 1 9.7906 86120 61254 74468 (50)9.3851 10346 ( 1)1.1532 56259 4.3204 5.7893 7.7577 l.O39!j 1.3920 1.8665 (’ 51) 1.1233 1)1.1575 68722 62690 18337 5.1172 3.4023 2.6633 5.1044 3.3959 2.6593 74347 36005 26458 10 I 12 14 17 I 19 21 06104 17936 00342 36459 10855 33255 74561 85912 50184 83690 29947 28159 05339 139 19321 26 85619 373'3 01041 26 28072 3626 73936 (51)2.2756 11258 ( 51)2.7061 70815 ( 1)1.1747 34012 1)1.178') 82612 5.1676 3.4274 2.6790 49252 39296 19145 _-1 5.18O:l 3.4336 2.6828 01467 31623 90577 144 20736 29 85984 4299 81696 .I 29 24207 4181 61601 10) 5.9797 10894 43237 69396 4.5177 29930 1)1.1916 5.2171 3.4520 2.6943 37529 03446 10326 72696 (51)5.3464 ( 1)1.1958 5.2293 3.4580 2.6981 141 21609 31 76523 4669 48881 10)6.8641 48551 21)4.7116 (52)1.0366 53533 11527 ( 35565 32088 04545 85417 1)1.2124 5.2776 3.4820 2.7130 3.1 (21) 1.1805 _ “_I. 146 21316 31 12136 4543 71856 (51)8.7997 43370 61224 07280 28 63288 4065 86896 513518 8,1510 26159 28871 34209 27863 16727 67070 12529 nk 25 71353 3522 75361 (10)4.8261 72446 12)6.6118 56251 ii i 25695 96823 79413 5>!314 84918 (50)5.4280 42767 57088 30000 09000 51700 07210 24 5.0265 3.3569 2.6348 127 16129 48383 44641 36941 72915 58602 23424 54749 33853 83316 132 17424 22 99968 3035 95776 10)4.0074 64243 12)5.2898 52801 14 1 6.9826 05697 39521 I 16 9.2170 49217 76966 (50)7.8302 26935 17161 22 48091 2944 99921 l/2 1)1.1269 97935 68959 16865 131 130 16900 21 97000 2856 10000 1.3785 12848 1.0085 00000 01525 27804 10)3.7129 12)4.8268 14 6.2748 16 1 8.1573 87975 6.3527 (18 (21) 29 METHODS 42484 26074 21532 71824 56943 51)6.319.7 1) 1.2000 5.2414 3.4641 2.7019 148 21904 32 41792 4797 85216 4'3715 00000 82788 01615 20077 149 22201 3'3 07949 49213 84401 (52)1.2197 79049 (52)1.4337 40132 ( 1)1.2165 52506 ( 1)1.2206 55562 5.2895 3.4879 2.7167 72473 11275 66686 5. 3014 3.4937 2.7204 59192 88147 28110 30 ELEMENTARY Table ANALYTICAL POWERS 3.1 AND METHODS ROOTS nk k 150 22500 33 75000 5062 50000 : 3 5" 151 22801 34 42951 5198 85601 1.1853 b 91159 ; 1: 24 l/2 1.6834 ( 1)1.2247 11220 44871 5.3132 3.4996 2.7240 92846 (52) l/3 l/4 l/5 35512 69927 155 24025 37 23875 5772 00625 I :;I ;: T% (52)1.9744 ( 1)1.2288 52704 20573 5.3250 3.5054 2.7276 74022 53712 92374 l/2 5337 I 21 6.5831 I19 4.3310 ( 1)1.2328 ( 13 1.2332 15)1.8745 10 I 8.1136 82490 05615 22977 47627 (52)4.3150 94990 ( 1)1.2449 89960 ( 1)1.2489 5.3832 99600 12612 3.5341 2.7455 85355 41525 92987 160 25600 40 96000 6553 60000 I (52)2.7076 61312 (52)3.lb59 82801 5.3368 3.5112 2.7312 I 75387 17 2.8493 52)2.3133 ( 1)1.2369 31688 ( 1)1.2409 10 9.5388 99256 07183 43278 5.3946 3.5397 2.7490 1.4976 I15 2.3512 17 ( I (52)5.0302 ( 1)1.2529 74186 24 l/2 (52)7.9228 ( 1)1.2649 5.4288 3.5565 2.7594 l/3 ::; 44 7412 11 L2229 13 I 2.0179 15)3.3295 17 5.4937 19 1 9.0647 (22) 1.4956 24 16251 11064 35233 58820 59323 165 27225 92125 00625 81031 18702 65858 83665 43047 82603 (53)1.6581 15050 ( 1)1.2845 5.4848 23258 06552 3.5840 2.7764 24634 94317 170 28900 49 13000 8352 10000 6.9757 1.1858 24 l/2 l/3 l/4 l/5 (53)3.3944 ( 1) 1.3038 5.5396 3.6108 2.7931 5.4401 3.5621 2.7629 03274 57754 6887 40481 58257 73137 21220 92093 25920 163 26569 43 30747 7059 11761 47536 71008 49033 I ( 2.9282 4.7437 7.6848 29434 31683 45327 I 1.2449 53)1.0674 ( 1)1.2727 44943 81480 21825 0296b 00056 5.4513 3.5676 2.7663 92206 61778 21345 23734 76411 09873 64660 42676 51635 53)2.2140 1)1.2922 5.5068 3.5948 2.7831 (53) 1.2373 ( 1)1.2767 5.4625 3.5731 2.7697 90189 84798 78446 36294 92813 68945 69683 99103 71571 99540 78329 14533 55571 14235 30547 159 25281 40 6391 19679 28961 7964: (53)2.5551 ( 1)1.2961 5.5178 3. b002 2.7865 87425 48140 48353 05744 18023 -._ 50 8752 2232.2661 (53)4.4945 ( 1)l 3114 5'5612 3:6214 2.7996 51 8957 1.5496 2.6808 29929 77717 45041 38921 75333 13878 1.3880 2.4013 (53)5.1654 81379 80785 29935 87705 ( %?::; ;%46 97766 46817 62559 (52)b.8160 22003 ( 1)1.2609 52021 5.4175 3.5509 2.7560 01515 88625 01343 __. 164 26896 44 10944 7233 94816 (53)1.4330 1)l. 2806 5.4737 3.5785 2.7731 168 28224 41632 94176 78216 07402 56436 22812 64632 88583 173 171 29241 50 00211 8550 36081 (53)3.9075 ( 1)1.3076 5.5504 3.6161 2.7963 08411 36670 45765 I (11)1.3382 (53)1.9168 ( 1)1.2884 5.4958 3.5894 2.7798 5.3601 3.5227 2.7304 I 167 Z-r&i9 46 57463 7777 96321 27556 45 74296 7593 33136 57441 78765 86713 3.5453 2.7525 162 15 17 19 22 (52)9.2007 ( 1)1.2688 79483 80509 20176 26244 98241 67296 (52)5.8582 ( 1)1.2569 5.4061 42 51528 1 1.8075 I131 11.1157 17)4.2949 00782 67365 51946 79555 59901 96409 90712 68931 32856 81241 03963 80069 158 24964 39 44312 6232 01296 157 161 25921 41 73281 6718 5.3484 3.5170 2.7348 03297 43086 95679 24649 38 69893 6075 73201 18843 21947 5.3716 3.5284 2.7419 154 23716 36 52264 5624 48656 40965 82267 69056 1 3.6914 19 5.7955 21 9.0990 (52)3.6979 l/3 l/4 l/5 94816 13 2"E 153 23409 35 81577 5479 81281 79543 84905 81203 156 24336 37 96416 5922 40896 (10)9.2389 57978 I 17)3.3316 15)2.1494 8.0041 24 152 23104 15 11808 3:bZbb 2.8029 99110 10436 20335 24847 03675 81908 20b81 lb9 28561 48 26809 8157 30721 (53)2.9463 ( 1)1.3000 5.5267 3.6055 2.7898 26763 00000 74814 51275 27436 174 30276 52 68024 9166 36176 (53)5.9317 ( 1)1.3190 5.5827 3. b319 2.8061 37979 90596 70172 28683 43329 ELEMENTARY ANALYTICAL POWERS 31 METHODS AND ROOTS nk Table 3.1 k 175 30625 53 59375 9378 60625 : : ; l/2 l/3 l/4 l/5 (53)6.8063 32613 ( 1) ;. 5';;; y;; 3:6371 2.8093 35763 61392 180 32400 58 32000 1.0497 60000 1.8895 68000 1.1019 24 l/2 l/3 l/4 l/5 58845 40786 16173 41501 34501 12576 42134 a6155 47634 43835 69315 49994 62212 9aii 06241 (53)8.9404 29702 49916 78661 ( 1)1.3304 5.6146 13470 72408 (54)1.0234 ( 1)1.3341 5.6252 81638 26328 66406 3.6423 2.0125 20574 64777 3.6474 2.8157 63337 53634 3.6526 2.8189 24271 28111 181 32761 59 29741 9)1.0732 a3121 (54)1.5285 ( l)l.3453 5.6566 3.6679 2.8283 185 34225 63 31625 24 l/2 l/3 l/4 l/5 (54)2.5829 ( 1)1.3601 5.6980 3.6880 2.8407 : 5 82606 47051 19215 17151 58702 71637 62405 52026 16217 66697 l/3 l/4 l/5 24 l/3 73122 08816 40794 43509 88352 184 33856 62 29504 33489 61 28487 70074 73756 51108 73940 85080 65 39203 (54)1.9898 ( 1)1.3527 5.6774 3.6780 2.8345 62931 04875 97079 07530 50791 (54)9.1375 ( l)l.3964 5.7988 3.7368 2.8708 69069 24004 89998 75706 26340 n”[‘-9’] (54)2.2679 ( l)l.3564 5.6877 3.6830 2.8376 laa 35344 66 44672 9)1.2491 98336 54)2.9397 51775 (54)3.3434 78670 (54)3.8000 41874 67473 18170 ( 1)1.3674 5.7184 79065 79433 ( 1)l 5.7,a6 3711 54316 30920 3.6929 2.8430 90888 23174 3.6979 2.0468 44609 74493 3:7il20 2.8499 78502 12786 (54)7.1346 ( 1)1.3892 5.7789 3.7212 2.8649 E7 88001 51842 54055 30326 22953 a7299 90487 95065 44399 96565 56899 13156 54496 92632 49854 54)5.5564 l';.:;;; 3:7175 93542 ff;;;; 63041 (54)6.2983 ( l)l.3856 3.7224 5.7689 ::::2" 74059 a6193 89130 40646 19436 98281 2.0589 50746 2.8619 38162 196 38416 75 29536 55)1.0331 ( ";.;;;; 3: 1416 2.8737 197 38809 76 45313 07971 (55)1.1673 18660 it;;; 57367 ( 1)1.4035 3.7464 5.0186 66885 20805 47867 3.0772 5.8159 (54)4.3160 ( 1)1.3747 5.7307 3.7077 2.8529 64756 2.8766 91203 j[(-751 ,a[,-,31 49187 24728 76683 66123 05790 4,,,2] 03640 14881 la526 72700 93540 92751 38178 194 37636 73 01384 (54)8.0768 (*)1.3928 5.7869 3.7320 2.0670 40718 38828 60372 75599 75844 199 . _ 198 39204 77 62392 ( 9)1.5369 53616 (55)1.3181 ( 1)1.4071 5.8284 3.7511 2.8796 20111 65997 33960 23210 80950 189 35721 67 51269 193 63361 49020 22627 84218 97266 86816 04818 (54)4.8987 ( 1)1.3784 5.7408 3.7126 2.8559 76639 74926 11371 08871 89786 ( 1)1.3638 5.7082 195 38025 74 14875 l/2 (54)1.7446 ( 1)1.3490 5.6670 3.6729 2.8314 186 34596 64 34056 7" w l--_ R1 182 33124 60 28568 190 _.. 36100 68 59000 : : 10 24 (54)1.1707 ( 1)1.3379 5.6357 3.6577 2.8220 96058 (54)1.3382 ( 1)1.3416 5.6462 3.6628 2.8252 9595 11 I 1.6887 13 2.9721 15 5.2310 17 I 9.2066 20 1.6203 22 2.8518 53)7.8037 1)1.3266 5.6040 2 1: 24 179 32041 39601 78 80599 (55)1.4875 ( 1)1.4106 5.8382 3.7558 2.8825 57746 73598 72461 93499 08624 32 ELEMENTARY Table 3.1 POWERS 201 40401 20601 40801 40000 80 00000 1.6000 00000 6.4000 l/2 l/3 (55)1.6777 ( 1)1.4142 5.8480 3.7606 2.8853 21600 13562 35476 03093 99812 l/2 l/3 :,/: %Z 77620 10032 00000 (55)1.8910 ( l);.;&; 3:7652 :::t: 60303 2;;;; 95059 2.8882 79450 205 42025 a6 15125 : : 2 i9 10 24 ANALYTICAL AND METHODS ROOTS 202 40804 a2 42408 1.6649 66416 3.3632 32160 6.1937 20964 1.3723 33251 2.7721 13166 5.5996 68596 1.1311 33056 (55)2,1302 61246 ( 1)1.4212 67040 5.0674 64308 3.1699 69549 2.8911 47666 206 42436 a7 41816 14096 77038 34698 38548 n-4 203 41209 83 65427 (55)2.3983 ( l)l.4247 5.8771 3.7746 2.8940 07745 80685 30659 26716 04537 __- 2na 207 42849 88 69743 43264 89' 98912 :EE 70372 (55)3.0345 ( 1)1.4317 5.8963 3.7838 2.8996 38594 82106 68540 a9674 84668 (55)3.4104 ( ";.,$$ 3:7804 62581 :;;;; 95756 2.9025 08125 210 44100 92 61000 : : (55)3.8307 ( 1)1.4387 5.9154 3.7930 2.9053 211 44521 93 93931 89523 49457 a1700 a5099 20638 (55)4.3005 ( 1)1.4422 5.9249 3.7976 2.9081 212 44944 95 isiia 10765 20510 92137 57844 22302 213 45369 96 63597 2 1 : 10 24 l/2 l/3 $5" (55)5.4108 ( 1)1.4491 5.9439 3.8067 2.9136 19838 37675 21953 54096 93459 215 46225 99 38375 (55)6.0642 ( 1)1.4525 5.9533 75557 a3905 41813 3.8112 2.9164 77876 63134 216 46656 100 71696 (55)6.7929 ( 1)1.4560 5.9621 3.8157 2.9192 102 a5105 21978 31958 85604 22328 217 47089 18313 (55)7.6051 ( 1)1.4594 5.9720 3.8202 2.9219 97251 51952 92620 77414 71130 218 47524 103 60232 204 41616 84 89664 (23)1.2482 (55)2.6985 ( 1)1.4282 5.8867 3.7792 2.8968 50286 09916 85686 65317 66709 50171 209 43681 91 29329 9 1.9080 29761 11 3.9877 82200 13 a.3344 64799 16 1.7419 03143 18 3.6405 77569 20 i 7.6088 07119 23)1.5902 40688 (55)4.8251 50531 ( 1)1.4456 83229 5.9344 72140 3.8022 14131 2.9109 13212 214 45756 98 00344 9)2.0972 73616 11)4.4881 65538 13)9.6046 74252 20)9.4129 (23)2.0143 (55)8.5100 ( 1)1.4628 5.9814 3.8247 2.9247 11168 62990 19601 73884 24030 53435 09627 219 105 47961 03459 10 24 l/2 l/3 l/4 l/5 (55)9.5175 03342 ( 1)1.4662 a7830 5.990f26415 3.8292 13796 2.9274 37906 1 : 4 106 220 48400 48000 56)l.o63a ";.;;;; 3:8336 73589 ;;;;; 58625 2.9301 56052 107 (56)1.1885 ( 1)1.4730 6.0092 3.8380 2.9328 94216 91986 45007 88048 64149 (56)2.0533 ( 1)1.4899 6.0550 3.8600 2.9462 222 49284 41048 12656 86096 65313 84996 16690 14905 67090 89736 66443 48947 08345 56780 221 48841 93861 5 6 7 : 10 24 l/2 :s: l/5 (56)1.6525 ( 1)1.4832 6.0368 3.8512 2.9409 10926 39697 10737 85107 28975 (56jl.8425 ( l';;;;$ 3:8556 2.9435 3oo"3 i;'$; 54127 97699 i56j1.3272 ( 1)1.4764 6.0184 3.8425 2.9355 59512 82306 61655 02187 62280 (56)2.2872 ( 1)1.4933 6.0641 3.8643 223 49729 89567 73441 30773 84962 20466 97640 78274 25550 66205 18452 26994 47878 (56)1.4813 ( 1)1.4798 6.0276 3.8469 2.9382 112 (56)2.5465 ( 1)1.4966 6.0731 3.8686 53665 64859 50160 01167 50529 224 50176 39424 51362 62955 77944 72841 ELEMENTARY ANALYTICAL AND 33 METHODS nk POWERS ROOTS Table 3.1 k : 113 : 225 50625 90625 115 226 51076 4317b 116 227 51529 97083 118 229 52441 1213 08989 5i352 2 i 1: 24 l/2 :s: l/5 (56)2.0338 73334 (56)3.1521 18526 ( 1)1.5000 6.0822 3.8729 2.9541 00000 01996 83346 76939 ( 29638 99349 79501 90210 121 24 l/2 1)1.5033 6.0911 3.0772 2.9567 230 52900 67000 123 56) 3.5044 1)l. 5066 6.1001 3.8815 2.9594 231 53361 26391 124 5568b (56)4.8025 07640 (56)5.3295 12896 (56)5.9116 89798 75089 25615 22905 91438 ( 1)1.5198 b..1357 3.8985 2.9697 68415 92440 48980 b7129 ( ;;;$; 1)l. if: l/5 5165 6.1269 3.8943 2.9bll 129 235 55225 77675 131 "lb.;::; 3:9027 2.9723 236 55696 44256 133 1)1.5099 6.1091 3.0050 2.9620 232 53824 87168 ( 62082 56)3.0943 51917 70200 61435 10235 126 ( 56) 4.325'b 51908 bb887 14744 29230 130b2 ( 1’;. y; y;; 3:89OQ 2.964b 233 54209 49337 56) 6.5545 ( l)l.5264 38267 33152 3.9069 6.1534 2.9148 128 (56) 60138 49494 91866 61357 33915 237 56169 12053 ( 1.2640 1'; :'6;; 3:9111 2.9774 238 134 i%:: 136 83026 06713 234 54756 12904 79321 g;:; 45426 41049 239 57121 51919 42736 31712 (56)9.861'3 93410 (57)1.0910 :x 74649 49664 56201 55818 "k:;;; "b;;i; ( 24862 ._.. 24 l/2 l/3 l/4 l/5 (56)8.0469 ( l)l.5329 6.1710 3.9153 2.9799 01671 70972 05793 17320 81531 (56) 8.9102 ( 1) 1.5362 6.1791 3.9194 2.9825 - .- 138 24 l/2 l/3 l/4 l/5 24 l/3 l/4 l/5 139 (57)1.3337 35777 (57)l. ( 1)1.5491 6.2144 3.9359 2.9925 93338 b5012 79343 55740 ( 147 l/2 57600 24000 (57)2.1876 ( 1); 4736 l)l.5524 6.2230 3.9400 2.9950 245 b0025 06125 148 12697 29150 46606 15921 13380 3: 9236 2.9850 241 58081 97521 141 99791 57)l.b276 17470 84253 12930 45390 1';;;;; 21327 36660 242 58564 72480 ;;6';: 150 ( 53798 26190 3: 9563 3.0049 (57)2.4123 62509 (51)2.6590 52293 ( 38714 26556 51896 11096 ( l)l.Sllb b.2743 3.9643 3:0098 23365 05351 70523 12147 20998 22094 1 5684 6.2658 3.9603 3.0073 7910 1y; 53635 54435 51438 243 59049 48907 84401 86094 11321 54510 66546 12701 79617 10300 pm; 3:9482 3.0000 247 b1009 69223 y"b 1)l. 143 (57)l. 91225 ;;;; 3.9277 6.1971 2.9075 79087 3:9441 2.9975 246 60516 86936 1)1.5427 152 (57)2.9298 (57)1.2065 61943 ( 62483 21795 72942 57776 1)1.5459 6.2058 3.9318 2.9900 _ 145 ?AA 59536 26704 23)7.4799 42569 (57)1.9831 ( l)l.5620 51223 49935 99710 71742 b5081 22039 00000 3.9522 6.2487 3.0024 248 61504 52992 42016 00200 37650 13371 13716 bbOlb 91719 154 15956 (57) 01515 b1305 76966 45305 ( 1';. 3.2268 ;;;I) 3.9723 3.0146 249 b2001 36249 91251 ;g 71312 70627 34 ELEMENTARY Table 3.1 ANALYTICAL POWERS METHODS AND ROOTS nk k : 156 250 62500 25000 251 63001 158 13251 160 252 63504 03008 253 64009 94211 52081 19416 46016 41511 65511 23159 91011 : 2 ; 1: 24 w (51) ( 3.5527 1'2. l/3 l/4 l/5 ;g; 319763 3.0170 165 13679 (51) y9" ( 1)1.5842 3.9099 97952 6.3079 3.9803 3.0194 53644 88168 65025 81315 33001 161 93549 24041 97986 (57)4.3014 ( 1)1.5874 6.3163 3.9042 3.0219 256 65536 17216 169 31119 50181 59598 02604 00136 (51) 4.7303 97372 03543 29397 94671 ;2’:; 3:9882 3.0242 257 66049 74593 ( 1)1.5937 6.3330 3.9921 3.0266 258 171 %:E 66096 37653 95144 81472 68820 75555 63693 113 51) 5.7143 11018 (57)6.2771 01735 (57)6.8927 88615 (51)7.5661 15089 (57)8.3022 1)1.5968 11942 25705 88015 61117 ( ii;;; ( 1)1.6031 21954 ( 1)1.6062 37840 ( l/2 6.3413 3.9960 3.0290 l/3 l/4 l/5 175 24 57)9.1066 85770 ( 1)1.6124 51550 6.3825 4.0155 3.0408 186 4:oooo 3.0314 00000 33133 6.3578 4.0039 3.0337 260 61600 16000 l/2 l/3 l/4 l/5 1';;:;; 61180 00541 91748 4.0077 6.3660 3.0361 262 68644 84128 98736 43669 04412 01560 1.4384 ( 1)1.6278 6.4231 4.0341 3.0524 (58) 196 181 91447 (58)1.0945 (58)1.3136 94086 41406 1.1993 ( 1)1.6217 27914 ( 1)1.6186 27474 ( 6.3988 4.0232 34278 27910 6.4069 4.0270 58577 67760 83226 3.0455 11602 3.0418 32879 07681 68660 90325 47105 188 266 70756 21096 190 261 71289 34163 (58) 192 9 5.1586 12 1.3825 14 I 3.7051 83974 68868 1)1.6340 6.4392 4.0422 3.0570 13464 16696 93240 47961 1) 1.6310 6.4413 4.0460 3.0593 10554 05721 72854 34462 58289 02045 54329 6.4312 4.0385 3.0547 27591 02994 54599 61673 04070 00464 87063 6.4633 4.0536 3.0638 58) 1.8846 ( 39954 l/3 l/4 l/5 40472 50643 (58) 199 (58)2.4618 ( 1)1.6462 6.4712 4.0573 3.0661 02511 51891 01763 13627 48596 53254 201 272 73984 23648 (5Q2.6893 89450 ( 1)1.6492 42250 6.4792 4.0610 3.0684 23603 86370 12165 203 (58)2.9369 ( 1) 1.6522 6.4811 4.0648 3.0706 1)1.6248 6.4150 4.0308 3.0501 268 11824 48832 86916 28110 15334 1.1229 82060 ( 1)1.6431 183 264 69696 99144 49442 89807 76528 (58) (58)2.2528 67601 11088 04982 54265 60235 24 47694 4.0116 6.3743 3.0385 89716 96760 55014 6155 4:0193 6.3906 1.5145 ( 1)1.6309 m 21920 3.0431 265 70225 09625 270 72900 83000 259 67081 13979 %; 98010 38372 57)9.9‘355 ( I)1 04299 34273 41703 10548 31145 25531 64507 01647 1) 1.6093 263 69169 1.9113 24 254 64516 81064 41643 ( 1’;. 163 194 (58)2.0608 ( 91176 11164 54111 13851 65640 89564 1) 1.6401 6.4553 4.0498 3.0616 273 14529 46411 205 (58) 269 72361 65109 21947 14811 41906 14141 274 15iC 70024 3.2063 69049 3.0129 11923 ELEMENTARY ANALYTICAL POWERS 35 METHODS AND ROOTS 212 9 5.8873 12 1.6307 14 4.5172 17 1 1.2512 19 3.4660 21 9.6010 24 I 2.6594 58)4.1640 1)1.6643 6.5186 4.0796 3.0796 277 76719 53933 39441 93025 96680 91180 76569 32097 85891 35828 31698 a3915 22161 11650 nk Table 3.1 k : 201 : 275 75625 96875 276 210 16116 24576 2 i 1: 24 l/2 l/3 l/4 l/5 (58)3.4993 ( 1)1.6583 6.5029 4.0722 3.0751 219 24 l/2 :;: l/5 (58)5.3925 ( 1)1.6733 6.5421 4.0906 3.0862 (58)3.8178 ( 1)1.6613 6.5108 4.0759 3.0773 28001 12395 57234 38199 51657 280 78400 52000 32264 20053 32620 23489 53577 221 (58)5.8742 ( 1)1.6763 63499 4.0942 3.0884 al225 231 49125 24 l/2 l/3 l/4 l/5 (5a)a.2466 ( 1)1.6aal 6.5808 4.1087 3.0971 243 24 l/2 l/3 l/4 l/5 233 II 14 6.6905 I 12 5.4726 9 1.9135 17 1.5651 19 4.4163 22 1.2802 24 3.6615 (58)8.9690 ( 1)1.6911 6.5085 4.1123 3.0993 32480 94302 44365 64171 98013 290 84100 R9000 9)7.1708 246 42160 24773 30071 35196 a4885 281 78961 a8041 39885 05461 11620 70950 54901 81796 93656 48566 10900 42039 53453 32215 63618 68441 291 84681 42171 71761 4.3544 (59)1.3596 15721 64428 ( 1) ;. y'9; ;g:; ( 1) t ‘6;;; ;g;;; 4:1302 3.1101 [J :: :::: 295 a7025 72375 20588 30396 ;:::: I 17 I 6.5907 19 5.7355 14 1.6919 1.9442 24 l/2 l/3 l/4 l/5 10930 56404 30232 41207 33956 287 82369 39903 58)6.9642 1)1.6822 6.5654 4.1015 3.0928 238 (58)9.7536 ( 1)1.6941 6.5962 4.1159 3.1015 13040 07435 02284 53637 32807 292 85264 748 97088 49696 25311 49909 05773 16858 a3323 a7302 (59)1.4763 46962 ( 1"6'W3; ;;I;; 411337 3.1122 64325 65011 297 88209 296 a7616 34336 63456 62783 (59)1.0602 ( 1)1.6970 6.6038 4.1195 3.1036 251 (59)1.6025 ( 1)1.7117 6.6418 4.1372 3.1143 2% z"o"8 7A4 --. 63969 58973 91371 08381 (59)1.8868 ( 1)1.7175 6.6569 4.1443 3.1186 259 ( 9 I 7.6765 (12 2.2722 I :;j;: 256 236 33126 a5562 72186 08689 49967 226 1 (59)2.0464 ( 1)1.7204 6.6644 4.1478 3.1207 01230 33200 10079 99156 31992 217 (58)4.9488 11121 ( 1) ;. ;;g ;g; 4:0869 3.0840 283 a0089 65187 51599 60384 14427 36766 38815 288 a2944 a7872 279 77841 17639 229 66245 45954 80656 06304 (58)7.5794 93086 ( 1) ;. y; $94;; 4:1051 3.0950 55240 21484 241 289 83521 37569 935’39 49008 67707 89906 (58)6.3970 ( 1)1.6792 6.3576 4.0979 3.0906 25168 (58)4.5402 ( 1)1.6673 6.5265 4.0832 3.0818 278 77284 a4952 31410 07486 a5616 72583 (59)1.2518 411266 3.1079 19524 224 214 49657 65053 43703 48904 45423 1 (59)2.2189 ( 1)1.7233 6.6719 4.1513 3.1228 a7131 68794 40272 47726 51191 1 (59)2.4054 ( 1)1.7262 6.6794 4.1548 3.1249 77893 56275 54498 34288 91148 24)4.0642 (59)1.1522 ( 1)1.7(100 6.6114 4.1231 3.1058 293 a5849 53757 91698 24277 52195 98970 93785 298 88804 63592 50416 72824 17015 58671 36838 02776 42278 16789 67650 20032 37723 51295 31407 54005 00000 89018 05626 43502 294 06436 (59)1.7391 ( 1)1.7146 6.6493 4.1408 3.1165 45550 42820 99761 24580 16755 299 a9401 :267 3089s (59)2.6068 ( 1)1.7?91 6.6868 4.1'583 3.1270 04847 61647 83077 la947 45768 36 ELEMENTARY Table ANALYTICAL POWERS 3.1 METHODS AND ROOTS 301 90601 nk 302 91204 43608 9)8.4288 12)2.5539 (59)2.8242 00000 95365 272 70901 215 _-. 303 91809 18127 92481 54422 24)6.5226 270 83188 270 92416 280 9)8.5407 ll/4 /3 3.1291 l/5 34645 283 15639 (59)3.3125 81949 (59)3.5861 05682 (59)3.8811 1)1.7349 l/2 59)3.0591 35157 ( 14720 72852 10496 95743 ( 1)1.7406 89519 ( 1)1.7435 6.7017 4.1652 3.1312 93025 72625 59395 55283 17958 286 l)i.7378 6.7091 4.1687 3.1332 306 93636 52616 289 6.7165 4.1721 3.1353 69962 57138 68030 99856 59577 6.7239 4.1155 3.1314 50814 95260 34853 308 307 94249 34443 94864 292 9 8.9991 12 2.7717 14 8.5369 17 2.6293 19 1 8.0985 i 18112 94464 17056 295 309 95481 03629 18496 46977 80688 90052 21360 2.4943 44579 24 ('X9)4.1994 86063 (59)4.5427 01868 (59)4.9127 08679 (59)5.3115 00125 (59)5.7412 10972 l/2 ( 1)1.7464 24920 ( 1)1.7492 85568 ( 9";;;; ( 1)1.7549 92877 ( 1y.;;;; ;'4;;; ll/4 /3 6.7313 4.1790 3.1394 l/5 15497 24910 96244 6.7386 4.1824 3.1415 64101 46136 52236 91000 301 10000 ( 9)9.2352 4: 1858 3.1436 80231 (59)6.2041 26610 (59)6.7026 93132 ( 1)1.7606 81686 ( 1)1.7635 312 97344 71328 04640 19209 6.7678 4.1960 3.1497 : : 2 i9 99452 41767 22833 312 6.7751 4.1994 3.1517 315 99225 55875 (59)7.2395 313 97969 64297 24961 50513 91105 36216 16356 61119 20304 9 12 I 14 17 19 I 22 24 (59)7.8174 3.1531 316 99856 54496 317 1 00489 318 55013 59756 88127 309 31800 6.7896 4.2061 3.1557 76544 411926 3.1476 59)8.4393 80601 ( 1)1.7691 68952 27591 52295 315 28072 13417 63512 48146 306 9.5979 3.0041 9.4029 2.9431 9.2120 2.8833 9.0249 02859 303 2.8015 6.7533 4.1892 3.1456 58988 311 96721 96100 297 1) ;. y;; l)l.7720 61336 62861 95609 99655 04515 6.7968 4.2095 3.1578 84386 18398 09519 318 1 01124 321 57432 314 98596 59144 324 319 1 01761 61759 10 24 l/2 (59)9.1086 34822 ( 1)1.7748 23935 1) 1.7716 92116 65931 18306 6.8112 4.2162 3.1618 6.8040 4.2128 3.1598 l/3 l/4 l/5 (59)9.8285 320 24 v2 l/3 l/4 l/5 68000 1.3292 ( 1)1.7888 6.8399 4.2294 3.1697 84208 49381 6.8184 4.2195 3.1638 61941 37156 20622 (60)1.1435 38734 ( 1)1.7832 55450 6.8256 4.2228 3.1658 322 1 03041 24197 60938 14209 03787 85054 86385 (60) 1.4325 ( 1)1.7916 6.8470 4.2327 3.1717 I 76161 86248 41287 21278 85474 65030 333 21862 35844 6.8541 4.2360 3.1137 24002 78192 38149 37808 57110 6.8327 4.2261 3.1678 71452 76889 02787 324 1 04325 336 1.6628 ( 1)1.7972 78568 20076 6.8612 4.2393 3.1757 12036 63249 07571 1 04976 98267 54024 06498 73199 01432 32163 84885 19218 86248 (60) 1.5436 ( 1)1.7944 1.2330 ( 1)1.7860 (60) 323 1 03684 10)1.0884 12)3.5157 15 1.1355 17 3.6679 20 1.1847 22 3.8266 Ii 25 1.2360 330 27996 54382 (60) 1.0602 ( 1)1.7804 (60) 321 1 02400 327 62028 38883 84608 05502 21997 (60) 340 12224 1.7909 ( 1)1.8000 36736 00000 6.8682 4.2426 3.1776 85455 40687 11523 (60) ELEMENTARY ANALYTICAL POWERS 37 METHODS AND ROOTS nk Table 3.1 k 325 3 343 10 1.1156 12 I 3.6259 15 1.1784 : 6 24 64063 08203 20166 6.8753 4.2459 3.1796 (60)2.2343 23554 47009 ( 1)1.8083 14132 ( 1)1.8110 6.8823 4.2491 3.1815 88750 72871 84924 21000 44298 67969 39300 08618 78985 48440 ( Q1.8165 6.9104 4.2621 3.1893 90212 23230 47595 54454 55434 60)2.9913 81825 3.1912 85058 335 95315 332 (60)3.9909 41565 (60)4.2868 ( 1) ;. ;:5"; y', ( 1y;; y;"o 9 01166 4:2813 3.2008 61118 90286 68669 340 (60)3.2159 84959 (60)4.6038 12427 ( 1)1.8357 6.9589 4.2845 3.2027 10 1.2444 74114 43539 85542 73711 15819 10837 85420 98859 (60)6.5558 l/2 ( 1)1.8439 08891 ( 1);;;;; ;"8;;; ( 1)1.8493 24201 412972 3.2103 29958 38860 345 40; 6.9931 4.3003 3.2122 76961 90657 19552 346 1 19025 410 63625 ._ I I I 66688 6.9313 4.2718 3.1951 00768 01446 32308 25) 1.9461 (60)4.9431 417 339 1 14921 389 58219 08291 3.2046 16051 (60)5.3063 11693 95264 6.9726 4.2909 3.2065 70186 __ . II II 1.6284 53607 1 18336 407 (105 1.6571 17 .8171 1 4.4003 12 5.7004 49439 07395 72660 40890 25 2.3205 22 6.7456 20 1.9609 15244 54607 83848 (60)7.0316 76479 (60)7.5405 43015 ( g;;;; ( 1)1.8547 23699 1) +. i;'o; 4:3035 3.2140 7.0067 4.3066 3.2159 17071 95850 348 1 21801 425 08549 I 15 1.8069 17 6.3063 48816 76738 25 7.6811 22 2.6807 20 2.2009 37377 95921 15737 30203 54169 80617 14906 62636 Ii 53376 (60)9.2876 83235 (60)9.9518 04932 ( 1)1.8574 7.0135 4.3097 3.2178 17562 79083 76748 35355 ( 1)1.8601 07524 ( 1)1.8627 93601 ( 1)1.8654 75811 1.0661 ( 1)1.8681 49656 15431 1222,6 7.0405 4.3222 3.2252 4-46)2] 96121 50321 67776 349 1 21104 421 44192 81923 (60)8.6661 4[(-46)5] 07584 13598 95243 48952 96386 98608 82649 15128 64201 144 343 1 17649 403 (60)8.0845 7.0203 4.3128 3.2196 32074 04899 49006 ( 1)1.8411 1 20409 21736 92066 44547 60213 30338 39497 76659 13924 6.9382 4.2750 3.1970 338 1 14244 386 14472 347 1 19716 414 26935 ( 1)1.8275 72295 43337 71684 (60)6.lloa 10 l/3 334 1 11556 37.x 59704 28759 55975 03680 i l/2 35942 15020 19165 (60)3.7146 12753 (60)5.6950 2 7 24 35715 99320 337 24 : $2 26037 34894 (60)3.4566 1 135bs 342 16964 01688 57730 57435 35043 61047 81952 10274 77114 12822 39: 1/3 369 11001 3.1932 341 16281 51821 27096 53398 66909 30158 47684 24602 04689 : 32047 76026 53751 4.258') 3.187,4 1 10889 94368 382 10 6.9795 4.2940 3.2084 34481 75061 79164 ( I I ( (60)2.5864 ( 1)1.813#3 6. 903,1 I(12 I 4.1565 15 1.3882 17 4.6363 20 1.5487 I2252 I 5.1727 ( 1.1276 37: : 8 4:2782 3.1989 6.8964 4.2556 3.1854 1 10224 365 336 12896 %E 90290 16737 60238 79440 50918 72309 93134 l/3 l/4 l/5 18774 27697 34426 329 1 08241 3515 11289 10)1.1716 11408 ( 1)1.8248 1 12225 375 6.8894 4.2524 3.1835 331 09561 64691 61272 95811 27813 73062 67184 70378 28495 37000 2 l/2 77028 76350 I 20 I 1.6411 25 1.5315 22 4.4064 (60)2.7818 24 15994 11326 93150 48153 45942 47069 09169 ( 1)1.8055 44335 10547 30632 1 87552 31706 (60)2.0759 Ii 12 1.1859 ~17 4.2618 15 3.9135 10 1.2914 : 352 75638 21030 359 l/3 l/4 l/5 1 07584 65183 12 3.7963 15 1.2452 II 17 4.0842 20 1.3396 22 4.3940 I 25 I 1.4412 (60)2.4042 330 l/2 349 10 1.1574 1 08900 24 1 06929 45976 65540 06300 95476 ( l)l.8027 1/3 1/4 l/5 ,346 328 327 1 06276 28125 15722 17 3.8298 20 1 1.2447 22 4.0452 25) 1.3147 60)1.9284 ; 9 10 10 326 1 05625 : 7.0271 4.3160 3.2215 05788 09269 57557 +j)l] 7.0338 4.3191 3.2234 (61) 4647)63 ELEMENTARY 38 Table ANALYTICAL POWERS 3.1 k 350 1 22500 428 75000 25000 87500 65625 : : z ii 9 10 24 ( "y;; l/3 413253 3.2271 ::: (61)1.2228 ( y:;1 4:3283 3.2289 z 6 .li 9 10 24 l/2 l/3 i//2 (61)1.6050 ( 1)1.8841 7.0806 4.3406 3.2362 20092 44368 98751 73183 76880 360 1 29600 466 56000 24 25771 ;61;2.2452 (61)1.3092 ( l)l.8761 7.0606 4.3314 3.2307 356 1 26736 451 18016 10 1.6062 01370 12 5.7180 76876 15 2.0356 35368 17 7.2468 61909 20 i 2.5798 82840 22 9.1843 82909 25 3.2696 40316 (61)1.7171 17251 ( 1)1.8867 96226 7.0873 41061 4.3437 26771 3.2380 98084 355 1 26025 447 38875 : 3 87825 95605 44363 70945 73933 15199 (61)2.5645 17652 l/2 l/3 l/4 l/5 3.2453 24 l/2 ::: l/5 42223 365 1 33225 486 27125 (10 1.7748 90063 12 6.4783 48728 15 2.3645 97286 80093 I 17 8.6307 (20 3.1502 34734 (23 1.1498 35678 (25 1 4.1969 00224 (61)3.1262 86296 ( 1)1.9104 97317 7.1465 69499 4.3709 23607 3.2543 07394 50; (10)1.8741 370 36900 53000 61000 95700 26409 87713 79454 3.2471 43191 3.2489 :/,: l/4 l/5 (61)4.3335 ( 1)1.9235 7.1790 4.3858 3.2631 25711 38406 54352 16237 74848 J[(-q5] 40172 361 490 10)1.7944 61)3.3384 1)1.9131 7.1530 4.3739 3.2560 1 34689 494 30863 1 33956 27896 20994 59019 12647 90095 14319 88625 (61)3.5643 ( 1)1.9157 7.1595 4.3768 3.2578 92671 24406 98825 98909 65967 ( 1)1.9287 7.1919 4.3917 3.2666 372 38384 78848 13146 48902 71791 67064 75348 26429 22317 85051 30152 66348 31039 95001 371 1 37641 510 64811 lg% 24 358 1 28164 458 82712 (61)1.9642 ( 1)1.8920 7.1005 4.3498 3.2417 362 1 31044 474 37928 (61)1.4999 ( 1)1.8814 7.0740 4.3376 3.2344 55202 88772 43955 13137 51567 359 1 2asa1 462 68279 (61)2.1002 ( 1)1.8947 7.1071 4.3528 3.2435 31355 88793 88459 14700 28247 363 1 31769 478 32147 29556 29532 93661 49104 37249 364 1 32496 482 28544 56304 (61)2.3997 54042 66304 96671 73541 88532 (61)1.8366 ( 1)1.8894 7.0939 4.3467 3.2399 354 1 25316 443 61864 353 1 24609 439 86971 40288 73217 54146 35134 02402 38480 25983 99442 (61)1.4014 29423 ( 1q.g; 76615 46600 4:3345 3.2326 22125 357 1 27449 454 99293 361 1 30321 10)1.6983 rzk 352 1 23904 436 14208 lb:::: 64375 84096 43263 99400 04063 93928 50768 (61)1.1419 l/2 AND ROOTS 351 1 23201 43551 432 48640 48727 04703 :%f 63867 47354 13124 28693 98732 07727 08809 METHODS (61)4.6235 ( 1)1.9261 7.1855 4.3887 3.2649 31606 36028 16151 76627 36822 4646121 +-p] 51)2.7400 1)1.9052 7.1334 4.3649 3.2507 1 498 10 1.8339 12 6.7489 15 2.4836 17 1 9.1397 20 3.3634 23 1.2377 25 I 4.5548 (61)3.8049 ( 1)1.9183 7.1660 4.3798 3.2596 53237 55888 92490 23697 33187 368 35424 36032 65978 94798 30086 58715 31207 42684 93078 38558 32609 95742 77406 39439 i61)2.9270 ( 1)1.9078 7.1400 4.3679 3.2525 70667 78403 36982 26743 22254 369 1 36161 502 43409 (61)4.0609 ( 1)1.9209 7.1725 4.3828 3.2614 98114 37271 80900 49839 09059 374 1 39876 523 13624 ( 1)1.9313 7.1984 4.3946 3.2684 20792 04996 79501 49404 $-;'5] (61)5.6094 ( 1)1.9339 7.2048 4.3976 3.2702 26383 07961 32147 22040 00047 ELEMENTARY ANALYTICAL POWERS 39 METHODS AND ROOTS nk Table 3.1 1 544 10 2.0632 12 7.8198 379 43641 39939 73686 07278 k : 375 1 40625 527 34375 1 531 10 1.9987 12 7.5151 15 I 2.8257 16 1.0624 20 1 3.9948 : 2 i 9 10 24 l/2 l/3 :/,: (61)5.9806 ( 1)1.9364 7.2112 4.4005 3.2719 88819 12334 (15)5.7998 (61)6.7950 06469 46060 ( 1)1.9390 7.2176 4.4034 3.2736 71943 52160 89461 90130 ( l';.;;:; ‘I;;;: 6' i 9 10 24 l/2 l/3 l/4 l/5 : : 14397 29605 (61)7.2410 ( 1)1.9442 7.2304 4.4093 3.2771 (61)9.3222 49236 ;61;9.9259 15535 ( 1)1.9519 7.2495 22130 04524 ( 1)l 7.2558 9544 41507 82029 4.4180 3.2823 60383 58869 56443 54436 25976 (62)1.1247 ( 1)1.9621 7.2747 4.4296 3.2892 : 414064 3.2754 54; 56280 50807 ( li1.9570 7.2621 4.4230 3.2857 38579 67440 42876 09631 53901 41687 06349 06853 14120 I 593 10 I 2.3134 12 9.0224 15 1 3.5187 18 1.3723 20 I 5.3520 23 2.0872 25 8.1404 (62)1.5330 ( 1)1.9748 7.3061 4.4439 3.2977 390 52100 19000 41000 19900 43761 10067 09260 83612 06085 29700 41766 43574 19178 13494 575 (62)1.1970 ( 1)1.9646 7.2810 4.4324 3.2909 386 1 48996 12456 03202 88270 79420 80423 21030 382 1 45924 557 42960 378 42884 10152 03746 86558 96519 62484 64160 28264 96838 77507 22210 26792 33520 65392 381 1 45161 553 06341 10 I 2.1071 71592 12 8.0283 23766 15 1 3.0587 91355 18 1.1653 99506 20 4.4401 72119 23 I 1.6917 05577 25 6.4453 90249 (6Q8.7538 56362 385 1 48225 570 66625 : 377 1 42129 535 82633 5.6477 (61)6.3754 78067 91673 47852 58684 46950 380 1 44400 548 72000 (61)8.2187 ( 1)1.9493 7.2431 4.4151 3.2806 376 41376 57376 17338 77109 06623 65690 70996 414209 3.2840 52418 72019 387 1 49769 579 60603 10 2.2430 75336 12 I 8.6807 01551 15 3.3594 31500 18 I 1.3000 99991 20 5.0313 66963 23 1 1.9471 46755 (25)7.5354 57941 (62)1.2736 88303 l)$.%;; 4:4353 3.2926 391 1 52881 597 76471 )b;;;: 48416 24406 1 46689 561 81887 1 50544 584 11072 ;62;1.3550 69013 ( lj1.9697 7.2936 4.4302 3.2943 33030 71560 10856 24265 392 1 53664 602 36208 9.1386 59392 04837 ;;f;; 65109 (62)1.7332 ( 1)1.9798 7.3186 4.4496 67559 98987 05586 11420 3.2994 02898 3.3010 88848 38586 40260 06958 98312 44931 90756 92233 97216 46'358 97510 1 47456 566 23104 (62)1.0566 ( !)1.9595 7.2684 4.4267 3.2075 94349 91794 82371 21679 03659 389 1 51321 588 63869 (25)7.9340 (62)1.4414 ( 1)1.9723 7.2998 4.4410 3.2960 611 69734 19629 08292 93662 67768 20622 394 1 55236 62984 86663 25)8.3515 (62)1.6302 ( 1':;;;; 414467 393 1 54449 606 98457 i 18 I 1.1232 25Y6.1149 23)1.6134 20)4.2570 15 2.9637 (61)7.7150 ( 1)1.9467 7.23b7 4.4122 3.2788 _._ 395 1 56025 616 29875 (62)1.8425 ; lil.9824 7.3248 4.4524 3.3027 397 _.. 1 56816 620 99136 10)2.4591 25786 58176 22760 29445 40634 71361 398 1 58404 630 44792 1 57609 625 70773 (62)1.9584 ( 1)1.9849 7.331Cl 4.4552 3.3044 35730 43324 36930 70277 50453 _.. 1 59201 635, 21199 2 i 9 10 24 l/2 l/3 62)2.0812 1)1.9874 7.3372 4.4580 3.3061 78965 60691 33921 94536 26138 I g (62)2.2114 87364 (62)2.3494 82217 i62i2.4957 07762 ( 1)1.9899 7.3434 4.4609 3.3077 74874 20462 13443 98433 ( 1)1 85885 96597 27013 67354 ; ljl.9949 7.3557 4.4665 3.3111 93734 62368 35273 32914 [ 41 (-(94 9924 7.3495 4: 4637 3.3094 (62)2.6506 ( 1)1.9974 7.3619 4.4693 3.3127 32365 98436 17821 38246 95131 40 ELEMENTARY Table 3.1 ANALYTICAL POWERS k 1 400 1 60000 640 00000 : 4 644 METHODS AND ROOTS 401 1 60801 81201 402 2 9" l/2 l/3 l/4 l/5 (62)2.8147 ( 1)2.0000 49767 00000 7.3680 4.4721 3.3144 (62)2.9885 ( 1)2,0024 62997 35955 54017 .-1 64025 664 30125 24 l/2 l/3 l/J l/5 (62)3.7924 56055 61180 36223 46344 99030 ( l/2 l/3 l/4 l/5 20 7.9849 23 3.2738 26 I 1.3422 (62)5.0911 ( 1)2.0248 7.4289 4.4998 3.3308 ( 1)2.0074 7.3064 4.4004 31295 85990 91129 61862 3.3194 24 l/3 l/4 l/5 (62)6.8101 ( 1)2.0371 7.4590 4.5134 3.3389 409 1 67281 1 66464 679 17312 684 (62)4.2684 06980 (62)4.5273 48373 (62)4.8013 ( 1)2 7.4107 0174 95055 24100 ( 1)2.0199 7.4168 00988 12948 14446 4.4943 3.3276 ( 1)2.0223 7.4229 4.4970 3.3292 411 1 68921 26531 65185 20)8.1420 414915 38251 694 3.3259 1 699 10 I 2.8813 13 1.1870 15 4.8908 18)2.0150 20 I 8.3019 23 3.4203 (62)5.3976 37632 ( l';.;;;; 4:5025 -M; 69814 3.3324 26 1.4091 7218 ( 1)2.0297 4.5053 7.4410 86236 719 73375 (62)s. 3.3341 416 1 73056 91296 74245 412 69744 34528 02554 96652 38207 25341 04405 84615 98461 06738 78313 18861 06108 06308 59539 30860 07026 06073 74842 14120 82211 36609 413 1 70569 704 44997 414 1 71396 10)2.9376 709 3.5727 1.4791 87127 40143 ( 62)6.064i 1)2.0322 7.4470 4.5080 3.3357 417 1 73889 725 11713 ( 62)6.4269 57944 58882 53630 20006 98328 1)2.0346 7.4530 4.5107 3.3373 34238 31426 23237 418 1 74724 730 17929 98995 39914 63768 37037 419 1 75561 34632 735 60059 47618 16532 43334 45063 03984 50201 13045 54879 35926 85215 47722 19812 49315 90153 72387 19056 54883 26518 48631 (62)7.2150 59801 (62)7.6430 25690 (62)8.0952 59269 ( 1)2.0396 7.4650 4.5162 3.3405 07805 22314 01729 55305 ( 1)2 57786 99115 13349 59799 ( 1)2.0445 7.4769 4.5216 3.3437 04830 66370 20097 61218 420 1 76400 740 88000 3.1116 96000 26)1.7080 (62)9.0778 ( 1)2.0493 7.4880 4.5270 3.3469 4.4832 3.3210 92707 3.7311 26)1.5521 l/2 75124 41792 74611 56568 1.3925 408 1 65649 19143 10)3.0528 il 10 2.9661 28 1.2309 15 8.7980 13 5.1084 2.1200 85306 ( 1)2.0099 10850 .-. 674 (62)3.5739 20630 44168 415 1 722% 714 15674 414777 3.3177 406 1 64836 23416 04703 4.4088 3.3243 25229 19344 65931 10945 45673 58841 20522 63008 pm; ;62;3.3676 404 1 63216 659 39264 10)2.6639 46266 1)2.0149 7.4047 410 1 68100 21000 61000 62010 04241 42739 10 2.6257 13 1 1.1585 15 4.7501 18 1.9475 1ym; 09590 (62)4.0236 ( 1)2.0124 7.3986 4.4860 3.3226 ;:8628; 00315 72718 62)3.1728 91940 28423 669 689 24 80393 98439 7.3741 4.4149 3.3161 403 1 62409 654 50827 t%: 85282 57283 26278 11364 1 10 24 nk 0420 7.4709 4:5u9 3.3421 421 1 77241 746 18461 422 78084 51448 91106 27047 40136 46338 72154 :9"::: (62)9.6110 38126 (63)1.0174 16609 ( 1)2.0518 7.4948 11226 28453 ( 1)2.0542 7 5oo7 63858 40668 4.5297 3.3485 11307 47155 4[(-46)4] +-y 415323 3.3501 96767 36405 +-;‘7] (62)8.5730 1)2.0469 7.4829 4.5243 3.3453 423 78929 06967 21 1.0249 762 73581 48949 24114 21992 59575 424 1 79776 25024 16974 62680 97814 I $2 I :: z (63)1.0768 83734 ( 1)2.0566 7.5066 4.5350 3.3517 96380 60749 81455 22644 (63)1.1396 ( 1)2.0591 7.5125 4.5371 3.3533 .[(-47141 73784 26028 71508 59390 05887 IGLEMENTARY ANALYTICAL POWERS AND 41 METHODS ROOTS Table d 3.1 k 425 1 80625 767 65625 : : I 773 10 3.2933 1'13 11.4029 115 I 5.9166 t 18 2.5460 I 21 1.0846 123 4.6204 I 26 1.9683 (63)1.2759 / 1)2.0639 7.5243 4.5431 3.3564 2 i 1'0 24 (63)1.2059 63938 w it: l/5 3.3548 86145 427 1 82329 778 54483 81476 08776 53858 68743 46847 51557 17963 21295 40370 76744 65204 01082 63431 63)1.3497 1)2.0663 7.5302 4.5457 3.3580 ::: l/4 l/5 (63)1.5967 ( 1)2.0736 7.5478 4.5537 3.3627 823 24 l/2 l/3 l/4 l/5 72093 44135 42314 28292 43107 (63)2.1073 ( 1)2.0856 7.5769 4.5669 3.3705 l/2 l/3 l/4 l/5 24 l/2 l/3 16666 65361 84852 08540 27318 I 63)2.2267 c 1)2.0880 7.5827 4.5695 3.3720 440 1 93600 84000 (63)2.7724 ( 1)2.0976 7.6059 4.5799 3.3782 1 881 10 3.9213 13 11.7450 15 1.7653 18 13.4555 21 1.5377 23 6.8428 26 I 3.0450 63)3.6361 1)2.1095 7.6346 4.5929 3.3858 18906 53949 88825 73502 05720 (63)1.7848 ( 1)2.0784 7.5595 4.5590 3.3658 436 1 90096 828 81856 1 89225 12875 851 24 ,63)1.6883 , 1)2.0760 7.5536 4.5563 3.3643 857 98685 97832 48212 64877 37758 71952 61302 86527 30941 75562 441 1 94481 66121 834 10 3.6469 13 1 1.5937 15 6.9644 18 1 3.0434 21 1.3299 23 5.8120 26 I 2.5398 (63)2.3526 ( 1)2.0904 7.5885 4.5721 3.3736 83700 60969 26299 14114 65436 437 1 90969 53453 15896 02247 78818 77243 99555 98057 86851 34640 54496 79338 48834 20969 863 10 3.8167 13 1.6869 15 I 1.4564 18 3.2957 21 1 1.4567 23 6.4387 26 1 2.8459 (63)3.0912 ( 1)2.1023 7.6174 4.5851 3.3813 442 1 95364 50888 09250 85488 75858 62329 26950 33117 20038 52385 79604 11603 71321 05834 53276 17696 04922 75651 40216 6312.9276 , 1)2.1000 7.6116 4.5825 3.3797 445 98025 21125 90063 18578 32671 73039 30002 98510 89837 37215 02311 06721 31864 83431 d4h 1 99804 893 14623 10 3.9923 63648 13 1.7845 86551 15 7.9771 01882 18 I 3.5657 64541 21 1.5938 96750 23 1 7.1247 18472 26)3.1847 49157 (63)4.0493 05610 ( 1)2.1142 37451 7.6460 27242 4.5980 83787 3.3889 21465, 14445 ._ 1 I [ 1 I3 (--6)4 4 44370 16087 22043 23998 09138 63)1.5099 1)2.0712 7.5419 4.5510 3.3611 433 1 87489 811 82737 63)1.8867 1)2.0808 7.5653 4.5616 3.3674 28946 65205 54712 50145 22267 817 (63)2.4852 ( 1)2.0928 7.5943 4.5747 3.3751 99040 44954 63318 62238 63549 93273 31518 86732 78463 77583 1 88%6 46504 (63)1.9941 ( 1)2.0832 7.5711 4.5642 3.3689 438 1 91844 840 21672 846 30189 66666 74278 81614 76223 439 1 92721 04519 26)2.6585 (63)2.6251 ( 1)2.0952 7.6001 4.5773 3.3767 443 ._ 1 96249 869 38307 10)3.8513 67000 875 52264 15920 32684 38502 71171 03314 444 1 97136 28384 26)2.9109 (63)3.2635 66867 43677 (63)3.4450 ( y.;;;; y;;; ( 1) ;. 16;;; ;;‘b;; 447 1 98916 887 16536 I 10 I 3.9567 57506 113 1.7647 13847 I 15 I 7.8706 23760 118 3.5102 98197 21 1.5655 92996 / 23 6.9825 44761 ,26 3.1142 14964 63)3.8373 95917 / 1)2.1118 71208 7.6403 21250 4.5955 09991 3.3874 03811, $ 97132 00000 62611 75695 (63)1.4277 ( 1)2.0688 7.5361 4.5484 3.3596 432 1 86624 806 21568 431 1 85761 800 62991 24 429 12523 ._- 430 428 1 83184 784 02752 415877 3.3828 62546 34454 16313 4:5903 3;3843 448 449 2 ooioi 899 15392 3.2567 (63)4.2724 ( 1)2.1166 7.6517 4.6006 3.3904 17891 04226 01049 24131 53268 36406, 49388 60316 2 01601 905 18849 26)3.3301 (63)4.5072 ( 1)2.1189 47041 55570 62010 7.6514 13148 4.6032 3.3919 18450 48644 C-47)4 I 42 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS nk k 450 2 02500 911 25000 : : 2 7 9” 10 24 l/2 l/3 l/4 l/5 (b3)4.7544 ( 1)2.1213 7.6630 4.6057 3.3934 1 50505 20344 94324 79352 58190 917 (63)5.0146 ( 1)2.1236 7.6687 4.6083 3.3949 455 2 07025 941 96375 : 4 451 2 03401 33851 946 08183 76058 66491 35988 b5055 456 2 07936 188lb 2 l 9 10 24 :g l/4 l/5 923 452 2 04304 45408 (b3)5.2883 ( 1): 77338 ;;ii 4:6108 3.3964 954 10 I 4.3617 13 1.9933 15 9.1095 1.9025 (63)6.1983 ( 1)2.1330 7.6913 4.6185 3.4009 13235 72901 71681 20218 65915 (b3)6.5336 ( 1)2.1354 7.6970 4. b210 3.4024 .__ 973 55778 59532 b3)6.8863 1)2.1377 7.7026 4. b235 3.4039 461 2 12521 979 72181 46” : 3 55383 15650 02263 2 11600 36000 10 (63)8.0572 70802 ( 1)2.1447 4.6311 61059 42629 56507 3.4084 w 07924 7.7194 l/4 l/5 (63)8.4883 ( 1)2.1470 7.1250 4.6336 3.4098 465 2 16225 : 1005 : 5 44625 29103 91055 32380 4.6361 3.4113 71390 88554 466 1011 ( 1’;. yo’ 08317 b9249 4:b134 3.3979 457 2 08849 43993 90480 38249 55800 96; 21619 45396 55833 24618 87171 50532 (63)7.2572 ( 1)2.1400 7.7082 4.6261 3.4054 2 17156 1018 94696 (63)5.8795 01000 ( 1) ;. y5”; y; 416159 3.3994 458 09764 71912 93570 42855 :'4::: 82342 57127 03764 39774 93456 38770 14413 38923 16664 967 00476 69669 459 2 lob81 02579 ( b3)7.6472 1)2.1424 1.7130 4.6286 3.4069 35292 28529 44772 37519 24718 464 2 15296 998 97344 73356 38903 (63)9.4176 18526 14052 82186 66616 ( 1)2.1517 1.7361 4.6386 467 2 18089 47563 37619 19665 85719 36534 70784 2 Oblib 935 463 14369 9922 52847 06816 462 (63)8.9414 ( 1)2.1494 7.7306 453 2 05209 59677 73368 lb236 01548 00201 13891 (63)5.5764 ;;y; 13444 11128 34114 95360 54565 59409 b2447 98505 63510 : 7" 8 9 10 24 924 3.4128 1025 27630 72096 76381 24642 93909 76852 43479 87677 88909 42121 (63)9.9181 ( 1)2.1540 7.7417 4.6411 3.4143 468 2 19024 03232 69666 65923 53281 91574 15079 469 2 19961 1031 61709 7" 9" 10 24 l/2 :;: (b4)1.0444 09634 ( 1)2.1563 7.7473 85865 4.6436 l/5 3.4157 10895 90198 85500 64)1.0996 1)2.1587 60547 84795 3.4172 53393 470 10 I 13 16 18 21 I 24 1038 4.8796 2.2934 1.0779 5.0662 2.3811 1.1191 26) 5.2599 24 ::: 69046 03314 7.7520 4. b461 24259 18278 7.7504 02264 75300 4.6486 3.4187 471 2 20900 23000 81000 50070 21533 31205 1044 2 21841 87111 1051 18768 472 2 22784 54048 (64) 1.2187 ( 1)2.1633 7.7639 4.6511 3.4201 10278 (b4)1.2827 30765 ( 36077 61968 81635 1)2.1656 7.7694 4. b53b 3.4216 .._ 2 23729 23817 44575 42003 474 2 24676 471 1058 68318 40783 62012 1064 96424 286b6 30473 13224 (64)1.3500 46075 (b4)1.4206 98007 ( 1) ;. ;;;; y;; ( 1)2.1702 53441 90361 97902 55283 4:b561 3.4230 1.1577 ( 1)2.1610 (64) 23215 99883 7.7804 4.6585 3.4245 (64)1.4948 ( 1)2.1725 85630 56098 7.7059 92032 3.4260 08213 4.6610 68652 (b4)1.5727 ( 1)2.1748 7.7914 4.6635 3.4274 77826 56317 87536 (64)1.6545 ( 1)2.1771 7.7969 51159 54106 74500 35480 4.6659 98399 50603 3.4289 06701 ELEMENTARY ANALYTICAL POWERS AND 43 METHODS ROOTS rzk Table 3.1 k 475 2 25625 71875 : 1071 : 2 .z 1: 24 l/2 l/3 l/4 l/5 I! 13 I 10 lb 18 21 24 2.4180 5.0906 1.1485 5.4557 2.5914 1.2309 26 5.8470 (64) 1.7403 ( 1)2.1794 7.8024 86060 55878 40422 90207 49472 53153 57424 3.4303 2 30400 92000 ;lO I 5.3084 l/2 l/3 l/4 l/5 lb000 39680 2.5480 1.2230 5.8706 2.8179 1.3526 59046 83423 28043 05461 ObZll 26 6.4925 (b4)2.2376 ( 31322 1)2.1908 1.0291 4. b8Ob 64) 1.8304 1)2.1817 7.8079 4.6709 3.4317 1112 10)5.3527 13 I 2.5746 lb 1.2384 1.3 5.9568 87912 42423 25322 12569 95422 l/3 l/4 l/5 (b4)2.8694 ( 1)2.2022 7.8568 (b4)2.0242 75033 3.4332 36143 ( 1)2.1863 7.8188 4.6758 21111 45511 11278 3.4346 74449 6. b290 1119 482 2 32324 80168 79188 41895 b4)2.3522 3.4389 41094 1147 1126 10 I 5.4423 13 2. b28b lb 1.2696 18 6.1323 21 I 2.9619 24 1.4306 57971 (b4)2.4724 7.8405 4.6855 3.4404 14973 2 3b196 91256 1155 94846 b2762 03713 487 2 37169 01303 4.6928 3.4446 1176 10)5.7648 13 2.8247 lb 1.3841 '18 6.1822 21 I 3.3232 4: b952 3.4460 94362 ( y. 2.9718 18 7.2817 21 3.6044 35163 85081 4:7072 3.4531 49066 25063 43906 62733 60531 l/2 l/3 l/4 l/5 3.4418 1162 10 1 5.6712 41683 64874 4-;)3] 12045 6.9783 3.4333 1.6892 8.3109 y;; ( 1';. ii,": 4: 7096 3.4545 2 42064 95488 98010 73021 73526 97749 71692 18873 56854 72689 ;;;"7; 10 13 lb 18 I 21 24 45383 19620 7.8514 4. b904 3.4432 30083 488 2 38144 14272 489 2 39121 1169 30169 73159 75702 09424 14999 61719 61191 75643 b4)3.4947 21879 72203 1)2.2113 34439 68425 94366 7.8783 4.7024 3.4503 76812 26700 l ZO!i 89849 58895 91354 97638 71635 60331 7.8997 91695 4.7120 3.4559 10) 13 lb I 18 21 53784 81640 1)2.2203 b4)4.4611 1)2.222B b99bO 1235 82790 39037 494 ;! 44036 2 43049 23157 32162 84825 62231 24411 15760 54092 56474 26 I 8.4814 64)4.2493 78653 23936 87226 84064 84096 7.9051 4.7144 65384 3.457'3 498 2 48004 05992 1242 49467 11077 29393 57633 66263 499 :! 49001 51499 3.0629 1.5253 ( 1’;. ;;y; y.9;; ( 1)2.2315 417215 3.4615 98967 55329 .:[(,5] 7.5963 3.7829 98402 10) 6.200'1 98004 13)3.0938 49800 74750 73006 57570 86070 18 7.1031 21 3.8441 79067 85754 24 1.9182 26 I 9.5720 6.1505 22124 61227 +p6)1] 92097 I_ 1198 5.9072 2.9122 1.4357 7.0782 3.4895 1.7203 8”;;;; ( 1’: 59893 67133 12960 21 3.2163 24 1.5695 26 7.6594 b4)3.3271 1)2.2090 7.8729 4.7000 3.4489 ;;;; 59546 1.9104 4.7168 3.4561 67488 46397 00000 417192 3.4601 1)2.2248 79904 92362 64)5.4138 (b4)4.9154 484 ; ! 34256 1133 ( 1)2.2000 55098 56201 06649 54870 10350 (b4)2.7307 (b4)5.1588 26)8.8318 64)4.6830 06863 94186 26098 13365 91145 61115 39113 17000 08320 15513 71463 13374 95451 50361 1)2.1977 7.8460 4.6879 46016 122; (b4)2.1283 ( 1)2.1886 7.824;! 4. b78:! 3.4361. 92764 .._ 2 45025 87375 02524 50214 07962 17248 3609 b4)2.5985 .._ (64)4.0472 83697 56794 / 18 1 2.2643 lb3 1.2070 10 5.5216 21 5.7856 2.7713 483 2 33289 78587 75752 497 1190 5.6594 2.8828 1.4183 71639 91376 '8;;; 24 I 1.7842 24 3.4475 68562 ( 1)2.2135 lb I 1.4710 4.6976 7.8676 40045 l/2 10 6.0037 53740 95065 63943 14959 30728 93057 (b4)3.8543 13 07649 94384 36822 1212 42798 ( 1)2.2068 04856 (b4)3.6703 7.8837 4.7048 3.4517 (64)3.lb73 491 41081 70771 2 40100 49000 01000 52490 28720 24 l/3 l/4 l/5 82996 ii-;; b4)3.0148 ( l';;?;; ; ! 29441 02239 26 6.9098 486 2 352% 84125 70250 71555 28008 36620 75750 99311 80708 55578 19966 01440 32967 89232 63849 419 1099 93826 96049 45935 26 b.2270 13 2.1675 lb) 1.3505 18 I 6.5908 l/2 478 2 28484 15352 1.9250 ( Q2.1840 7.8133 4.6733 (64) 481 2 31361 84641 91232 92583 27132 34506 1.3781 43855 1140 1092 10 I 5.2204 13 2.4953 lb 1.1927 18 5.7015 21 I 2.7253 24 1.3027 49036 90230 35282 94639 3.4375 477 2 27529 1085 31333 10 1 5.1769 44584 13 2.4694 02567 lb 1.1779 05024 18 5.6186 06966 21 I 2. b800 75523 6.0979 52278 1105 24 476 2 265% 50176 81079 65430 b4063 60126 4.6604 13 :lb 18 '21 '24 I 1078 48691 16)1.5438 25162 91360 7.9264 4.7239 47190 60970 (64)5.6808 12221 3.4629 43500 08444 1 47029 ( 1)2.2338 7.9317 4.7263 3.464'3 30790 10391 4191b q-37)3] 36816 . 44 ELEMENTARY Tahlc 24 l/2 l/3 l/4 l/5 3.1 POWERS 1250 10 6.2500 13 I 3.1250 16)1.5625 18)7.8125 21)3.9062 24)1.9531 26)9.7656 64)5.9604 1)2.2360 7.9370 4.7287 3.4657 500 2 50000 00000 00000 00000 00000 00000 50000 25000 25000 64478 67977 05260 08045 24216 505 2 55025 1287 87625 10)6.5037 75063 13)3.2844 06407 16 1.6586 25235 18 1 8.3760 57438 21)4.2299 09006 24 l/2 l/3 l/4 l/5 64)7.5682 1)2.2472 7.9633 4.7404 3.4726 08268 20505 74242 85740 28104 510 2 60100 1326 51000 (10)6.7652 01000 52510 13)3.4502 24 (64)9.5&?70 3.4794 1365 24 l/2 l/3 l/4 l/5 (65)1.2116 ( 1)2.2693 8.0155 4.7637 3.4862 1406 24 l/2 l/3 l/4 l/5 ANALYTICAL (65)1.5278 ( 1)2.2803 8.0414 4.7753 3.4930 33090 77522 515 2 65225 90875 39706 61144 94581 81212 73428 520 2 70400 08000 48342 50850 51517 01928 16754 +;)a] 2 AHD 126; 1257 METHODS ROOTS rth- 502 2 52004 06008 1272 503 2 53009 63527 (10)6.4013 64)6.2532 1)2.2383 7.9422 4.7310 3.4671 44659 02929 93073 70628 09398 506 2 56036 1295 54216 (10)6.5554 43330 (13)3.3170 54325 (16 1.6784 29488 (18 1 8.4928 53211 (21)4.2973 83725 24)2.1744 76165 27)1.1002 84939 (64)7.9361 96349 ( 1)2.2494 44376 7.9606 27129 30775 4.7428 3.4740 02314 511 2 61121 32831 1334 6.8184 17664 3.4842 11426 32039 16)1.7804 9.0980 07719 81944 4.6490 (65)1.0048 50848 3.4808 1373 (65)1.2693 ( Q2.2715 8.0207 4.7660 3.4076 1414 10 I 7.3680 13 3.8387 16 I 1.9999 19 1.0419 21 5.4287 40954 516 2 66256 88096 83471 63338 79314 92045 26271 521 2 71441 20761 21648 39279 83164 91229 74301 1.4735 (65)1.5999 ( 1)2.2825 91925 46126 42442 8.0466 02993 4.7775 3.4943 96092 59190 .:[(-,P] (27)1.0163 (64)6.5597 ( 1)2.2405 7.9475 4.7334 3.4684 (64)6.8806 ( 1)2.2427 7.9528 4.7357 3.4698 35678 79050 35650 73855 29676 92370 507 2 57049 1303 23843 10)6.6074 18840 64)8.3212 1)2.2516 7.9738 4.7451 3.4753 1342 27)1.2379 65)1.0531 1)2.2627 8.0000 4.7568 3.4822 1381 65)1.3297 1)2.2737 8.0259 4.7683 3.4889 1422 (65)1.6752 ( 1)2.2847 8.0517 4.7790 3.4956 (64)8.7242 ( 1)2.2538 7.9791 4.7475 3.4767 512 2 62144 17728 16 1.8226 18 9.3502 21 4.7966 24 2.4606 ii27 1.2623 (65)1.1036 ( 1)2.2649 8.0052 4.7591 3.4835 517 2 67289 88413 53822 14108 59837 06497 32173 12886 50331 04946 49431 61427 522 2 72484 36648 1430 (65)1.7540 ( 1)2.2869 8.0568 4.7821 3.4970 ni[(-;)S] 16064 (64)7.2166 ( 1)2.2449 7.9581 4.7381 3.4712 04000 94432 14416 37221 51715 509 2 59081 1318 72229 (10)6.7122 96456 523 2 73529 55667 91340 69982 57150 93433 04452 72252 524 __. 2 74576 1438 (65)1.8363 ( 1)2.2891 44200 19325 18034 56810 03133 67011 18483 519 2 69361 1397 98359 10 7.2555 34832 13 3.7656 22576 16 1.9543 58118 19 1.0143 11863 21 I 5.2642 78570 1.4179 (65)1.4588 ( 1)2.2781 8.0362 4.7730 3.4916 81704 61335 28718 03654 25675 06897 02835 44383 45086 11950 514 2 64196 1357 96744 65)1.1564 1)2.2671 8.0104 4.7614 3.4849 518 2 68324 1389 91832 (65)1.3928 ( 1)2.2759 8.0311 4.7707 3.4903 3.2520 (64)9.1459 ( 1)2.2561 7.9843 4.7498 3.4781 69942 85534 12176 10436 44229 513 2 63169 1350 05697 6.9257 92256 40039 22917 41700 00000 28460 02253 98008 31932 47881 86957 99566 84448 66149 47628 85203 73139 2 58064 1310 96512 97020 66050 73099 72336 74353 59294 63400 57353 99522 77017 55408 17824 30669 04628 86203 8.0620 17979 74532 37889 4.7844 3.4983 58829 74167 1 $[‘-;‘“I ELEMENTARY ANALYTICAL POWERS AND 45 METHODS nk ROOTS Table 3.1 k : 3 4 5 6 7 8 9 10 24 1447 10)7.5969 13)3.9883 16 2.0938 19 11.0992 21)5.7713 525 2 75625 03125 14063 79883 99438 97205 10327 (65)1.9223 ( 1)2.2912 8.0671 4.7867 3.4997 l/2 l/3 l/4 l/5 09365 87847 43230 39859 08406 526 2 76676 1455 110 7.6549 113 1 4.0265 I16 2.1179 I19 1.1140 I 21 5.8598 124 I 3.0822 127)1.6212 , 65)2.0121 / 1)2.2934 8.0722 4.7890 3.5010 530 2 80900 1488 77000 10)7.8904 81000 : : 2 7 9" 10 24 l/2 l/3 l/4 l/5 27)1.7488 (65)2.4133 ( 1)2.3021 8.0926 4.1980 3.5063 74704 53110 72887 12335 96379 49267 535 2 86225 1531 30375 (10)8.1924 75063 24 l/2 l/3 l/4 l/5 (65)3.0233 ( 1)2.3130 8.1180 4.8093 3.5129 1574 66304 06701 41379 72829 40196 :65)2.5250 : 1y;; 4:8003 3.5076 31576 60898 09432 43961 385'24 42634 17226 77821 38448 68988 61977 17632 40614 531 81961 21291 00552 56493 46498 14290 68R82 22076 53022 41417 43724 58868 58033 71420 1463 10)7.7133 (65)2.1059 ( 1)2.2956 8.0113 4.7912 3.5023 94141 24056 37598 02002 (65)2.6416 ( 1)2.3065 8.1028 4.8026 ' 3.5089 ! j36 540 2 91600 64000 1583 57'189 49669 67381 96201 18626 52463 82534 48057 74241 92160 70797 532 83024 68768 58458 57499 95390 13716 12519 39019 16494 91583 (65)3.3066 ( 1)2.3173 8.1281 4.8138 3.5155 541 2 92681 40421 1592 09101 26045 44739 61283 62774 542 2 93764 20088 28710 I 13 I 4.6773 16 2.5351 10 8.6297 19)1.3740 24 l/2 l/3 :/,: 65) 3.1796 1)2.3237 8.1432 4.8205 3.5194 : : 5 1618 38253 (65)3.9512 48669 II 21 1.4472 21 2.0363 24 4.1877 (65)4.1303 90008 52850 70514 82029 ( 1';;:;; y; ( 1y;; 545 2 97025 78625 4:8228 3.5207 00711 84516 546 2 98116 1621 71336 1471 10 7.7720 13 4.1036 16)2.1667 19)1.1440 21 6.0404 24 I 3.1893 27)1.6839 (65)2.2040 ( 1)2.2978 8.0824 4.7935 3.5036 4:8250 3.5220 1636 10)8.9526 12993 03642 26174 21864 94250 25684 12169 89345 93862 27819 85199 528 2 78784 97952 51866 43385 23707 30117 79020 72923 88903 12944 25059 80041 63454 98962 (10)7.8310 (65)2.7634 ( 1)2.3086 8.1079 4.8048 3.5103 1601 10 8.6935 13 4.7206 16 2.5632 19 1.3918 21 I 7.5578 2.2284 65)4.3171 1)2.3302 8.1583 4.8272 3.5233 547 2 99209 67323 02568 1645 83632 10416 50910 55831 07963 00000 79399 31523 25117 534 58943 79276 12808 71774 09762 98937 82701 87014 00810 71134 529 2 79841 35889 98528 2 85iG 1522 73304 10 8.1313 94434 13 14.3421 64628 16 2.3187 15911 19 j 1.2381 94297 21 6.6119 57543 24 1 3.5307 85328 27)1.8854 39365 65)2.8906 14446 1)2.3108 44002 8.1129 80255 4.8071 23882 3.5116 25964 538 2 89444 1557 20872 (65)3.4575 1)2.3194 8.1331 4.8161 3.5168 1480 19 1.1592 21 6.1326 I 24 I 3.2441 (2711.7161 (65)2.3064 ( 1)2.3000 8.0875 4.7958 3.5050 533 2 84089 1514 19437 10)8.0706 55992 13)4.3016 59644 16 2.2921 84590 19 1.2220 54187 21 I 6.5135 48814 537 2 88369 1548 54153 2 '37296 1539 90656 8.2538 99162 (27)1.9572 (65)3.1619 ( 1)2.3151 8.1230 4.8116 3.5142 527 2 77729 63183 39744 539 2 90521 1565' 90819 10)8.4402 45144 (65)3.6151 ( 1)2.3216 8.1382 4.8183 3.5181 543 2 94849 03007 93280 21151 97285 70426 56412 26405 37789 36040 05107 51847 83903 1609 (65)4.5120 ( 1)2.3323 8.1633 4.8294 3.5246 548 3 00304 66592 1654 94612 39982 69477 26138 48871 (65)5.6199 f 112.34311 ' -'8; i882 4.8405 3.531:l 83652 37353 23044 31217 77550 544 2' 95936 89184 46770 80758 10204 72806 80696 549 3 01401 69149 7" : 10 24 l/2 l/3 (65)4.7153 ( 1)2.3345 8.1683 4.8316 3.5259 73024 23506 09170 90704 75582 +;)3] (65)4.9274 ( 1)2.3366 8.1733 4.8339 3.5272 63602 64289 02026 05553 68570 (65)5.1486 ( 1)2.3388 8.1782 4.8361 3.5285 ;[(-47181 79188 03113 88788 17361 59664 (65)5.3793 ( 1)2.3409 8.1832 4.8383 3.5298 +37)41 $-?3] 99369 74903 44110 31895 36198 46 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS nk k 550 3 02500 1663 (10)9.1506 13)5.0328 551 3 03601 1672 84151 10 19.2173 56720 13 5.0787 63553 16 2.7983 98718 19 1.5419 17693 21 I 8.4959 66491 24 1 4.6812 17536 75000 25000 43750 16)2.7680 64063 27 24 l/2 l/3 l/4 l/5 (65)5.8708 98173 ( 1)2.3452 07880 8.1932 12706 4.8427 34641 3.5324'21650 1709 24 l/2 l/3 l/4 l/5 (65)7.2951 ( 1)2.3558 8.2179 4.8537 3.5388 (10)9.8344 1756 m l/3 l/4 l/5 l/3 l/4 l/5 l/3 l/4 l/5 (65)7.6171 ( 1)2.3579 8.2228 4.8558 3.5400 560 3 13600 16000 96000 1765 (66)1.1198 ( 1)2.3769 8.2670 4.8754 3.5514 ( 65)9.4429 l);.:t%: 4:8667 3.5464 565 3 19225 62125 57461 72865 29409 20869 82586 556 3 09136 79616 06650 17697 60240 68693 93672 65225 98519 88409 95340 561 3 14721 58481 24 27 1 553 3 05809 12377 14448 08690 99605 24482 30384 4.7582 2.6265 98682 80873 24 4.8364 27)2.6745 (65)6.4052 ( 1)2.3494 8.2031 4.8471 3.5349 76258 68025 31859 31136 86956 (65)6,6896 1)2.3515 ( 8.2080 4.8493 3.5362 46227 95203 82453 24905 66821 1700 3 06916 31464 44203 53644 1728 10)9.6254 (65)7.9528 ( 1)2.3600 8.2278 4.8580 3.5413 1775 557 3 10249 08693 44200 84664 84744 25361 70341 67840 562 3 15844 04328 44725 (66)1.3835 ( 1)2.3874 8.2913 4.8861 3.5577 55344 67277 44342 71586 46263 1 q-y] 71309 7";;;; (65)9.8553 ( 1)2.3706 8.2523 39138 71525 53918 68801 39637 4.8689 3.5477 36145 03064 566 3 20356 1813 21496 11 1.0262 79667 13 5.8087 42917 16 3.2877 48491 19 1.8608 65646 22 1.0532 49956 24 5.9613 94749 27 3.3741 49428 (66)1.1684 07534 ( 1)2.3790 8.2719 4.8775 3.5527 570 3 24900 93000 00100 3.4296 w 1718 10 9.5565 13 ! 5.3134 16)2.9542 1.6425 54891 67858 31913 70600 98558 74407 1851 11)1.0556 24 38919 75283 34384 05234 1691 10 9.3519 13 1 5.1716 16 2.8598 19 1.5815 21 8.7458 558 3 11364 1737 41112 10 9.6947 54050 13 1 5.4096 72760 16)3.0185 97400 19)1.6843 77349 21)9.3988 25608 65)8.3027 1)2.3622 8.2327 4.8602 3.5426 1784 27311 02362 46311 49337 38514 (65)6.9860 ( 1)2.3537 8.2130 4.8515 3.5375 1746 (10)9,7644 1.7056 9.5344 24)5.3297 27)2.9793 (65)8.6672 ( 1)2.3643 8.2376 4.8624 3.5439 563 3 16969 53547 1794 92851 20459 27082 15700 44836 559 3 12481 76879 37536 21474 24040 43038 26358 91224 18084 61384 25407 07368 564 3 18096 06144 %"6 94850 1803 l/2 11516 ( 1)2.3473 8.1981 4.8449 3.5337 05803 43798 65765 03532 21007 3.0330 (65)9.0471 ( 132.3664 8.2425 4.8645 3.5451 83922 (65)6.1325 555 3 08025 53875 I ::I :: F% 1.7270 24 2.5793 552 3 04704 1681 96608 10)9.2844 52762 13 I 5.1250 17924 16 2.8290 09894 19 1.5616 13462 21 I 8.6201 'I6308 1861 11 l.Ob30 13 6.0698 16 3.4659 19 r 1.9790 22 1.1300 24 6.4524 27 3.6843 (66)1.4430 ( 1)2.3895 8.2961 4.8883 3.5589 75451 03838 76704 38859 571 3 26041 69411 27337 86093 04959 31732 27119 54848 51718 00887 b0629 90248 13236 93720 .;[,,)8] 1822 3.4342 (66)1.2189 ( 1)2.3811 8.2767 4.8797 3.5539 187: I 22 I 1.1459 27 3.7493 24 6.5548 (66)1.5048 ( 1)2.3916 8.3010 4.8904 3.5602 567 3 21489 84263 39578 71112 76180 72529 29685 93358 572 27184 49248 93699 23956 84103 20907 56759 72660 87161 89774 52149 30501 52074 39430 ;[‘-p”] 27)3.1995 66)1.0284 1)2.3727 8.2572 4.8711 3.5489 1832 (6631.2716 ( 1)2.3832 8.2816 4.8818 3.5552 13511 93323 62104 63270 00598 64695 568 3 22624 50432 27927 75058 35499 79820 46087 66)1.0732 1)2.3748 8.2621 4.8732 3.5502 44065 68417 49226 62170 24533 1842 11)1.0482 13 5.9643 16 I 3.3937 19 1.9310 22 j 1.0987 569 3 23761 20009 11851 25433 01172 15967 48085 3.5573 (66)1.3264 ( 1)2.3853 8.2864 4.8840 3.5564 573 3 28329 1881 32517 27)3.8154 66)1.5693 ( 1)2.3937 8.3058 4.8925 3.5614 53980 17896 41841 65115 88109 8340; 17788 60719 72088 92764 27117 97054 574 1 79476 1891 19224 (66)1.6363 ( 1)2.3958 8.3106 4.8947 3.5627 1$[‘-321 84728 29710 94107 21351 25633 ELEMENTARY ANALYTICAL POWERS 47 METHODS AND ROOTS nk Table: 3.1 k : 3 1901 575 3 30625 09375 1911 576 3 31716 02976 11 13 16 I 19 5" 6 i 1: 24 l/2 l/3 l/4 l/5 (66)1.7061 ( 1)2.3979 8.3155 4.8968 3.5639 93459 15762 17494 51807 66137 l66)1.7788 I 1)2.4000 8.3203 4.8989 3.5652 : 51122 00000 35292 79486 04916 1961 192: 1.1084 6.3955 3.6902 2.1292 66)1.8544 1)2.4020 8.3251 4.9011 3.5664 577 32929 00033 17190 67189 42268 69789 88668 56614 27966 68735 82430 47517 04396 41976 582 3 30724 1971 37368 3 375'61 225141 : 193: (66)1.9331 ( 1)2.4041 8.3299 4.9032 3.5676 570 34084 00552 21191 80481 82318 36180 26512 99239 72960 61432 63056 54185 26546 17321 583 3 39889 1981 55287 1.1552 45323 579 3 35241 04539 65281 1941 11)1.1238 ;66;2.0150 ( 1)2.4062 8.3347 4.9053 3.5689 48620 41883 55313 45944 10958 584 3 41;56 1991 76704 6' ; 9 10 24 l/2 :g l/5 : 3 (66)2.1002 ( 1)2.4083 8.3395 4.9074 3.5701 2002 1.1711 54121 18916 50915 62599 42892 (66)2.1889 ( 1)2.4103 8.3443 4.9095 3.5713 585 3 42225 01625 79506 2012 06331 94:159 41009 76!j18 73'L27 66)2.2811 1)2.4124 8.3491 4.9116 3.5726 586 3 43396 30'356 2022 38380 67616 25609 87710 01670 587 3 44569 62003 5" ; : 10 24 l/2 : ;: l/5 1 (66)2.5807 ( 1)2.4186 8.3634 4.9180 3.5762 2053 (66)3.1655 ( 1)2.4289 8.3672 4.9284 3.5823 2106 II (11 1.2533 13 7.4573 16 4.4371 19 2.6400 24 ::: l/4 l/5 I ::I k :'42 (27j5.5612 (66)3.8762 ( 1)2.4392 8.4108 4.9388 3.5884 19397 77324 46607 05007 77194 590 3 48100 79000 43453 91560 06527 80050 69695 595 3 54025 44875 37006 55187 26336 90170 :z 14639 08928 62184 32585 80725 21030 (66)2.6887 ( 1)2.4207 8.3682 4.9201 3.5774 2064 (66)3.2968 ( 1)2.4310 8.3919 4.9305 3.5835 2117 (66)4.0356 ( 1)2.4413 8.4155 4.9409 3.5896 02707 43687 09391 05372 99018 27)4.8571 (66)2.8010 ( 1)2.4228 8.3729 4.9222 3.5787 591 3 49281 25,071 52680 49156 42387 67063 83235 596 3 5'5216 089736 19703 11123 41899 62581 26411 2074 (66)3.4333 ( l)?;;;; 4:9326 3.5847 2121 11 1.2702 13 7.5835 '16 I 4.5273 '19 2.7028 ;22 I 1.6135 24 9.6331 27 5.7509 (66)4.2013 ( 1)2.4433 8.4202 4.9490 3.5908 44372 08521 08288 66760 03051 19175 592 3 50464 74688 72793 '3;;;; 51429 95134 3 56409 76113 73753 34304 69980 39878 95407 64580 99254 02448 58345 45948 33830 30176 (66)2.3770 ( 1)2.4145 8.3539 4.9137 3.5738 88299 39294 04732 96184 28526 2032 'ii i.1953 :13 7.0288 16 '4.1329 '19 I 2.4301 22 1.4289 '24 I 8.4022 27 4.9405 66)2.9178 1)2.4248 0.3777 4.9242 3.5799 588 3 45744 97472 89135 88116 86212 95893 55185 56487 26815 02055 71131 18728 98052 37670 2085 11 1.2365 13 7.3328 16 I 4.3483 19 2.5785 22 1.5291 24 I 9.0675 27)5.3770 (66)3.5753 ( 1)2.4351 593 3 51649 27857 70192 61239 06715 93322 05840 97630 85394 01250 '59132 4.9347 8.4013 3.5860 213'3 33156 98104 05396 598 3 57604 47192 4.6145 (66j2.4768 ( 1)2.4166 8.3506 4.9159 3.5750 40597 99188 09195 70393 01946 53698 589 3 46921 2043 36469 (66)3. 039:! ( 1)2.4269 6.3824 4. 926'3 3.5811 2095 54545 32220 65312 90382 54508 594 3 52836 84584 7.3948 98628 27)5.4684 (66)3.7228 ( 1)2.4372 8.4061 4.9368 3.5872 52572 42640 11521 17992 12252 14026 2149 599 3 58801 21799 . (66)4.3734 ( 1)2.4454 0.4249 4.9451 3.5920 92798 03852 44747 02478 32329 (27)5.9465 (66)4.5524 ( 1)2.4474 8.42'?6 4.9471 3.5932 93118 34829 47650 38310 68534 32875 48 ELEMENTARY Table ANALYTICAL POWERS 3.1 2160 (66)4.7383 ( 1)2.4494 8.4343 4.9492 3.5944 2214 (6635.7826 ( 1)2.4596 8.4576 4.9595 3.6004 2269 (66)7.0455 ( 1)2.4698 8.4809 4.9697 3.6063 3 60000 00000 81338 89743 26653 32004 31819 77757 74775 90558 10838 02669 610 3 72100 81000 68477 17807 26088 26156 34171 33286 19354 34993 70868 26906 620 3 84400 2385 28000 (67)1.0408 ( 1)2.4899 8.5270 4.9899 3.6180 66)4.9315 1)2.4515 8.4390 4.9512 3.5956 605 3 66025 45125 615 3 78225 2326 OLi375 (66)8.5704 ( 1)2.4799 8.5040 4.9798 3.6122 2170 79722 79920 18983 69859 81437 (66)6.0164 ( 1)2.4617 8.4623 4.9615 3.6015 2280 (66)7.3280 ( 1)2.4718 8.4855 4.9717 3.6075 601 3 61201 81801 61624 16360 50833 82950 41953 87314 53756 94142 '0% 92896 29165 606 67236 45016 22797 54150 28415 92819 83448 82770 27505 86963 06725 47078 58954 92098 611 3 73321 99131 60494 41419 57944 61679 15802 616 3 79456 2337 44896 (27)7.8669 (6638.9112 ( 1)2.4819 8.5086 4.9819 3.6134 63254 18488 34729 41730 01975 00850 621 3 85641 2394 83061 (67)1.0819 ( 1)2.4919 8.5316 4.9919 3.6192 28109 87159 00940 80728 47808 METHODS AND ROOTS 218: (66)5.1323 ( 1)2.4535 8.4436 4.9533 3.5968 223: rzk 602 624 4 672 i! 8 66592 66885 93065 ::z: 08947 21860 44384 68829 87734 51218 24918 2192 (66)5.3409 ( 1)2.4556 8.4483 4.9554 3.5980 607 68449 48543 46656 08202 67079 --. 603 3 63609 56227 12849 05832 60500 06978 19083 2203 (66)5.5575 ( 1)2.4576 8.4530 4.9574 3.5992 608 69664 55712 2258 3 64816 48064 90288 41145 28104 60182 11665 609 3 70881 66529 '3:::: (66)6.2593 ( 1)2,4637 8.4670 4.9636 3.6027 %i 40623 36999 00076 04536 79959 2292 ;ll 1.4028 13 8.5853 '16 5.2542 '19 1 3.2155 '22 1.9679 '25 I 1.2043 '27 7.3707 (66)7.6213 ( "UN',; 612 3 74544 20928 32079 32326 23383 84711 37843 77960 93114 89047 0~;; 4;9737 3.6086 94704 95885 (66)6.5115 3.6039 2406 (67)1.1245 ( 1)2.4939 8.5361 4.9939 3.6204 68280 48470 43484 22621 73271 622 3 86884 41848 25305 92783 77980 89170 12677 (66)6.7735 66255 (66)7.9259 ( 1)2.4758 8.4948 4.9758 3.6098 z:::: 46261 45058 10121 05604 20352 51097 83681 06516 25239 74428 66)9.6321 1)2.4859 8.5178 4.9859 3.6157 618 81924 29032 59418 15202 70395 59704 87297 10750 48432 53659 60579 40269 40813 44173 2418 11 1.5064 13 1 9.3851 16 5.8469 19 1 3.6426 2.2693 8.8080 (67)1.1687 ( 1)2,4959 8.5407 4.9959 3.6215 231: (66)8.2421 ( 1)2.4779 8.4994 4.9778 3.6110 614 76996 75544 59840 35419 92747 68947 79533 67433 42041 57465 02339 23260 53291 51433 I 22 I 2.1553 19 3.4820 16 5.6252 86668 76757 46313 (67)1.0013 ( 1)2.4879 8.5224 4.9879 3.6169 26192 71061 32097 56556 13560 242: (67)1.2145 ( y;‘: 1 50991 619 3 83161 2371 76659 623 3 88129 04367 41206 28716 35190 40623 65108 64101 27115 96795 50116 95191 76049 29447 3.6051 613 75769 46397 617 3 80689 2348 85113 66)9.2649 1)2.4839 8.5132 4.9839 3.6145 72833 4:9979 3.6227 624 89376 70624 36694 z; 66786 70474 70376 71145 91262 99199 17363 98799 37928 1 LEMENTARY E ANALYTICAL POWERS 625 2441 24 l/2 l/3 $2 (67)1.2621 ( 1)2.5000 8.5498 5.0000 3.6238 3 90625 40625 77448 00000 79733 00000 98318 630 3 96900 2500 47000 I 16 I 1.5752 13 6.2523 11 9.9243 m l/3 l/4 l/S 02919 81416 57803 80639 75339 80080 18882 70139 8.5726 5.0099 3.6296 2560 24 l/2 l/3 l/4 l/5 14376 67994 81641 14307 14356 76187 80893 18391 47419 99201 37239 98801 3.6250 2512 57224 631 3 98161 39591 (67)1.8474 ( 1)2.5199 8.5952 5.0198 3.6354 78090 635 4 03225 47875 36020 20634 38034 81108 21280 (67)1.5874 ( 1)2.5119 0.5771 5.0119 3.6308 66692 71337 52262 57040 29638 636 2572 11)1.6361 4 04496 59456 70140 (67)1.9185 ( 1)2.5219 8.5997 5.0218 3.6365 _. 2621 24 l/2 l/3 l/4 l/5 (67)2.2300 ( 1)2.5298 8.6177 5.0297 3.6411 2683 39634 04043 47604 56273 65574 2464 67)1.3627 1)2.5039 6.5589 5.0039 3.6262 2524 11 I 1.5953 14 1.0082 16 6.3723 19 I 4.0273 22 2.5452 25 1.6086 28 1.0166 67)1.6489 1)2.5139 8.5816 5.0139 3.6319 74520 22128 (67)2.3152 22362 ( 1) 2 ‘6;;; ‘2:;;; 4 10881 74721 2646 67)2.4034 1)2.5337 645 4 16025 36125 5:0316 3.6422 2695 97308 65548 5.0336 8.6267 3.6434 646 4 17316 86136 2708 (11)1.7523 24 w l/3 l/4 l/5 (67)2.6880 ( 1)2.5396 8.6401 5.0395 3.6467 2476 65028 96805 89894 95209 14650 (67)1.4158 ( 1)2.5059 8.5635 5.0059 3.6273 632 3 99424 35960 2536 95318 89841 91794 51614 86220 20891 48403 59081 61018 80854 41581 79727 1.6316 (67)1.7127 ( 1)2.5159 24057 85020 22598 28767 99973 4--97 (67)2.7898 ( 1)2.5416 8.6445 5.0414 3. b479 47292 53005 85472 80939 30063 .;[(-;I”] 2488 73152 3.1 629 3 95641 58189 (67)2.8953 ( 1); 5:0434 3.6490 73988 80891 71892 04672 23768 28361 (2Q1.1173 67)2.0686 1)2.5258 8.6087 5.0257 3.6388 2658 (67)2.4949 ( 1)2.5357 8.6311 5.0356 58602 06237 01272 3.6445 1 2548 634 4 01956 40104 95163 94164 66188 52582 99626 49851 (67)2.1479 ( 1)2.5278 8.6132 5.0277 3.639') 643 4 13449 47707 58638 2670 (67)2.5897 ( 1)2.537'7 8.6356 5.0375 3.6450 44467 82992 17605 35581 2720 (67)3.0046 ( 1)2.5455 8.6534 5.0453 3.6501 “;[(-;)3] 4 19904 97792 93247 84412 97422 78492 86051 1 32334 44932 48015 67827 89842 644 4 14736 89984 67740 15508 55108 74325 68481 -.. 2733 4’-37 82860 61719 35662 23720 03608 75544 639 4 08321 260') 17119 _. 648 647 4 18609 40023 34949 y; 20469 22049 76765 06675 09545 87241 80703 80871 25079 (28)1.0492 (67)1.7788 ( 1)2.5179 8.5907 5.0179 3.6342 30535 49125 638 4 07044 2596 94072 642 4 12164 09288 30845 58755 633 4 00689 36137 5.0159 3.6331 61105 p9'; 70600 50275 18009 39823 II 19 3.8954 22 9.6940 27 1:5411 25 2 4502 (67)1.4710 ( 1)2.5079 8.5680 5.007') 3.6285 96309 92817 37711 89230 8.5062 30760 33719 28406 620 3 94384 3 93129 91883 637 4 05769 2584 74853 11 1.6464 84814 14 1 1.0488 10826 16 6.6809 24963 19 4.2557 49202 22 2.7109 12241 25 1.7268 51098 28 I 1.1000 04149 67)1.9922 61654 1)2.5238 85893 8.6042 52449 5.0238 29110 3.6377 08430 - .- 2633 Table I 11 I 1.565:) 13 6.1930 16 9.8458 64.1 4 09600 44000 nk 621 3 91876 50221 65430 96100 II 19 3.9389 27 2.4815 25 1.5633 22 9.6493 (67)1.5281 ( 1)2.5099 AND ROOTS 626 2453 11 1.5356 13 9.6132 16 I 6.0179 19 3.7672 22 2.3582 25 1.4762 27 I 9.2415 67)1.3115 1)2.5019 8.5544 5.0019 49 METHODS (67)3.1179 ( 1)2.5475 8.657') 5.0473 3.6513 4 21201 59449 75679 47841 46522 23886 11957 50 ELEMENTARY Table 3.1 ANALYTICAL POWERS METHODS AND ROOTS nk k 651 2146 4 23801 4 22500 25000 2758 94451 2771 10364 ;U; 1.4097 (67)3.6134 ( 1)2.5553 14868 66091 17457 19800 30929 16597 02582 86468 46611 81368 5.0550 8.6756 3.6558 83054 97359 01749 :lb 1.1531 19 5.0628 ‘22 I 3.3060 ;25 ,28 44710 09757 91053 5.0492 3.6524 67033 36416 (b7)3.3569 41134 ( 1)2.5514 70164 8.6668 5.0512 3.6535 31029 07939 59612 (67)3.4829 ( 1);;::; 5:0531 3.6546 656 2810 (67)3.8885 ( 1)2.5592 8. b845 5.0589 3.6580 : : 5 : 8 1: 24 112 l/3 l/4 l/5 4 29025 11375 2823 93654 1)2.5612 8.6889 5.0608 3.6591 49271 38399 4 35600 96000 73600 32576 95002 60701 ObOb3 68001 36881 78950 46516 87691 11 '14 lb i 19 22 25 28 2888 1.9089 1.2618 8.3408 5.5132 3.6442 2.4088 1.5922 (67) 4.8398 ( 1)2.5709 8.7109 5.0704 3.6647 76246 06215 665 : 4 42225 2940 2835 2954 79625 79069 54676 3.6602 661 4 36921 04781 69592 4.3393 17689 3.6613 (b7) 83152 662 2901 99602 15727 (b7)5.0187 05901 664 40896 4 _-. 54944 - 2927 34241 09058 1);;;;; 5: 0124 3.6658 666 4 43556 08296 2967 48947 (67)5.2038 ;;;;t 11720 ( 1)2.5748 8.7197 5.0743 3.6669 23896 b67 4 44889 40963 298; (67) 5.3955 27431 19745 59553 78638 26200 30727 ( 1)2.5768 8.7241 5.0762 3.6680 41343 38514 3b224 --. bb8 46224 77632 4 47561 2994 18309 58582 93933 27470 98350 12498 27948 49870 i 10 l/3 l/4 l/5 87920 99531 88202 55239 95358 18918 31842 65311 00901 7” 24 (67)4.5003 ( 1)2.5670 8.7021 5.0666 3.6624 4 34281 91179 ~24063025 4’ 5 10 17214 20758 3.6569 4 39569 2914 79482 68738 62236 82739 95071 5.0570 663 4 38244 17528 48737 20153 82121 84834 92026 72888 42371 23736 2861 80288 (67)4.1837 26264 (67)3.7485 ( 1)2.5573 8.6801 4 31649 93393 49695 62971 45603 --- 2874 lQ1.8974 14 1.2523 lb 1 8.2653 19 5.4551 22 3.6004 25 2.3762 28 i 1.5683 (67)4.6671 ( 1)2.5690 8.7065 5.0685 3.6636 2.1588 657 4 30336 0041b 67) 4.0335 81447 96778 2797 2784 11 '14 i 1.1873 (67)3.2353 ( 1)2.5495 8.6623 654 4 27iid 653 4 26409 45077 1.8182 46353 4 25104 67808 67) 5.5939 61683 (67) 5.7993 79113 1)2.5787 8.7285 5.0781 3.6691 59392 18735 48670 40389 ( 1)2.5806 8.7328 5.0800 3.6702 97580 91741 11)2.0151 14)1.3501 lb) 9.0458 19 6.0607 22 4.0606 25 I 2.7206 28) 1.8228 43226 3021 11 2.0271 '14 I 1.3602 12100 25107 38217 11605 76776 53440 ‘lb '19 :22 9.1271 6.1243 4.1094 5.0838 8,741b 3.6724 672 (b7)b.4599 15340 03431 84552 66242 24639 ( 1)2.5865 8.7459 5.0857 44934 3.6735 673 4 51584 64448 25) 2.7946 28) 1.8719 09844 69597 4 52929 21217 b7819 43810 674 4 54276 67120 93105 67366 24470 1.8502 27799 8.7503 40123 (67)6.9396 ( 1)2.5903 8.7546 96605 66769 91362 5.0895 3.6757 64588 1 2321 ( 1)2.5845 44740 lb I 9.2090 19 6.1884 22 4.1586 88867 35821 “[C-36)2] (67)b. 62528 11711 69581 30789 48592 lb705 lb509 37805 67266 41314 5:0819 3.6713 4 50241 4 48900 63000 67) 6.6956 1)2.5884 5.0876 3.6746 14426 ;U;; 671 _.I 3007 56673 (67)6.0120 ( 1'2%;; 37627 46;)“I 3034 ( 1);:;;; 5:0914 87644 13208 '9;;;; 59790 3.6768 32575 3048 (67)7.1922 (67)7.4535 ( 1)2.5942 3061 22063 (67)l. 24354 15552 50997 43858 19196 18565 7239 5.0933 8.7633 t 4637)3] 82024 52878 80887 ( 1)2.5961 8.1677 5.0952 3.6779 26219 3.6790 4’-37’21 ISLEMENTARY ANALYTICAL POWERS METHODS AND ROOTS 51 nk Table k 675 4 55625 3075 46875 (11)2.0759 (14)1.4012 (16)9.4585 (19)6.3844 24 l/2 l/3 l/4 l/5 41406 60449 08032 92922 (22)4.3095 (25)2.9089 (28) 1.9635 95322 76211 53215 32735 09614 ( 1)2.5980 8.7720 5.0971 3.6801 3144 I 14 I 2.1381 16 1.4539 11 9.8867 2.1139 (67)9.5546 22820 a0962 59344 5.1065 3.6855 45762 45546 30685 685 4 69225 3214 19125 : 3 4 5 6 7 8 9 72571 (67)8.5926 68325 00000 82955 19514 99371 ( lJ2.6019 22366 ( 21241 42651 (11 (14 116 119 I22 125 1.4646 9.9743 6.7925 4.6256 3.1500 55745 05627 02132 93952 2.1452 97581 16453 167)9.8976 I 1)2.6095 8.7979 3.1919 14196 17949 97670 67850 68) 1.0252 5.1084 22134 38701 12971 72141 3.6866 128 28893 5.1102 3.6871 03562 168) 1.1797 ( 1)2.6172 8.8151 5.1159 3.6909 50466 59819 07022 49595 1)2.6191 a.8194 28446 19551 60171 47349 5.1177 3.6920 73120 26615 690 4 76100 l/2 l/3 l/4 1/5 09000 12100 31349 81631 53253 3357 55922 17173 22179 695 4 83025 02375 96441 10968 3299 (2aJ2.3418 91886 ( lj2.6210 a.8237 68484 30714 5.1196 3.6931 42816 a7886 3371 22729 73128 92956 11 2.1761 14 1.4862 17 I 1.0151 19 6.9333 22 4.7354 25 I 3.2343 28)2.2090 (6aJ1.0619 ( 112.6134 a.8065 5.1121 3.6887 32441 26869 12225 68688 91774 10 68)1.6130 1)2.6362 a.8578 5.1344 3.7016 03502 a5265 48911 76863 63101 (68) 1.6696 ( 1)2.63al 8.8620 5.1363 3.1027 35809 al192 95243 22801 28321 3200 13504 11)2.1888 92367 6a)l.o998 1)2.6153 a.8108 5.1140 3.6898 a2878 39366 68115 38880 71315 689 3256 4 73344 60672 45423 (68)1.2650 ( 1)2.6229 8.8280 5.1214 3.6941 02381 692 4 78864 73888 3328 93189 75410 3270 (68)1.3099 ( i)2.6248 4 74721 82769 a0950 a4991 5.1233 3.6952 59200 50159 693 4 a0249 12557 3342 07305 30255 a6536 39271 a9288 a5422 5.1289 3.6984 27069 62494 694 4 81636 55384 72050 1.4539 ( lj2.6305 a.8450 69927 a.8322 09925 99204 76894 (68) 3386 41112 1.5052 ( 1)2.6324 a.8493 11857 a9316 44010 5.1307 3.6995 79001 30796 (68) 08873 (6a)1.7281 ( 1)2.6400 8.8663 5.1381 70846 75756 37511 66751 3.7037 91713 (6a)1.5582 14678 ( 1)2.6343 a.8535 5.1326 3.7005 a7974 98503 28931 97866 698 4 a7204 4 a5809 2.3736 1.6568 1.1564 a.0721 68392 77376 26809 65112 26484 22 5.6343 3.9321 24 4 67856 97694 44925 57584 699 4 a8601 32099 44286 12312 3400 11 14 17 19 l/2 46612 67319 60923 58832 696 4 84416 53536 62844 a.7893 5.1046 3.6844 - -a 37179 3.6387 8.8408 5.1270 3.6973 50418 ( lj2.6057 94737 I 19 7.5987 (22 5.2583 '68)1.4043 (67)9.2230 11987 19871 89872 35983 78761 4 71969 42703 (66)1.2216 3313 679 4 61041 46639 '38037 74277 4 66489 3186 (11)2.2405 11 2.2931 14 1.5868 17 1.0980 : 1)2.6286 (11)2.1255 (14) 1.4432 687 3242 691 4 77481 39371 a3744 08784 70007 a5107 lJ2.6115 8.8022 4 70596 28'356 05952 19683 84703 87060 79523 72953 31118 68) 1.3563 1)2.6267 8.8365 5.1252 3.6963 14 1.4754 686 (28)2.2746 24 11 2.1634 4 65124 14568 3130 43313 29644 86801 75023 682 3172 17J1.0062 19 6.8626 22 1 4.6803 (68)1.1391 22)5.1379 3.5452 a7040 678 4 59684 65752 93799 11595 54097 89678 65402 82142 65092 13744 1)2.6038 a.7850 5.1027 3.6833 08428 04200 03354 41087 50822 30603 14071 24 3285 11 2.2667 14 1 1.5640 17)1.0791 19)7.4463 8.7807 5.1009 3.6822 4 63761 3158 2.1507 10 l/3 l/4 l/5 3116 t 1)2.6000 8.7763 5.0990 3.6811 3228 111)2.2146 114 1.5192 117 1 1.0421 119 7.1493 I22 4.9044 I25 3.3644 28 I 2.3080 l/2 677 4 58329 88733 54722 43247 09783 94923 50263 31928 91415 t 67)8.2931 4 62400 32000 33568 37600 48262 3102 11)2.1006 14)1.4221 I (16 9.6279 (19 6.5180 (22 4.4127 (25 2.9874 (28 2.0224 (11)2.1130 (14 1.4326 (16 1 9.7135 (19 6.5057 (22 4.4651 (25 3.0273 (28 I 2.0525 (67)8.9025 30847 ( 1)2.6076 8.7936 24 676 4 56976 15776 32722 34587 (67)8.0036 3089 3.1 3415 (68)1.7886 69670 (68) ( lJ2.6419 68963 ( 8.8705 5.1400 3.7048 15722 08719 53884 1.8511 a.8748 95210 60813 09888 5.1418 3.7059 48708 14839 1)2.6438 52 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS nk k : 3430 4 90000 00000 3444 701 4 91401 72101 3459 702 4 92804 48408 3414 703 4 94209 28927 3489 4 95616 13664 : i 9" 10 24 l/2 ;g : 3 4 (6a)1.9158 ( 1)2.6457 a.8790 5.1436 3.7069 3504 12314 51311 40017 (68)1.9825 ( 1)2.6476 8.8832 5.1455 3.7080 86124 74581 705 4 97025 02625 3518 a7808 40459 66120 22771 33112 (68)2.0515 90555 2q2.9481 (68)2.1228 74939 91511 ( 1’;;;;: gg; ( 1) 2 "8:;; '0;;;; 511473 3.7090 706 4 98436 95816 56056 90435 511491 3.7101 a8981 46554 (68)2.1965 ( 1)2.6532 8.8959 5.1510 3.7112 708 5 01264 3548 94912 707 4 99849 3533 93243 63787 99832 20362 19154 01473 709 5 02681 3564 00829 2 i 1'0 24 l/2 ::: l/5 (68)2.2726 ( 1)2.6551 8.9001 5.1520 3.7122 82709 a3609 30453 47377 55193 (68)2.3513 ( 1)2.6570 a.9043 5.1546 3.7133 710 3594 25887 66051 36564 73657 07718 (68)2.4325 13215 ( 1)2.6589 8.9085 5.1564 3.7143 47160 38706 97998 59051 711 5 05521 25431 3609 (68)2.5165 ( 1)2.6608 8.9127 5.1583 3.7154 712 5 06944 44128 3624 07242 26939 36887 20404 09195 (ba)2.6032 ( 1)2.6627 8.9169 5.1601 3.7164 713 5 08369 67097 3639 12640 05391 31117 40881 58153 714 5 09796 94344 \ 2) :: ZE I 19 I 9.4599 17)1.3249 22 6.7543 4.8226 24 v2 l/3 l/4 l/5 (68)2.6927 1)2.6645 a.9211 5.1619 3.7115 3655 24 (bQ2.7852 ( 1)2.6664 a.9253 5.1637 3.7185 76016 a2519 21404 59433 05928 715 5 11225 25875 3670 89985 58325 07760 76065 52523 716 5 12656 61696 : 4 (68)3.1867 ( 1)2.6739 a.9420 5.1710 28051 48391 14037 23488 (6CQ3.2954 ( 1)2.6758 8.9461 5.1728 33372 17632 80866 30591 3.7221 1 27165 3.1231 67905 720 5 18400 3732 48000 68)2.8808 1)2.6683 8.9294 5.1655 3.7195 3686 (68)3.4076 ( 1)2.6776 8.9503 5.1746 3.7248 721 5 19841 3140 05361 44702 32813 90191 90782 97942 (68)2.9795 ( 1)2.6702 6.9336 5.1674 3.7206 717 5 14089 61613 87302 85568 43817 35801 07483 3701 (68)3.5236 ( 1)2.6795 8.9545 5.1164 3.7258 722 5 21284 3763 67048 36544 05985 68708 03588 42186 :;zt a3171 20408 18825 29584 (ba)3.0814 ( 1)2.6720 8.9376 5.1692 3.7216 63889 77843 43321 14489 85260 ._ 718 5 15524 46232 00491 52201. 02899 39125 45902 719 3716 (68)3.6432 86875 3.7260 723 5 22729 3719 3795 33067 5 16961 94959 a3164 724 5 24176 03424 2 7 a 1: 24 (68) 3.7668 ( 1)2.6832 a.9628 5.1800 3.1279 63712 a1573 09493 40128 19273 1 (68)3.8944 ( 1)2.6851 a.9669 5.1818 3.7289 R2[(-36)2] 51981 44316 57022 37817 54232 1 (68)4.0261 ( 1)2.6870 a.9711 5.1836 3.7299 qc-,7,5] 75870 05769 00718 33637 a8042 (68)4.1621 ( 1)2.6888 a.9152 5.1854 3.7310 4637)2] 63488 65932 40590 27593 20708 (ba)4.3025 t 112.6907 'a;4793 5.1872 3.7320 4’137’2] 46659 24809 76646 19688 52232 ELEMENTARY ANALYTICAL POWERS 3810 (11)2.7628 725 5 25625 70125 16406 3826 11)2.7780 14)2.0168 17)1.4642 AND ROOTS 726 5 27076 57176 91098 94137 65143 3842 (11)2.7934 53 METHODS Table nk 727 5 28529 40583 29038 728 3858 (11)2.8088 (14)2.0448 (17 1.4886 30403 28533 35172 26405 28230 76552 23730 09683 47513 24 (68)4.4474 61095 (68)4.5970 46501 (68)4.7514 46686 20 1 1.0837 22 7.8895 25 1 5.7435 I 28)4.1813 (68)4.9108 l/2 ( 1)2.6925 82404 ( 38717 37347 98317 11864 ( 1)2 a'9917 6962 93753 62009 ( 5:1925 3.7351 84860 39979 l/3 0.9835 5.1890 3.7330 l/4 l/5 : 3890 08896 09928 82616 1)2.6944 8.9876 5.1907 3.7341 730 5 32900 17000 3906 : 731 5 34361 17891 3922 5 29984 28352 1)2.6981 5.1943 8.9958 3.7361 732 5 35824 23168 3938 (11)2.8242 (14)2.0589 5 37284 32637 3954 11)2.9025 1.563;' 20 1.1478 38047 (68)5.4202 21655 (68)5.6010 04807 (68)5.7875 58467 (68)5.9800 51217 ( 1)2.7037 9.0082 5.1997 3.7392 01167 22937 12653 41158 ( 1):;;;; ;;;'8; ( 1)2.7073 97274 ( 1)2 9.0041 5.1979 3.7382 13346 33452 17550 735 5 40225 65375 3986 5:2014 3.7402 736 5 41696 88256 4003 90029 63647 9.0164 5.2032 3.7412 737 5 43169 15553 4019 (68)6.1786 86185 ( 2";;;; 1) ;. ;;:; 5:2068 3.7433 (68)6.3836 81700 11253 24423 512085 3.7443 740 5 47600 24000 4068 74542 (68)6.8132 ( q.;;;; '0;;;; ( l'y& 61314 42461 741 5 49081 69021 5:2103 3.7453 4085 11)3.0312 49693 59393 ( 1)2.7202 94102 9.0450 5.2156 3.7484 4134 41696 43874 03580 (68)7.5099 ( 1)2.7221 9.0491 5.2174 3.7494 745 5 55025 93625 49065 31518 14206 05023 16115 98844 l'$.;:f 5 56516 4151 60936 l/2 :5: l/5 (68)8.5457 ( 1)2.7294 9.0653 5.2244 3.7534 57129 68813 67701 31847 55355 (68)8.8253 ( 1';. SW& 16213 52049 72407 E: 14035 33628 49854 98142 93019 4168 (68)8.0118 ( 1)2.7258 9.0572 26396 02634 48245 64391 27557 5.2209 3.7514 21982 37909 747 5 58009 32723 748 5 59504 4185 08992 739 II 46121 a3419 81466 53804 95761 79698 y; 5:2138 3.7473 75222 .5;5;; 5:2191 3.7504 (68)7.0382 ._ 410: 01490 28)5.4101 24 5:2121 3.7463 07181 ( 68)7.7569 4035 ( 1';. 742 5 50564 18488 28)5.0587 49690 24254 (68)6.5950 29268 39324 05277 (11)2.9824 (14)2.2040 (17)1.6287 16095 24678 56123 y7’; 43437 730 5 44644 47272 70867 27605 1) $.-p; 7092 9:02OI1 5.205(1 3.7423 30890 65584 85019 (14)2.1891 (17 1.6156 (20 1.1923 09120 30506 46422 36620 07703 24153 68916 (68)7.2704 :;: 58576 (68)5.2450 ( 1)2.7018 4052 l/3 80275 93922 82539 16384 72255 29635 04352 25 6.183(1 28 4.5390 (22 I 8.7993 l/2 734 Ia 38756 46904 t 22 8.4249 (11)2.9663 l/4 l/5 00000 00000 52423 92819 69560 82891 66963 (17 I 11 I 2.9184 17 1.5766 14 2.1450 II 20 4.6012 28 6.5172 25 8.2601 22 1.1588 l/3 87861 1)2.7OOll 9.000[1 5.1961 3.7371 I 14)2.1304 3970 24 46353 89891 44308 73700 15828 ( 9” l/2 95365 11321 (68)5.0752 10 24 729 5 31441 20489 (17)1.5009 (20)1.0941 (22)7.9766 (25 5.814(1 (28 1 4.2391 2 1 ://: l/4 l/5 3874 3.1 4118 2;;;; 80938 89950 744 II 53536 30784 (28)5.196 20453 (68)8.2746 ( 1)2.7270 9.061:) 5.2226, 3.7524 65623 36339 09792 77799 47174 (11)3.1472 749 ! j 61001 420: 89749 ,21220 09038 48404 ;;;; 5:2261 3.7544 (68)9.1136 94019 (68)9.4110 55807 (68)9.7177 03069 2";;;; ( u;.;;;; y;; ( l)$m~ ;9"6";; ( 1)2.7367 9.081!5 5.2314 3.7574 a6437 84131 62453 512279 3.7554 34653 68472 1 5:2296 3.7564 a3419 7341; 63122 30432 77202 . 54 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS nk k : 3 4 5 ? 4218 11)3.1640 14)2.3730 17 1.7797 ;20 1 1.3348 23 1.0011 10 ; 25 7.5084 28 I 5.6313 24 69) 1.0033 l/2 1)2.7386 l/3 ::: : 3 .__ 751 5 64001 750 5 62500 9.0856 5.2331 3.7584 4303 75000 62500 46875 85156 38867 29150 11 3.1809 14 2.3889 17 1.7940 20 1 1.3473 23 1.0118 64751 71280 09431 70983 47308 57828 68628 25 1 1.5990 2Q5.7068 52291 51471 91278 12788 02964 75697 80079 755 5 70025 68875 11)3.2492 14 2.4532 17 1.8521 20 1.3983 85006 10180 73686 91133 10 23 i 1.0557 25 7.9711 28 I 6.0182 l/2 69) 1.0359 1)2.7404 9.0896 5.2349 3.7594 88271 96977 37921 39217 425; 9.0936 5.2366 3.7604 19217 81806 11 3.2665 14 I 2.4694 17 1.8669 2 6 1 a 9 l/3 $2 69)1.1768 69) 1.2148 1)2.7477 26333 1)2.7495 1 : 4 4389 11)3.3362 2 14)2.5355 17)l. 9269 i 1: 24 20 1.4645 23 1 1.1130 8.4590 48491 75936 78075 ::: l/4 l/5 09750 9.1258 5.2505 3.7684 05271 33069 49662 9.1097 5.2436 3.7644 24 24 10 ::: l/5 20 1 1.5333 23 1.1729 25 1 8.9733 28)6.8645 69)1.6138 1)2.7658 9.1457 5.2591 3.7733 97125 83006 35500 27157 10275 82361 15059 86020 91907 63337 14274 47590 95151 770 5 92900 4565 33000 11)3.5153 04100 14 2.7067 84157 17 2.0842 23801 20 1.6048 52327 23 1.2357 36292 25 i 9.5151 69445 28)7.3266 80473 (69)1.8870 23915 ( 1)2.7748 9.1656 5.2677 3.7783 87385 56454 19986 14849 1 .i[,-,2] 1.1042 5:2384 3.7614 a2467 51214 45417 b6916 10795 69) 1.2540 1)2.7513 9.1137 5.2453 3.1654 74495 10313 63298 81798 43934 69862 69)1.4230 ( 1)2.7586 9.1298 5.2522 3.7694 4494 11)3.4428 14 2.6372 17 2.0200 20 1.5473 23 / 1.1853 6.9548 (69) 1.6652 ( 1)2.7676 9.1497 5.2b08 3.7743 22845 Ob063 1)2.7604 766 5 86756 55096 26035 04743 98833 95706 05111 48857 92289 70501 57625 65424 81144 ._ 771 5 94441 4583 14011 460: (69)1.9467 ( l';.;:;; 5:2694 27094 ;i;;; 29452 (69)2.0082 ( 1)2.7784 5.2711 9.1735 38127 88798 37257 85227 3.7792 95720 3.7802 75573 4-37151 .i[,,,2] 39353 80599 4372 5 76681 45479 97112 (69)1.2943 ( 1)2.7531 9.1177 5.2470 3.7664 79980 93146 753568 b4176 763 82169 ;;;;z 6528:l 91509 68822 72711 (69)1.3359 ( 1)2.7549 9.1218 5.2488 3.1614 4459 88198 95463 00968 05067 57442 764 5 83696 43744 12784 5:2557 3.1714 772 95984 99648 26517 17168 14132 ( uy;;: 767 88289 45125 17663 11)3.4608 39475 63877 73794 91900 40987 73373 78477 69)1.7182 59425 1) '9. ;g;': 76485 37512 5:2625 81576 3.7753 66108 9.1017 5.2401 3.7624 72453 53520 (69)1.5156 9.1338 5.2539 3.7704 59366 40838 19555 06044 38329 50728 74547 63605 26467 11368 84062 32555 87807 69) 1.4686 1.1400 ( 1)2.7459 (69) :;;A: 762 88020 a4547 00985 01041 82064 758 80644 442: 11)3.3714 38985 03656 80565 14564 75634 07855 28546 68109 41772 761 165 5 85225 4476 11 3.4248 14 I 2.6200 17 2.0043 99699 19457 34787 64385 1)2.7568 ( y.;;;; 98093 51564 17600 25376 99286 79182 (69) 61840 71888 60997 17667 760 5 77600 76000 69)1.3788 16698 46588 40186 65520 9.1057 5.2418 3.7634 17395 757 73049 81216 33993 61064 89022 29433 65863 35750 69) 1.0696 4286 92061 06051 ( 1)2.7422 754 5 68516 753 5 67009 4269 57711 56701 756 5 71536 4320 152 c. 65504 59008 47740 52239 56084 86975 85305 79054 24 4235 (11)3.4789 4529 16435 15056 45463 97144 06863 2006N8 768 5 a9az14 84832 23510 13255 52500 99582 z;t 86072 (b9)1.7728 ( y.;;;; 5: 2642 3.7763 (69)1.5640 ( 1)2.7640 9.1417 5.2574 3.7724 4547 11 3.4970 14 1 2.6892 17 2.0680 20 1.5903 23 1.2229 25 9.4045 28 i 7.2320 69)1.8290 1)2.7730 9.1616 5.2660 3.7773 (69)2.0716 ( 1)2.7802 9.1775 5.2728 3.7812 97529 a9917 ;;s:;; 22t54 35044 82084 65S13 92641 09310 87755 44479 43403 54412 .[C,,l] 769 5 91361 56609 78323 53231 35734 19480 55680 29178 82938 77701 84925 86919 08854 32958 774 5 99076 4636 84824 713 461; 13890 54992 a7449 28071 08126 (69)2.1368 ( 1)2.7820 9.1815 5.2745 3.7822 94378 85549 00317 47894 32239 ZLEMENTARY ANALYTICAL POWERS 55 METHODS AND ROOTS nk Table 3.1 k 465i 715 00625 04375 03906 15527 57034 36701 08443 111 I 114 ( 17 I 20 I 23 4672 3.6261 2.8138 2.1835 1.6944 1.3149 91544 (69)2.2041 l/2 l/3 l/4 l/5 ( 1';. ;;;i 5:2762 3.7832 82181 52750 50735 09055 780 08400 52000 05600 74368 96007 56885 14371 89209 24 (69)2.5719 ( 1)2.7928 9.2051 5.2847 3.7880 75831 97041 48009 64083 40305 78066 785 6 16225 4837 36625 24 l/2 l/3 l/4 l/5 (69)2.9982 ( 1)2.8017 9.2247 5.2931 3.7929 : : 5 4930 77060 85145 91357 89157 22172 790 6 24100 39000 86133 62839 03163 64854 84463 24 776 6 02176 88576 59350 99655 169)2.2734 I 1)2.7856 9.1894 5.2719 3.7841 476; I 11 3.7205 114 I 2.9057 I 17 2.2693 I 20 1.7723 123 I 1.3842 I 26 1.0810 f 28 8.4432 169)2.6523 I l'$;;W; 5:2864 3.7890 31271 28553 17655 01784 51928 84864 781 09961 79541 24215 29412 74671 81618 30044 83664 63416 13239 ;'6;;; 33318 48871 4855 11)3.8167 ?I4 2.9999 ,117 2.3579 (20 1.8533 123 I 1.4567 1,26)1.1449 '128)8.9996 / 69)3.0912 1)2.8035 9.2287 5.2948 3.1938 786 6 17796 87656 18976 41115 53117 51621 34374 93218 46695 99652 69154 06804 74081 88029 4949 791 6 25681 13671 : 8 777 4690 11 3.6448 14 2.832U 17 1 2.2005 20 1.7098 23 1.3285 26 I 1.0322 28 8.0206 (69)2.3447 ( 1)2.7874 9.1933 5.2796 3.7851 6 03729 97433 87054 77241 24016 07161 20164 60167 61501 92689 71973 47428 51478 59667 4782 6 11524 11768 (69)2.7350 ( 1)2.7964 9.2130 5.2881 3.7900 29868 26291 25029 24706 18681 787 6 19369 4874 43403 1.4716 84488 52028 18931 57399 52904 11)3.6636 778 6 05284 10952 (69)2.4183 00846 ( y.;;;; i;;;$ 5:2813 3.7861 (69)2.4940 ( 1)2.7910 9.2012 5.2830 3.7871 49388 33467 .__ 4800 11 3.7587 14 1 2.9431 17 2.3044 20 I 1.8043 23 1.4128 26 1.1062 28 8.6619 69)2.8202 1)2.7982 9.2169 5.2898 3.7909 24 l/3 l/4 l/5 (69)3.4918 ( 1)2.8106 9.2443 5.3015 3.7971 06676 93865 35465 91145 41656 69)3.5994 1)2.8124 9.2482 5.3032 3.7987 .,- 795 : 5024 6 32025 59875 5043 45514 72222 34384 74670 02623 6 13089 48687 81219 25695 67419 97989 43625 56559 88854 15463 13716 50477 14473 87500 4893 11)3.8557 11)3.9346 14 3.1162 17 2.4680 20 1.9546 23 I 1.5481 93088 03872 14511 (69)3.2857 ( 1)2.8071 9.2365 5.2982 3.7958 09926 33770 21746 39113 16799 49% __ 7775-l-. (69)3.3872 ( 1)2.8089 9.2404 5.2999 3.1967 (11 3.9545 (14 3.1359 (17 2.4867 (20 1 1.9720 (23)1.5638 60118 49456 30018 50005 62619 (69)3.8243 ( 1)2.8160 9.2560 5.3066 3.8006 10648 26944 90066 24523 15447 69)3.7102 1)2.8142 9.2521 5.3049 3.7996 I 5062 24 l/3 l/4 l/5 (69)4.0626 65702 :69)4.1871 02820 ( 1’;. y; ;‘i:;; ( 1’;. ;‘6;; y& 5:3099 3.8025 66512 36800 5:3116 3.8034 35526 92932 40422 00000 72584 02622 55329 (11)3.9744 (14 3.1557 {17{2.5056 20 1.9894 (23 1.5796 26 1.2542 28 I 9.9587 (69)3.9417 ( 1)2.8178 9.2599 5.3082 3.8015 5100 11 4.0755 14 1 3.2563 17 2.6018 20 2.0788 23 1.66101 26 I 1.3271 29)1.0603 (69)4.5827 ( 1)2.8266# 9.279? 5.316b 3.8063 6' 38401 82399 58368 71136 40538 70590 17601 53063 95298 13463 58805 08064 33150 55574 7911 5081 6 36804 69592 07803 19075 04737 17219 49232 52362 45624 87922 18843 59160 02968 48104 56439 14381 33255 19227 79716 794 6 30436 66184 95501 49428 65046 98046 61449 51190 54451 77065 00561 11460 95923 79705 5005 ,- 797 6 35209 61573 (69)4.3151 ( 1)2.8231 9.2715 5.3133 3.8044 39997 25568 22375 23755 21646 1: l/2 6 14656 90304 19983 789 6 22521 4911 69069 793 6 28849 01257 04196 33723 82708 08705 I 11 4.0349 17 2.5630 14 3.2158 II 20 2.0427 26 1.2975 23 1.0341 29 1.6280 7” 8 ._. 4818 (69)2.9079 ( 1)2.8000 9.2208 5.2915 3.7919 h __... _ ,“%I4 14558 57147 28569 45663 06266 784 (11)3.7780 788 792 6 21264 4967 796 6 33616 58336 : 5 779 6 06841 29139 87207 lo’ l/2 4127 27378 (69)3.1870 ( 1)2.8053 9.2326 5.2965 3.7948 4709 (69)4.4470 23172 ( 1) $ ;;;i "3;;;; 5:3149 3.8054 68841 02317 56 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS nk I; _._ : 3 5120 801 41601 22401 18432 6 40000 00000 : 6 72400 52492 39346 1'0 l/2 l/3 l/4 l/5 (69) 4.7223 ( Q2.8284 9.2831 5.3182 3.8073 : 5216 4.1993 : 66483 27125 77667 95897 07877 (69)4.8660 92789 ( 1)2.8301 9.2870 5.3199 3.8082 94340 44047 57086 59229 805 6 48025 60125 5236 64006 2 i 9 10 24 (69)5.4840 46503 l/2 ( 1)2.8372 9.3024 5.3265 3.8120 l/3 l/4 l/5 71468 86329 55159 (69)5.6499 03151 ( ";.;;;; 513282 3.8130 l/4 l/5 5413 85441 49694 97518 38230 78910 24 l/2 49576 20485 38634 52016 79391 l/3 l/4 l/5 : : 5 (69) 25463 67164 74803 59085 ( lj2.8354 (b9)5.8205 60843 :% 19524 78976 52346 (69)5.9961 74542 91690 75012 47468 513298 9 3101 3.8139 89376 23915 31338 07593 809 _. _ 5294 6 54481 75129 5:3315 3.8148 5.3364 3.8177 84023 20859 61370 63391 28295 61880 6 5353 I 17 2.8663 14 3.5300 11 I 4.3473 816 65856 (69)7.1611 98588 ( 1)2.8513 9.3331 5.3397 3.8196 15486 9160H 71049 01974 ( 1)2.8530 9.3370 5.3414 3.9205 68524 16687 12288 41144 (69)8.0552 81B 69124 43432 69274 06266 48326 03930 94015 57904 21966 54907 ( 1y;g m!; h 5473 30157 (69)7.8222 04529 14300 07941 ( "yp; ;g;; ( ";A:%: nm; 5:3463 3.8233 5554 11)4.5654 3.7528 59849 90912 36029 (69)6.9530 i286j (69)7.5956 92531 9.3178 5.3331 3.8158 42067 :;o"l"i 92216 RI7 :% 51976 67212 68445 18251 26093 85292 13927 813 60969 67797 00190 34554 41493 52533 41510s 25547 52870 13847 67489 38513 15651 74587 40938 09746 543; (69)6.1768 ( "t% ( 1)2,8495 9.3293 5.3381 3.8186 821 6 74041 5533 87661 61548 ( lj2.8442 (69)6.7506 _-. 804 6 46416 18464 49732 11384 32030 06173 6 72400 5513 68000 11)4.5212 17600 9.2986 5.3249 3.8111 14112 14025 10420 90236 16717 5.3228 808 52864 ( 1)2.8478 9.3255 5:3446 3.8224 5197 22264 525; (69)6.5539 21104 33200 (69)7.3753 ( 1)2.8548 9.3408 5.3430 3.8214 ( 1)2.8337 9.2947 5.3232 3.8101 812 59344 87328 45103 44224 95910 13479 40945 32047 21222 36166 28962 1.2929 (29 ( 1)2.8319 9.2909 5.3216 3.8092 60452 07211 16720 09631 811 57721 815 6 64225 43375 (23 1.9465 (26 I 1.5864 I 23 I 1.3881 29 1.1146 26 1.7287 (69)5.1662 11731 80748 22686 00899 06129 59670 (6936.3626 ( 1)2.8460 9.3216 5.3348 3.8167 05879 ;;;;; 39778 01783 %:f l/3 803 44809 81627 86465 02531 78133 25440 18829 61219 93459 ( 1)2.8407 810 6 56100 24 (69)5.0140 806 6 49636 06616 69325 37076 38883 60940 67318 40258 45448 52192 l/2 802 6 43204 49608 '6::f;: 71036 ; 24 5158 26950 53125 5:3479 3.8242 822 6 75684 12248 88679 31694 5393 11)4.3903 5493 814 6 62566 53144 34592 819 6 70761 53259 I 11 1 4.4992 23 2.0242 20 2.4716 17 3.0178 14 3.684& 26 i 1.6578 29 1.3578 (69)8.2949 82936 51936 90032 47414 03191 87724 10046 47511 ( lj2.8618 17604 9.3560 5.3495 3.8252 62163 88616 '6 77% 5574 41767 5594 95237 95877 23193 824 6 78976 76224 7" 9" 10 24 w l/3 l/4 l/5 (29)1.3913 (69)8.5414 66801 ( lj2.8635 64213 9.3599 5.3512 3.8261 01623 28095 56858 34555 ( 69) 8.7949 ( "29';;;; 5:3528 98523 ;i;;; 58822 3.8270 89612 1 (69) 9.0557 ( 1)2.8670 9.3675 5.3544 3.8280 33244 54237 05121 88059 21458 1 ( 69) 9.3238 ( 1)2.8687 9.3713 5.3561 3.8289 66467 97658 02245 15810 5239; (69) 9.5995 98755 ( 1)2.8705 9.3750 5.3577 3.8298 40019 96295 42079 02432 ELEMENTARY ANALYTICAL POWERS k 1 2 3 4 2 i 1: 24 l/2 l/3 l/4 l/5 825 5615 11 4.6325 14 3.8218 17 3.1529 20 I 2.6012 23 2.1460 26 I 1.7704 29)1.4bOb 69)9.8831 1)2.8722 9.3788 5.3593 3.8308 15625 03906 : 5 828 _-. b a5564 827 3.8683 5676 09244 57626 27542 35853 81323 87277 (70)1.0174 68882 ( 1)2.8740 21573 9.3826 5.3609 3.8317 bb869 11564 75196 90182 39795 23 2.1879 26 1.8094 i 29 I 1.4964 (70)1.0474 ( 1)2.8757 9.3864 5.3626 3.8326 831 6 90561 87000 5738 56191 l/3 l/4 l/5 (70)1.1425 47375 ( 1)2.8809 9.3977 96375 72058 5.3674 3.8354 l/2 68731 5: 3690 84709 43756 3.8363 b7514 ;70;1.1760 ; 1; g. m; 835 5821 II 14 4.0591 I 11 4.8612 17 3.3893 20 2.8301 23 2.3631 26 1.9732 29 1.6476 (70)1.3197 513755 3.8400 1 2 5759 46709 ;;"o';; (70)1.2104 ( 1)2.8844 9.4053 5.3706 3.8372 22789 5697 83671 25484 47415 b0769 60060 12021 67128 (70)1.0782 ( 1)2.8774 9.3902 5.3642 3.8335 832 6 92224 30368 b 98896 TA42 77056 b8999 27006 24550 5863 (70)1.1099 ( 1)2.8792 9.3940 5.3658 3.8345 86167 41020 38751 99229 90383 76253 5084 63591 36010 20643 51293 19107 834 b 95556 5800 11)4.8379 1.6280 (70)1.2822 ( 1)2.8879 9.4128 5.3739 3.8391 838 7 02244 80472 93704 81491 22889 86929 05816 b9049 23907 33463 839 7 03921 I 23115 52801 32589 I 49211 00592 (70)1.3581 59133 (29)1.6'375 (70)1.3976 41959 90431 ( ;;;;; ( 1’;. i;:f ;:q2:; 5:3787 3.8418 50067 91464 34071 53677 71392 98914 41873 32391 93565 833 6 93889 5780 09537 11 4.8148 19443 14 1 4.0107 44596 17 3.3409 50249 20 2.7830 11557 23 I 2.3182 48627 26 1.9311 01106 29 I 1.6086 07222 70)1.2458 90957 1)2.8861 73938 9.4091 05407 5.3723 12294 3.8382 12366 837 7 00569 836 b 97225 82075 840 7 05600 5927 04000 : 829 b 87241 63552 62496 y; ( 1y;; 3.1 b5913 ! i 10 24 Table 23326 6 88900 5717 57 nk b 83929 5656 09283 15723 97971 830 : AND ROOTS 826 b 82276 5635 59976 6 80625 METHODS 1y;;; 5:3?71 3.8409 42790 73010 841 7 07281 5948 23321 70)1.4383 1)2.8948 9.4278 5.3803 3.8428 842 7 08964 5969 I (70) 1.4800' 1) 2.8965 9.4316, 5.3819 3.8437 b4626 86372 49672 42272 60304 25741 -., Rdd .< 7 10649 5990 77107 47688 ( 70) 1. b124 ( 1)2.9017 9.4428 5.3867 3.8464 23072 22965 93606 55904 09040 1 12336 6012 11584 2 7 i 10 24 l/2 l/3 l/4 l/5 : 3 (70)1.5230 10388 ( 1)2,8982 9.4353 5.3835 3.8446 75349 87961 b3271 6033 41568 045 7 14025 51125 (70)1.5671 ( 1)2.9000 9.4391 5.3851 3.0455 25939 00000 30677 64807 56523 7 15716 95736 23626 70428 64916 70609 -.. 846 6054 29)1;812i 70)1.6590 1)2.9034 9.4466 5.3883 3.8473 Ad7 6076 7 17409 45423 6098 90531 58848 46228 07220 63600 70)1.7069 1)2.9051 9.4503 5.3899 3.8482 83826 848 7 19104 00192 6119 41821 b7809 41057 60862 96177 849 7 20801 60049 5" I ; 8 2.2436 1;: 24 10 l/3 l/4 l/5 70)1.7561 1)2.9068 9.4540 5.3915 3.8492 47601 88371 71946 56705 07664 [email protected]] (70)1.8067 ( 1)2.9086 9.4577 5.3931 3.8501 11101 07914 99893 51133 18288 ;[‘-;I”] 29)1.9003 (70)1.8586 ( 1)2.9103 9.4615 5.3947 3.8510 27182 52223 68111 26442 24903 44148 28051 I I (70) 1.9120 ( 1)2.9120 9.4652 5.3963 3.8519 ‘y-37)2] 55324 43956 46902 35753 36956 1 q-37)1] (7O)l. 9669 ( 1)2.9137 9.468') 5.3979 3.8528 10351 60457 66137 25951 45003 ELEMENTARY Table : 5” F i 3.1 POWERS 3 10 24 l/2 1/q ANALYTICAL 6141 850 7 22500 25000 6162 _.___ 7 7A?“l 95051 62500 53125 95156 11)5.2446 14)4.4632 (17 3.7981 (20)3.2057 (23)2.7249 70883 05250 64710 57268 09335 28744 79034 (70)2.1406 90429 ( 1)2.9189 I 1 (70)2.0232 71747 20 3.2322 23)2.7506 26)2.3408 I 29)1.9920 (70)2.0811 ( 1)2.9154 75947 ( 1)2.9171 1.9687 , l/5 9.4726 5.3995 3.8537 6250 44043 82372 9.4763 5.4011 3.8546 14744 52195 7 31025 26315 6272 70884 14922 95899 99058 I 11)5.3690 14 4.5958 20457 81511 I 20)3.3401 17)3.9065 23)2.8558 6)2.4417 I l/3 75506 (14 79694 25639 15345 07421 20 1 3.9340 17 3.3675 (23 2.8826 67035 14574 38067 (29)2.0876 (70)2.3290 66620 92589 ( 1)2.9240 9.4912 38303 19958 5.4074 3.8582 31751 75391 ::z 4.5690 6360 (11)5.4700 14)4.7042 1 57569 47768 1)2.9274 9.4949 5.4090 3.8591 (11)5.4955 6382 18797 18180 77490 861 7 41321 77381 68250 75660 9.5096 5.4153 3.8627 6472 (11 85413 26084 77475 :;: 1/q l/5 5.5984 21628 67708 70568 15541 96443 98423 36164 88234 79435 80095 866 7 49956 6494 61896 06506 [14{4.8426 17 4.1888 20 3.6233 23 3.1342 26 2.7110 29 2.3450 (70)3.0788 ( 1)2.9410 24 08410 80150 69910 99621 75378 865 7 40225 14625 9.5280 5.4231 3.8672 6585 3.6527 9.4986 5.4105 3.8600 6405 11 5.5211 14 4.7592 17 I 4.1024 94533 nk 857 7 852 25904 70208 66172 99979 53982 45993 21986 6206 7 27609 50477 (11)5.2941 (14)4.5159 (17)3.8520 (20 3.2858 (23 1 2.8028 (26)2.3907 (29)2.0393 15835 00907 89174 43165 (70)2.2017 94325 03904 06107 88131 ( 1)2.9206 16373 64021 857 7 34449 22793 53336 89409 30523 03059 89021 03491 18192 27165 56234 14756 97225 78746 9.4838 5.4042 3.8564 I 14)4.6498 (17)3.9895 (20)3.4230 1 36164 28712 74349 23191 48298 32440 24 l/2 l/3 l/4 l/5 48329 ( 1)2.9291 63703 ( 1)2.9308 9.5023 5.4121 3.8609 6427 (11)5.5468 07842 74889 79161 3.5363 3.0483 2.2650 70)2.8325 23218 29097 (70)2.9124 54150 1)2.9359 83651 ( 1)2.9376 9.5170 5.4184 3.8645 51555 71787 72447 9.5207 5.4200 3.8654 867 7 51689 6517 14363 11 5.6503 14 14.6900 17 4.2473 86373 ( 1)2.9461 83973 9.5354 5.4263 3.8690 17196 12167 45344 9.5390 5.4270 3.8699 872 7 60384 6630 11)5.7818 54848 38275 6653 81845 76171 37445 9.5244 5.4216 3.8663 (70)3.6344 25075 76241 ( 1) ;. p; ;;g;: 5:4325 3.0726 859 7 37881 39779 28654 57114 58061 69474 17279 69182 70178 98059 51174 78737 87691 06312 12022 64090 869 7 55161 6562 34909 29)2.4558 70)3.4393 ( 1)2.9478 9.5427 5.4294 3.8708 873 7 62129 38617 5.8084 6676 26970 36231 80595 43681 38824 20725 874 1 63816 27624 06126 (17 I 4.4267 62010 91351 72440 23067 60799 1)2.9444 (70)3.5355 18234 55935 37938 1)2.9393 868 7 53424 6539 72032 63527 65178 16109 20 3.6824 23 I 3.1926 (70)2.9945 68684 87794 871 7 58641 76311 27839 864 95291 6607 86409 7 46496 86164 30354 42587 (70)3.3455 49727 46809 52418 854 7 29%6 35864 18279 41610 45135 75345 95545 14329 02952 05644 870 7 56900 03000 1 9.5059 5.4137 3.8618 863 7 AA769 .._. 35647 08634 ( 70)3.2543 9.5317 5.4247 3.8681 4.0175 (70)2.5333 ( 1)2.9427 02709 00130 19185 (17 I 05907 ( 1)2.9495 9.5464 5.4310 3.8117 6338 lZOj3.4510 23 2.9644 (26 2.5464 (29 2.1874 (70)2.6051 54753 I 20 3.8645 23 3.3137 17)4.3362 5.4058 3.8573 858 6316 11)5.4193 862 1 43044 05928 43859 26007 52818 70)2.2645 1)2.9223 9.4875 13619 72729 68659 (70)3.1654 58668 6228 11)5.3190 14 4.5424 17 1 3.8792 20)3.3128 23)2.8291 48569 08729 70146 1 20 23 29381 83597 (70)2.7547 ( 1)2.9342 9.5133 5.4168 3.8636 39031 ::i l/5 %9" ;: '02":; 26)2.6003 I 29)2.2388 15789 ( 1)2.9325 i%;; 1:;j l/2 1 I ( 1)2.9257 81600 70176 (70)2.6789 11)5.3941 14)4.6227 17 3.9617 (70)2.3953 860 7 39600 56000 (29)2.2130 6294 20 3.3952 23 2.9096 26 2.4936 29 2.1370 70)2.4634 I :;I 44: '0% 24 9.4801 5.4026 3.8555 _-._. 856 7 327% 22016 ROOTS 72719 I 95693 02137 58534 (11 1 5.3439 24 6184 11)5.2693 14 4.4094 17 3.8250 20 3.2589 23)2.7766 11)5.2200 14)4.4370 (17)3.7714 l/3 l/2 AND METHODS 58173 56899 1 03403 (70)3.8400 93943 (70)3.9470 65953 1)2.9529 60090 08827 70)3.7359 64612 ( 1) ;. ;;y; ;;;g ( 1)2.9563 49100 10846 31924 72857 9.5537 5.4341 3.8734 12362 18707 97651 514356 3.0743 75984 85661 9.5610 5.4372 3.8752 21,EMENTARY ANALYTICAL POWERS 59 METHODS AND ROOTS 6745 11)5.9155 14)5.1879 (17 4.5498 (20 13.9902 (23)3.4994 69129 26133 94186 76101 55041 22871 25458 nk Tabile 3.1 k a75 6699 (70)4.0568 ( 1)2.9580 9.5646 5.4387 3.8761 6814 11)5.9969 24 l/2 l/3 :s: 70)4.6514 1)2.9664 9.5828 5.4465 3.8805 7 65625 21875 90376 39892 55914 86530 59242 7 74400 72000 53600 04745 79395 39714 39631 19041 6122 ;11)5.8886 14)5.1584 '17)4.5188 '20 3.9584 23 3.4676 26 I 3.0376 29)2.6609 70)4.1696 1)2.9597 9.5682 5.4403 3.8770 a76 7 67376 21376 59254 65506 15784 82626 30781 44564 76638 39882 29717 98205 39803 44816 [:~~::E (70)4.2853 ( l’p; 514418 3.8779 881 7 76161 6831 11 6.0242 14 1 5.3073 17 4.6151 20 1 4.1193 23 3.6291 26 1 3.1972 29)2.8168 70)4.7799 1)2.9681 9.5864 5.4480 3.8814 97841 58979 72161 94874 75284 69625 98440 19925 32920 64416 68204 86284 60596 879 7 72641 6791 51439 (11)5.6697 41149 E: 88904 18579 37725 91747 29583 (11)5.9426 (14 5.2176 (17 4.5810 (20 4.0221 (23 i 3.5314 (26 3.1006 (29 12.7223 (70)4.4042 ( 1)2.9631 9.5755 5.4434 3.8788 a78 7 70884 36152 21415 21602 71767 81011 74928 34987 57518 13682 06478 74480 42365 13542 a82 11924 28968 57498 61913 29607 17514 55847 09657 56718 60716 48481 93948 31621 41346 6884 11)6.0791 14)5.3678 17 4.7398 20 1 4.1852 23 3.6956 26 3.2632 29 I 2.8814 70)5.0472 ( 1)2.9715 9.5937 5.4511 3.8832 883 7 79689 65387 49367 88891 45891 83922 05703 19836 23115 74047 31592 16954 75645 21296 884 7 81456 07104 (11)6.1067 34799 14 5.3983 53563 17 4.1121 44549 75782 I 20 I 4.2185 7002 11 6.2180 14 5.5215 17 I 4.9031 20 4.3540 23 1 3.8663 7 88544 27072 16399 98563 19524 23417 72794 877 686: (11)6. 0516 (29)2.8489 (70)4.9118 ( l’y& 5:4496 3.8823 t : : 5 885 7 83225 6931 54125 t t 10 24 l/2 ::: l/5 (70)5.3289 ( 1)2.9748 9.6009 5.4542 3.8849 : : 7049 11365 94956 54166 59763 78808 890 7 92100 69000 886 7 84996 6955 06456 11)6.1621 87200 114)5.4596 97859 17 4.8372 92303 20 1 4.2858 40981 123 3.7912 55109 126 3.3643 68021 29 I 2.9808 30072 70)5.4753 17719 1)2.9765 75213 9.6045 69584 5.4551 99862 3.8858 56313 7073 697: 17587 98020 45144 (70)5.6255 14442 ( 1’;. 2;;; 54522 81682 38658 33146 5:4573 3.8867 891 7 93881 47971 2 .l 1: 24 l/2 :,': l/5 (70)6.1004 ( 1)2.9832 9.6190 5.4619 3.8893 25945 86778 01716 47252 58728 70)6.2670 1)2.9849 9.6226 5.4634 3.8902 895 7169 24 l/2 l/3 (70)6.9783 ( 1)2.9916 9.6369 5.4696 3.8937 8 01025 17375 51604 55060 81200 02417 19006 .y-,$l] 7193 111 1 6.4451 14 5.7748 17 I 5.1742 120 4.6361 23 4.1539 26 3.7219 29 3.3348 70)7.1679 1)2.9933 9.6405 5.4711 3.8945 75070 62311 02990 80860 32348 896 8 02816 23136 35299 41228 57740 34935 76902 63304 79120 04854 25909 69057 29599 88722 887 86769 64103 54594 78425 43063 16896 (70)6.4380 ( u;.;;;; 5:4650 3.8911 7717 892 95664 32288 12009 84312 99206 81692 18069 62918 56123 82017 36905 01570 13179 05185 a97 8 04609 34213 56429 38917 03608 76231 E2” 86240 (70)7.3623 86846 ( 11;. ;;i; 95826 54244 55504 57662 4637)3] 5:4726 3.8954 4-;~2] 676.3 (29j2.7535 (70)4.5261 ( 1)2.9647 9.5192 5.4449 3.8796 6908 29)2.9142 (70)5.1862 ( 1)2.9732 9.5973 5.4527 3.8841 888 3.0488 (70)5.7797 ( 1)2.9799 9.6117 5.4588 3.8876 05069 78281 32885 91067 76153 09128 23268 92303 93416 08475 91658 96696 7025 11116.2460 (70)5.9380 ( 1)2.9816 9.6153 5.4604 3.8884 22119 60897 13749 37224 18358 00450 889 7 90321 95369 72830 28303 10303 97744 12350 84321 R9d II 7121 11 6.3592 14 1 5.6788 17 5.0711 20 1 4.5285 23 4.0440 26 1 3.6112 29)3.2248 (70)6.6135 7 91449 21957 49076 09425 76816 60897 04881 96359 87648 55666 ( 1) ;. ;;;; ;;:;; 514665 3.8919 7241 11)6.5028 14 5.8395 17 1 5.2439 20 4.7090 23 4.2287 26 I 3.1974 29)3.4100 (70)7.5619 ( 1)2.9966 9.6477 5.4741 3.8963 44210 71239 898 8 06404 50792 74112 80953 43696 61439 31112 7 99236 7145 16984 (70)6.7936 ( 1)2.9899 9.6333 5.4680 3.8928 7265 07487 83278 90671 73955 48512 -.. 8 08201 72699 05980 70570 20026 64813 36769 a0133 25828 .:[,-,,l] (70)7.7666 ( 1)2.9983 9.6513 5.4151 3.8971 29743 32870 16634 03489 93220 60 ELEMENTARY POWERS Table 3.1 k 1 2 900 8 10000 7290 00000 (70)8.1920 901 11801 32701 08636 77981 37961 94103 84987 19573 20735 95066 ( 1) 2. pi y”o; 4' 5 7" 8 lo' 24 10 l/3 l/4 l/5 (70)7.9766 ( 1)3.0000 9.6548 5.4772 3.8980 44308 00000 93846 25575 59841 l/2 v3 l/4 l/5 24 l/2 l/3 l/4 l/5 24 l/2 l/3 l/4 l/5 7412 11)6.7080 14)6.0707 17)5.4940 20 4.9721 23 4.4997 26 4.0722 29 I 3.6854 (70)9.1109 ( 1)3.0083 9.6727 5.4848 3.9023 905 8 19025 17625 19506 57653 35676 02287 52570 76076 09848 96943 21791 40271 17035 81426 7535 24 910 8 28100 71000 (71)1.0399 ( 1)3.0166 9.6905 5.4923 3.9066 7660 11 7.0094 14 6.4136 17 I 5.0684 20 5.3696 23 4.9132 26 I 4.4956 29)4.1134 71)1.1860 1)3.0248 9.7082 5.4999 3.9109 04400 20626 21083 77104 83951 915 8 37225 60875 57006 53161 92642 70767 48752 22608 94687 58902 96692 36884 06083 67606 97n .__ 8 46400 7786 88000 24 l/2 l/3 l/4 l/5 ANALYTICAL 29)4.3438 (71)1.3517 ( 1)3.0331 9.7258 5.5074 3.9152 84542 85726 50178 88262 04268 32576 +-;P] 514787 3.8989 7436 MFlTIODS AND ROOTS 7338 (11)6.6195 14)5.9708 17)5.3856 902 8 13604 70808 14688 02249 63628 (70)8.4131 ( 1)3.0033 46393 25692 9"h .__ 8 20836 77416 7461 ( 1';. y; 5:4863 3.9032 7560 (71)1.0676 ( 1)3.0182 9.6940 5.4938 3.9075 768; (71)1.2175 ( l);.;;;; 5:5014 3.9118 7812 (71)1.3874 ( 1)3.0347 9.7294 5.5089 3.9160 908 20)5.0886 70117 14616 (70)9.8641 y; ( "6 3.9041 58031 7585 04712 912 8 31744 50528 7610 11)6.9483 (71)1.0961 ( 1)3.0199 9.6976 5.4953 3.9083 916 39056 15296 49711 77136 79856 85148 70796 35649 72654 62793 ;;;;i 08174 917 40889 77180 95213 ii ;. 07:; 43103 (ii 54826 17 519458 78275 20 5.4523 70378 23 4.9998 23637 26)4.5848 38275 (29)4.2042 96698 (71)1.2498 67732 ( 1);;;;; 00;;;; 5:5029 09036 3.9126 29961 94035 98181 10859 00236 83344 7837 1 23 5.2221 26 4.8148 29 I 4.4392 (71)1.4241 ( 1)3.0364 9.7329 5.5103 3.9169 q-37)3] 65476 33774 15172 92410 99668 (71)1.1253 ( 1)3.0215 9.7011 5.4968 3.9092 7736 30769 y;;;; 56824 65216 48497 72778 09448 70729 93183 51482 35404 08671 3.9135 28819 77448 7063 1 (71)1.4616 ( 1)3.0380 9.7364 5.5118 3.9177 q-37)2] 923 8 51929 30467 41363 91506 48410 88520 82664 909 8 26281 89429 (11)6.8274 02910 (14)6.2061 09245 17)5.6413 53304 20)5.1279 90153 43049 I 23)4.6613 26)4.2371 60832 79196 I 29)3.8515 (71)1.0128 22166 ( 1)3.0149 62686 9.6869 70141 5.4908 67587 3.9058 24962 7510 914 8 35396 51944 64760 82398 15712 25761 55345 (71)1.1553 1)3.0232 9.7046 5.4984 3.9101 918 8 42724 20632 (29)4.2503 (71)1.2829 ( 1)3.0298 5.5044 9.7188 904 8 172ib 63264 19907 91596 11602 71289 29245 56837 88981 24888 59276 76254 01264 18640 7635 11 1 6.9788 14 6.3786 17 I 5.8301 20 5.3287 23 4.8704 78622 88986 58327 98203 56397 5.0436 75026 26266 00417 45985 05308 45290 30906 94986 33373 13312 08873 18909 47257 65825 ;'8;: 5:4893 3.9049 79852 77655 69425 85370 42186 22089 744h4 _ _.._. R 7486 I 11 I 6.1720 17 5.6042 14 6.7974 24265 31551 43449 7387 11 6.6784 14 16.0372 17)5.4577 20 4.9337 23 1 4.4601 26)4.0319 29)3.6448 70)8.8724 1)3.0066 9.6691 5.4833 3.9015 46120 95840 09608 84235 55089 5.4817 9.6656 3.9006 907 8 22649 42643 3.7676 (70)9.6067 09844 9n3 8 15409 14327 18373 73291 87881 93857 11553 (70)8.6398 ( 1)3.0049 40328 65946 90774 4.5799 (70)9.3557 7363 11 6.6489 6.0039 14 17 I 5.4215 20 4.8956 / 23 1 4.4208 16465 31484 5.4802 9.6620 3.8997 nk 1 “[‘-.;‘l] 37042 43292 98896 02760 12376 7761 11 I 7.1328 14 6.5550 17 6.0241 20 I 5.5361 23 5.0877 26 4.6756 29 4.2968 (71)1.3169 ( 1)3.0315 9.7223 5.5059 3.9143 919 ._. 8 44561 51559 32827 73368 12425 59319 30414 24251 98666 59057 01278 63112 07081 81068 (11)7.2893 (14)6.7353 7888 (71)1.5001 ( 1)3.0397 9.7399 5.5133 3.9186 924 8 53776 89024 34582 45154 24518 36831 63373 80842 31220 :ELEMENTARY ANALYTICAL POWERS 11)7.3209 14)6.7718 17)6.2639 20 5.7941 23 1 5.3596 1) ;. y;; (11)7.4805 (14)6.9568 24 930 8 64900 57000 20100 83693 (71)1.7522 l/2 l/3 l/4 l/5 ( 1)3.0495 9.7610 5.5223 3.9237 28603 90136 00077 09423 07185 l/3 l/4 l/5 71)1.9928 1)3.0577 9.7784 5.5297 3.9279 8305 24 :;: l/4 l/5 : 29)5.3861 71)2.2650 1)3.0659 9.7958 5.5370 3.9321 68584 76970 61652 16964 17180 :71)2.0446 [ 1)3.0594 9.7819 5.5311 3.9287 940 8 83600 84000 51141 01461 41943 61087 94855 09204 (71)2.3235 ( 1)3.0675 9.7993 5.5385 3.9329 945 8 93025 8436 08625 l/2 :;: l/5 71)1.6214 1)3.0446 9.7504 5.5178 3.9211 8095 (11)7.5450 (14)7.0320 (71)2.5725 ( 1)3.0740 9.8131 5.5444 3.9362 932 8 68624 57568 76534 11329 (29)4.9449 (71)1.8449 ( 1';;;;; 10997 29260 97390 93317 50630 18334 38512 fJ;J;; 5:5252 3.9253 71)2.0977 1)3.0610 9.7854 5.5326 3.9295 56558 11708 46493 94905 57017 8 85481 8358 11 7.8741 14 I 7.4174 17)h.9872 20)6.5819 23 6.2002 26 15.8406 29)5.5018 (71)2.3835 ( 1';.;06;; 37621 66014 60819 49330 21220 61168 43259 25707 44328 72330 33566 66899 45467 515400 3.9337 04376 47511 85230 98931 43371 83427 ( 1)3.0757 9.8166 5.5459 3.9371 1 q--y] 11300 59156 09574 16151 nk 5:5193 3.9220 (11)7.5775 (14)7.0698 (17)6.5961 (20)6.1541 (23)5.7418 8492 6.1256 q-3”3] (11)7.4483 (14 6.9195 (17 1 6.4282 (20 5.9718 (23 I 5.5478 (26)5.1539 (29)4.7880 (71)1.7075 ( 1)3.0479 9.7575 5.5208 3.9228 934 8 72356 80504 49907 86613 72697 (7131.9423 ( 1)3.0561 9.7749 5.5282 3.9270 57521 35348 38996 41358 14326 37837 76625 019 938 8 79844 8252 93672 11 7.7412 54643 14 1 7.2612 96855 17 6.8110 96450 20 1 6.3888 08471 23 5.9927 02345 26 1 5.6211 54800 29)5.2726 43202 71)2.1521 28115 1)3.0626 78566 9.7889 08735 5.5341 47239 3.9304 34540 8385 11 7.9076 14 1 7.4569 17)7.0318 20 6.6310 23 6.2530 26 I 5.8966 29)5.5605 (71)2.4450 ( 1)3.0708 9.8062 5.5415 3.9346 929 8 63&l 65089 97677 61442 72579 65226 62795 64537 33055 64573 50131 00256 24332 63013 8147 11 I 7.6100 14 7.1077 17 6.6386 38514 ;iJ;o" 3.1 I/ 8 81721 8279 36019 (11)7.7743 19218 (14)7.3000 85746 (17 6.8547 80516 (20 16.4366 38904 (23 6.0440 03931 (26 5.6753 19691 (29 I 5.3291 25190 71)2.2078 73640 1)3.0643 10689 9.7923 86145 5.5356 21636 3.9312 72229 943 8 89249 61807 37840 02483 59042 43076 73621 48424 39464 09921 30507 71149 07472 15863 8412 11)7.9412 14 7.4965 17 7.0767 20 I 6.6804 5.6197 (71)2.5080 ( 1)3.0724 9.8091 5.5429 3.9354. 948 8 98704 8519 71392 (11)8.0766 88.796 8546 8 91136 32384 33705 24617 19239 22962 88134 01911 58299 36263 76005 49998 $1 00601 70349 16600 (71)2.7064 46809 :;;:: 5:5473 3.9379 8017 933 8 70489 66237 10991 17755 39965 98588 67282 5:5267 3.9262 8 96809 78123 ( I';.;;;; I 8121 (71)1.8930 ( 1y;: 32860 45573 28852 71663 96137 37771 81020 928 61184 78752 78819 99544 66776 12369 67478 48220 42348 92748 09242 97922 38042 18115 ( "y;; 76015 93351 942 8 87364 9688.3 48685 48061 36074 76381 21751 08890 53574 35733 ;;;;; Table 799; (11)7.4163 (14 1 6.8823 (17 6.3868 (20 I 5.9270 (23 5.5002 (26 5.1042 (2934.7367 (71)1.6639 937 8 77969 8226 56953 (11 7.7082 95650 jl417.2226 73024 17) 6.7616 44623 946 8 94916 8465 90536 11 8.0087 46471 14 7.5762 74161 17 7.1671 55356 20 1 6.7801 20967 23 6.4140 02003 26 6.0676 45895 29 I 5.7399 93016 (71)2.6386 83331 ? 5.6796 927 59329 97983 63302 97481 83465 48572 29826 58649 46668 87554 61470 93072 50550 72488 941 8332 : 5 i 10 24 AND ROOTS 796: 11 i 7.3844 14 6.8453 17 6.3456 20 5.8824 23 5.4530 936 8 76096 8200 25856 11 7.6754 42012 '14 1 7.1842 13723 6.7244 24045 6.2940 60906 5.8912 41008 F l/2 931 8 66761 54491 71)1.7980 1)3.0512 9.7644 5.5237 3.9245 : 3 4 5 t 10 24 926 57476 22776 50906 54739 21688 72283 47534 92617 41763 23988 24811 85700 61854 26131 8069 %: 71952 79042 5:5148 3.9194 8043 794: (11 I 7.3526 (14 6.8085 (17)6.3047 (20)5.83'31 (23 I 5.4061 (26 5.0060 (29)4.6356 (71)1.5800 ( 1)3.0430 9.7469 5.5163 3.9203 925 55625 53125 41406 70801 80491 81954 18307 46934 23414 77607 61 METHODS 74614 48170 I q-37)2] 29)5.8625 (71)2.7758 06988 76218 ( 1)3.0789 9.8235 5.5488 3.9387 60864 72299 38494 79481 I q-y (71)2.8470 ( 1)3.0805 9.8270 5.5503 3.9396 10693 84360 25224 01217 10103 I 62 ELEMENTARY Table 3.1 ANALYTICAL POWERS AND METHODS ROOTS I& k : : 5 8573 111\8.1450 950 9 02500 75000 62500 8600 b fi 2b)b.3024 1'0 24 l/2 l/3 l/4 l/5 (29)5.9873 (71)2.9198 ( 1)3.0822 9.8304 5.5517 3.9404 94097 62784 93712 55775 28789 23805 23198 69392 90243 07001 75725 40019 (71)2.9945 ( l'y& 5:5532 3.9412 69236 873; 96006 90686 29105 53295 39397 13936 22816 90121 b7356 15592 23306 10280 l/2 1/3 1/4 l/5 (7u3.3119 ( l)3.0903 9.8476 5.5590 3.9445 8847 28238 07428 92005 53362 79145 960 9 21600 36000 20)7.5144 7.2138 6.9253 (29) 6.6483 32467 8986 11)8.6718 24 l/2 l/3 :,/2 a875 (71)4.2526 ( 1)3.1064 9.8819 5.5735 3.9528 9126 86677 48297 15367 00972 965 9 31225 32125 00006 09649 44913 45122 49061 05659 970 9 409on 73000 49474 69948 24967 28046 08040 04889 (71)3.0710 ( 1'9'. 8";;; 49109 49724 5:5546 3.9420 82461 97756 03681 69469 957 9 15849 8764 (li)a.3877 (14)8.0271 (17)7,6819 20 7.3516 23 7.0355 26 I 6.7329 I (29)6.4434 (71)3.4824 ( 1)3.0935 9.8545 5.5619 3.9462 67493 93908 la770 52663 28698 08664 al792 63575 b2966 41660 61691 61578 29943 a902 f11)8.5644 __.. 77128 65971 36200 05288 (7u3.8491 ( 1)3.1000 9.8682 5.5677 3.9495 9014 11 8.7078 14 1 a.4117 17)8.1257 20)7.8494 23 7.5825 26 7.3247 29 I 7.0757 (71)4.3596 ( 1)3.1080 9.8853 5.5749 3.9536 9154 lab99 (71)3.9464 05693 12484 94135 00000 72403 64363 23275 12228 44894 966 9 33156 01203 35962 36940 61884 971 9 42841 98611 (ll)a.2484 (14 7.8607 (17 7.4913 (20 7.1392 (23 I 6.8036 (71)3.1494 ( i)3.0870 9.8408 5.5561 3.9429 (71)4.4692 ( 1)3.1096 9.8887 5.5764 3.9544 9183 2::: 57504 62361 67316 34668 42771 972 9 44784 30048 23177 35877 59391 03699 954 9 1Olib 8682 50664 11335 88213 92155 lb916 (71)3.2296 ( i)3.0886 9.8442 5.5575 3.9437 12425 (11)8.2831 (14)7.9020 (17)7.5385 (20)7.1918 91146 a9042 53565 97541 52709 b9441 12996 69808 12721 40574 25580 __- 959 9 19681 8819 74079 (11)8.4581 31418 (14)8.1113 48029 9 17764 8792 17912 (11 a.4229 07597 (14 1 a.0691 45478 (17)7.7302 41368 (20)7.4055 71230 (23)7.0945 37239 1 :g: ::z? (23)7.1539 6.5793 96686 (71)3.5708 ( 1)3.0951 9.8579 5.5634 3.9470 (71)3.6613 ( 1)3.0967 9.8614 5.5648 3.9478 94899 72513 71813 55021 57508 92945 13977 54307 .-_ 9 27369 '3930 56347 (26)7.1225 (29)b.a590 (71)4.0460 ( 1)3.1032 9.8751 5.5706 3.9511 967 35089 31063 14379 65205 38153 18994 03867 28696 80180 72454 30190 44069 54054 57396 92425 24554 953 9 08209 8655 962 961 9 23521 10374 a2870 27838 39352 31217 74781 95790 39958 26360 (71)3.7541 ( 1)3.0983 9.8648 5.5663 3.9487 71)3.3961 1)3.0919 9.8511 5.5605 3.9454 952 Ob304 01408 69404 38087 65059 20336 20160 956 26228 24 862: 77556 53956 955 9 12025 a709 83875 (li)a.3l7a (14)7.9435 (17 7.5861 (20 1 7.2447 (23) 6.9187 951 9 04401 a5351 11688 20515 ball0 92173 9070 (29)7.2235 (71)4.5815 ( 1)3.1112 9.8921 5.5778 3.9552 9211 11)8.9629 14 8.7209 17 I a.4854 20)8.2563 23)8.0334 67671 z: 98708 65240 77983 ._. 9 29296 a958 41344 (11 8.6359 10556 (14 1 8.3250 17776 (17 a.0253 17136 (20 1 7.7364 05719 (23)7.4578 95113 32667 46699 24130 13495 58964 65831. 968 9 37024 39232 98067 09331 69837 74886 75794 60312 (71)4.1480 ( i)3.1048 9.8785 5.5721 3.9519 96142 34939 30490 04575 86085 ,-_ 9h9 9 38961 9098 53209 11)8.8164 77595 14 8.5431 66790 (17 a.2783 28619 (70 a.0217 00432 (23 I 7.7730 27719 (71)4.6964 ( 1)3.1128 60232 9.8955 971 .._ 9 467;!9 673:L7 57994 58129 922'59 a3968 61601 76483 80110 5.5793 3.9560 15803 77177 914 9240 9 486% 10424 29)7.ba40 24 l/2 l/3 l/4 l/5 (71)4.8141 ( 1)3.1144 9. a989 5.5807 3.9568 72219 82300 a2992 54698 93368 (71)4.9347 ( 1)3.1160 9.9023 5.5821 3.9577 08664 87290 83537 92482 08886 (71)5.0581 ( 1)3.1176 9.9057 5.5836 3.9585 34323 91454 a1747 29155 23732 (71)5.1845 ( 1)3.1192 9.9091 5.5850 3.9593 15371 94792 77627 64719 37908 1 39148 (71)5.3139 f i)3.1208 ,~-9.9125 5.5864 3.9601 19427 97307 71181 99178 51415 ELEMENTARY ANALYTICAL POWERS k 63 METHODS AND ROOTS Table nk .._ 975 : : 2 7 (11 (14 17 20 23 9268 I 9.0368 8.8109 8.5906 18.3759 8.1665 9” 10 24 (71)5.4464 ( 1)3.1224 9.9159 5.5879 3.9609 m l/3 l/4 l/5 15584 98999 62413 32533 64254 (71)5.5820 ( 1)3.1240 9.9193 5.5893 3.9617 74443 99870 51328 64785 76427 (71)5.7209 ( 1)3.1256 9.9227 5.5907 3.9625 68141 99922 37928 95938 87934 9354 11)9.1486 14)8.9413 17)8.7505 20)8.5579 23 8.3697 26 8.1855 29 I 8.0055 71)5.8631 1)3.1272 9.9261 5.5922 3.9633 978 9 56484 41352 16423 46861 05230 94115 18245 84443 01586 70383 99154 22218 25992 98776 9498 9 50625 59375 78906 56934 83010 15935 18037 983 9 66289 62087 .__ WI” : 9 60400 92000 (11)9.2236 81600 (14)9.0392 07968 (17 8.8584 23809 20 1 8.6812 55332 23)8.5076 30226 9411 : 2 l3 lo' 24 l/2 l/3 l/4 l/5 I (71)6.1578 ( 1)3.1304 9.9328 5.5950 3.9650 03365 95168 83884 82813 18474 i 11 I19.1784 23 8.7182 20 8.8689 17 9.0223 14 9.3371. 71)6.3103 1)3.1320 9.9362 5.5965 3.9658 89657 91953 61267 09584 27331 (71)6.4665 ( 1)3.1336 9.9396 95666 87923 36356 5.5919 3.9666 35265 35529 986 : 9615 11 9.4900 I4 9.3666 17 I 9.2449 : l/2 l/3 l/4 l/5 : 3 : ! 8 9 10 24 ://: $2 (71)6.9577 ( 1)3.1384 9.9497 5.6022 3.9690 9702 (11 9.6059 14 9.5099 I 17 9.4148 20 9.3206 I 23 I 9.2274 26)9.1351 I 29)9.0438 (71)7.8567 ( 1)3.1464 9.9665 5.6093 3.9730 61406 70965 47896 05785 56179 990 9 80100 99000 60100 00499 01494 53479 46944 72475 20750 81408 26545 54934 01690 77521 (71)7.1292 ( 1)3.1400 9.9531 5.6036 3.9698 84708 63694 13846 27123 61152 ( 71)7.3048 1)3.1416 9.9564 5.6050 3.9706 ( 1)3.1480 9.9699 5.6107 3.9738 15248 09547 17644 79839 (71)8.2466 ( 1)3.1496 9.9732 5.6121 3.9746 991; (11)9.8805 (14)9.8508 l/2 : :: l/5 59857 43069 988 9 76144 9644 30272 (71)8.8665 ( 1)3.1543 9.9833 5.6163 3.9770 35105 62059 05478 70767 82648 +y] [ ::I z: :::: (17 9.3012 (71)9.0828 ( 1)3.1559 9.9866 5.6177 3.9778 91413 46768 48849 81384 81740 ,:[(-231 (71)9.3043 ( 1)3.1575 9.9899 5.6191 3.9786 984 9 68256 63904 69325 14296 56083 55614 77521 47381 66671 98779 03150 61904 32527 81509 9791 11 i 9.7229 14 9.6548 17 9.5872 20 9.5201 23 9.4535 26 9.3873 29 9.3216 (71)8.4485 ( 1)3.1511 9.9766 5.6135 3.9754 9673 96812 77428 19667 83363 49952 9 78iil 61669 66691 42406 I 29 8.9704 26 8.8621 (71)7.4845 ( 1)3.1432 9.9598 5.6064 3.9714 (71)6.7901 ( 1)3.1368 9.9463 5.6007 3.9682 EZ 57495 I 996 24 987 9 74169 04803 52406 81724 14862 5: 5993 3.9674 I 20 9.1896 23 9.0193 2 i 9 10 24 03443 ;;3;; 9527 56477 97569 04202 54950 08956 79843 12862 44315 26396 99386 (71)6.6265 ( l';.;:;; (71)6.0087 ( 1)3.1288 9.9295 5.5936 3.9642 3.1 997 94009 26973 38921 97304 44612 80578 04937 11422 17769 02025 30681 89983 90939 80191 “q-p] (11 (14 (17 66822 46729 38925 66560 70939 993 9 86049 46657 26304 65820 81759 70787 29591 54884 43400 45822 90251 12009 46340 82534 99490 9.9202 9.9003 9.8805 I (29)9.8017 (71)9.5308 ( 1)3.1591 9.9933 5.6205 3.9794 8.9528 (71)7.6685 ( 1)3.1448 9.9631 5.6078 3.9122 83144 10178 37039 98061 84662 74555 994 9 88036 (71)8.6551 ( 1)3.1527 9.9199 5.6149 3.9762 22630 76554 59866 59086 82913 (71)9.7627 ( 1)3.1606 9.9966 5.6220 3.9802 39866 96126 65555 06871 75173 998 96004 11992 39680 99201 98402 37206 15531 iEi 79767 13800 28884 99434 78001 .:[(-y] . 4. Elementary Logarithmic, Expwential, Transcendental Circular Functions and Hyperbolic Functions RUTY ZUCKER l Contents Page 67 Mathematical Prcbperties ..................... 4.1. Logarithmic Function ................... 4.2. 4.3. 4.4. 4.5. 4.6. Numerical Methods References 4.1. log,, 5, 71 79 83 86 ....................... 4.7. Use and Extension Table 67 69 Exponential Function ................... Circular lknctions .................... Inverse C rcular Functions ................. Hyperbol:.c Functions ................... Inverse Hyperbolic Functions ............... of the Tables 89 .............. 89 ......................,,... Commcln Logarithms (100&11350) 2=1(0(1)1350, 10D ........... 93 95 Table 4.2. Natural Logarithms (0 5s<_2.1) In z, s=O(.O101)2.1, 16D . , , . , . . . . . , , . 100 Table Radix ‘Table of Natural Logarithms . . . . . . . . . . . . 114 -in (l-cc), r=lO-n(lO-n)lO-n+l, n=lO(-1)1, 2511 ln (l+z), Table 4.4. Exponential ez, fz=o(.o31)1, x=5(.1)10, *tz=0(1)100, Table 4.5. Radix Table of the Exponential Function . . . . . . . . . 140 ez, eez, x=1(1-n(lO-“)lO-“+l, n=lO(-l)l, 25D Table 4.6. Circular Sines and Cosines for Radian Arguments sin 2, co9 5, ~=0(.001)1.6, 23D Table 4.7. Radix Table of Circular Sines and Cosines . . . . . . . . . 174 sin 2, cos 2, ;c=lO-n(lO-“)lO-n+‘, n=lO(-1)4, 25D 4.3. Function (0 5 1215 100) , , . . . . . . . . . . 116 18D, s=O(.1)5, ’ 15D 12D, --2=0(.1)10, 20D 19s (0 <z < 1.6)1 . 142 4.8. Circular Sines and Cosines for Large Radian Arguments (0~z~1000). . . . . . . . . . . . . . . . . . . . . . . . . 175 sin 2, cos 2, ;e=O(l)lOO, 23D, ~=100(1)1000, 8D Table 1 National Bureau of Standards. ELEMENTARY Table TRANSCENDENTAL .FUNCTIONS Tangents, Cotangents, Secants and Cosecants for Page (0 <z 5 1 .S) . . . . . . . . . . . . . . . . . . 186 tm 5, cot x, set 2, csc r, 2=0(.01)1.6, 7 to 9D 4.9. Circular Radian 5 Arguments -1 -cot csc x, x=0(.01).5, x-x-l, 8D Table 4.10. Circular Sines and Cosines to Tenths of a Degree (0’ 5 8 5 90”) . 189 sin 8, cos 8, 8=O”(.lo)900, 15D Table 4.11. Circular Five Tenths tan e, cot set 8, csc Tangents, Cotangents, Secants and Cosecants to of a Degree (0” 5 0<90’) . . . . . . . . . . . . . . 198 8, e=o”(.50)900, l5D 8, e=O”(.50)900, 8D Table 4.12. Circular Functions for the Argument fz sin %, cos %, tan %, cot %c, set %, csc L, 2 2 2 Table 4.13. Harmonic sin *‘T, cos z, Analysis. r=l(l)[s/2], 2 s=O(.Ol)l, 2 20D . . . . . . . . . . . . , . . . . . 202 s even s odd T&&2], s=3(1)25, 2 (05 z < 1) . . . . 200 10D Table 4.14. Inverse Circular Sines and Tangents (O<z<l). arcsin 2, arctan 2, x=0(.001)1, 12D f(z)=[2(1-x)1-f&r-arcsin z], z=.95(.001)1, l2D . . . . , . 203 Table 4.15. Hyperbolic Functions (0 5 az 10) . . . . . . . . . . . . 213 5 sinh x, cash x, x=0(.01)2, 9D, 2=2(.1)10, 9D tanh z, coth 5, x=0(.01)2, 8D, 7D, 2=2(.1)10, 10D Table 4.16. Exponential and Hyperbolic Functions for the Argument KZ (O<x<l) . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 e**=, sinh ?rx, cash ?rx, tanh ?rz, z=O(.Ol)l, 10D Table 4.17. Inverse Hyperbolic Functions (0 5~5~)) . . . . . . . . . 221 arcsinh x, arctanh 5, z=O(.Ol)l, 9D arcsinh x, (arccosh X)/(X?- l)‘, 2= 1(.01)2, 9D, 8D arcsinh z-ln z, arccosh z-ln 2, s-‘=.5(-.01)0, 10D Table 4.18. Roots z, of cos x, cash z~= f 1, n= 1(1)5, Table 4.19. Roots 2, of tan zn=kzn -x=0(.05)1, n=1(1)9, 5D x-*=-1(.05)1, n=1(1)9, (--00 <AlO, 7D . . . . . . 223 l<Xlm) . . . . . 224 5D Table 4.20. Roots 2, of cot x,,=kr,, (0 <Xla X=0(.05)1, n=l(1)9, 5D X-‘=1(-.05)0, n=1(1)9, 5D ) . . . . . . . . . . . 225 The author acknowledges the assistance of Lois K. Cherwinski io the preparation and checking of the tables. and Elizabeth F. Godefroy . 4. Elementary Logarithmic, Transcendental Exponential, Circular Mathematical 4.1. Logarithmic Integral 4.1.1 and Hyperbolic IFunctions Properties Function Representat In z= Functions Logarithmic 4.1.6 ion Identities Ln (z,z2)=Ln zl+Ln z2. (i.e., every value of Ln (z1z2) is one of the values of Ln zr+Ln z2.) z dt 17 S 4.1.7 In (z1z2)=ln q+ln z2 iY (--?r<arg 4.1.8 n 0 1 4.1.9 x -I Ln z=Ln zl--Ln In z=ln z,-ln z=r+iy 4.1. (a not an integer or zero.) where the path of integration does not pass through the origin or cross the negative real axis. In z is a single-valued function, regular in the z-plane cut along the negative real axis, real when z is positive. 4.1.3 In z=ln r+iO 4.1.11 In P=n In 2 (n integer, Special r= (x2+y2)*, 4.1.13 In 0=- In (&i)=f& In e=l, S 6 dt 1 t=’ “=2.71828 18284.. . (see 4.2.21) The general logarithmic function is the manyvalued function Ln z defined by Logarithms 4.1.18 S 03 e is the real number such that x=r cos 0, y=r sin 0: Lnz= arg 25~) ln (-l)=*i e=arctan f. 4.1.4 -s<n In l=O 4.1.12 4.1.16 z dt 1t (n integer) (see chapter 1) Values 4.1.15 (-*<OS*). z2<n) Ln zn=n Ln 2 4.1.14 z=x+iy=reie. 4.1.2 zl-arg ;z2 (-r<arg Brunch cut for, In z and 9. z217r) z2 4.1.10 FIGURE zl+arg to General log, z=ln Base z/in 1s 4.1.19 where the path does not pass through the origin. 4.1.20 4.1.5 Ln (re@)=ln (refe)+2kti=ln k being an arbitrary integer. principal branch of Ln z. r+i(e+akr), In z is said to be the 4.1.21 4.1.22 log, z=ln 2 log,, z=ln z/in lO=log,, e In z = (.43429 44819 . . .)ln 2 67 68 ELEMENTARY TRANSCENDENTAL 4.1.23 ln z=ln 10 log,, z= (2.30258 50929 . . .) loglo z (log, x=ln x, called natural, Napieriun, or hyperbolic logarithms; log,, x, called common or Briggs logarithms.) Series 4.1.24 Expansions FUNCTIONS 4.1.35 4.1.36 4.1.37 In (1+2)=2-~2z++23-. .. and z#-1) 111 In x<n(x 4.1.38 ,.. (/z/11 Iln (l--x)\< I’*-- 1) for any positive /In ($J+ (1+2)(:2-ln In z=(z-1)-~(z-1)*+Q(z-l)3-. (Iz-1111, ZZO) (sy+. . .] In (%)=2 (k+&+&,+ (z in the cut plane of Figure Polynomial zffl) a+2 (&> (a>O, Limiting * t=(x-1)/(x+1) al= .86304 (&J5+ * . .] a,=.36415 4.1.42 921--a#z) log10 x=a~t+a~t~+a~t~+a,t’+u~t”+~(x) Values lim x-0 In x=0 z+m (a constant, 4.1.31 4.7.) le(x)l56XlO-’ [(&)+f +f 4.1.30 Approximations log10 x=alt+u3t”+t(x), 4.1.29 In (z+a)=ln .. 4.1.41 . . .) jl421, 2 42 42 92 422 =---2;: 22 --. 922 . . l-- 3- 5- 7- z#O) 0, Fractions 4.1.40 4.1.27 (922 W<1) (z in the plane cut from - 1 to - co) .. (sy+i n (l-120 In (1+z)=*&GG5+6+. 4.1.26 In 2=2 [(s)+i (XX) 4.1.39 (92 13) 4.1.28 2_<2-1 Continued (?J+a (O<zI.5828) (XX) 4.1.25 In z=(G)+: g lim xa In x=0 z--10 (CYconstant, t=(x-.1)/(x$1) je(x)/_<lO-’ 9&~>0) aI=. 1718 u3= .28933 5524 2&~>0) a’= .09437 6476 f&=.19133 7714 as= .17752 2071 4.1.32 4.1.43 lim $I i-ln m-+m( m =y (Euler’s > =.57721 56649 . . . (see chapters 1, 6 and 23) Inequalities 4.1.33 ex <In z<-In (l--2)< 1 In (1+x)=u1x+~x2+u3x3+u4x4+usx5+~(x) Is(x)/ 5 1 x 10-s UI= .99949 556 x#O) & (x<l, ‘XZO) a4= - .13606 275 a2= - .49190 896 U+X)<J: (x>-1, 4.1.34 01:x< constant) as= .03215 845 a3= .28947 478 * The approximations ings, Jr., Approsimations ,Univ. Press, Princeton, 4.1.41 to 4.1.44 are from C. Hastfor digital computers. Princeton N.J., 1955 (with permission). ELEiMENTARY 4.1.44 TRANSCENDENTAL O_<z<l 4.1.52 In (1+2)=a~z+u~22+~~23+uq~4S~u~25+ug~6 4.1.53 fw’ + fws+f (2) Approximation 64239 41238 90258 38084 a$= . 16765 ue= --. 09532 u7= . 03608 us= --. 00645 in Terms 4.1.45 S ln [z+(z2&1)(]dz=z le(z)1<3XlO-* uq= . 99999 u2= -. 49987 a,= . 33179 u4= -. 24073 69 FUNCTIONS of Chebyohev S zn In [z+(z2&l)i]dz= 2 S -&+f)+ Definite In (l+d=n$o (see chapter 22) In [z+(z2*l)+] -- & 3 O<z<l Z’,*(X)=COS n8, cos ~=!kc-l [2+(22&1)+]-(22511)) 4.1.54 40711 93897 84937 35442 Polynomials In dz (n#--1) Integrals 4.1.55 -4,T,,*@) 4.1.56 n n AZ 0 . 37645 2813 1 34314 5750 2 -: 02943 7252 3 .00336 7089 4 -. 00043 3276 5 . 00005 9471 A, 6 -.. 00000 00000 ; -: 00000 9 00000 10 -: 00000 11 . 00000 Differentiation 8503 1250 0188 0029 0004 0001 4.1.57 rdtT ln t-Zz(x) 4.2. Exponential Beries Function Expansion 4.2.1 Formulas z 22 e’=exp ~.=l+~+~+~+ 4.1.46 5.1.3) (see 0 z3 ... (z=z+iy) & In z=Jj where e is the real number defined in 4.1.16 4.1.47 &In Fundamental z=(-l)n-l(n-l)!z-fl 4.2.2 Integration 4.2.3 4.1.49 4.1.50 s S 2 4.2.4 In z dz=z In z---z p+l 4.2.6 n integer) S zn (ln z)“-‘dz Clenshaw, functions, (with per- z Powers then z=Log, N a”= exp (2 In cc) If u=juj exp (i arg a) l&l = ((+-Y arg (u’)=y (--?r<arg ul?r) *rg = In lul+z arg a 4.2.11 Ln u’=z In a 3 The approximation 4.1.45 is from C. W. Polynomial approximations to elementary Math. Tables Aids Comp. 8, 143-l‘L7 (1954) mission). of General If N=u*, 4.2.9 4.2.10 (n#-1) (--?T<:J~IT) d z exp z=exp 4.2.7 4.2.8 4.1.51 S (k any integer) exp (ln 2) = exp (Ln 2) = 2 Definition (n#-1, z?+l (In 2)” --- nt n-t-l n+l In (exp z)=z 4.2.5 zn In zdz=- Zn+l ln z-n+l (n+02 z” (In 2)‘” dz= Ln (exp 2)=2+2k7ri Formulas s dz T=ln 4.1.48 Properties 4.2.12 4.2.13’ for one of the values of Ln uz In uz=z ln a (a real and positive) le*[=ez , 70 ELEMENTARY TRANSCENDENTAL arg (eZ) y = &l&Zz=&3 4.2.14 4.2.15 4.2.16 u%*= (cd)" (--?r<arg FUNCTIONS Special a+arg bS?r) 4.2.22 4.2.23 4.2.24 Values (see e=2.71828 18284 . . eO=l em= co 4.2.25 4.2.26 e-- =0 efd=-l 4.2.27 e*?= fi 4.2.28 e2*ki = (k any integer) 1 Exponential Inequalities If 2 is real and different -z 4.2.29 e l-r<l-x<e-z 4.2.30 4.2, Logarithmic FIGURE Periodic 4.2.17 and exponential junctions. Property 4.2.32 Exponential 4.2.18 4.2.19 (k any integer) (- ?r<921< 7r) 4.2.33 (x>-1) x<(e"--I)<& l+x>t!G 4.2.35 (x<l) (x>-1) e">l+$ ( Values 4.2.37 (Jarg 21_<p-t<+n, (n>O, z>O) > ‘>,ezs P<l-; (O<xI 4.2.38 $4<le' -li<~l4 4.2.39 6.5.11) (~>O, Y>O) 1.5936) (Yconstant) Continued e,(z) see +x<(l-“-q<x 4.2.36 eZ> 1-t; 4.2.21 (For @a 5 r) can be removed 4.2.20 /iy zae-z=O *m e'<j-$ 4.2.34 (ezl)Zs=~rlr2 Limiting @-a Identities &z22=ez1+z2 The restriction (- n<Yq if z2 is an integer. from zero ez>l+x 4.2.31 &Wkl= ,f 1) chapter (ez-l(<e~“-ll(z~e’r~ Fractions @<14<1) (all z) ELEMENTARY TRANSCENDENTAL 4.2.42 Approximations 2a nrctnn e tzl 2a z--a+ I a?+4 a2+9 32+ !iz+ 72+ * ” a2+1 71 FUNCTIONS in Terms 4.2.48 of Chebyshev 4.2.43 Approximzltions O<x<ln T:(x) =cos no, cos 8=2x-- 1 (see chapter 22) e-=2 ’ 2=.693 1.75338 7654 .85039 1654 . 10520 8694 .00872 2105 * 00054 3437 .00002 7115 . 00000 1128 . 00000 0040 e-==1+u~x+u2x2$~E(x) le(x)J<3XlO-” a,=-.9664 u2= .3536 4.2.44 O<x<ln 2 . 00000 4.2.45 O<x<ln 2 d dz ez=e” 4.2.50 d” dz” ear= anen’ d d=d dz e-z=1+a~x+a2x2+~3x3+~4x4+~~~5 +u6x*+u,x’+E(x) le(x)112X10-10 a’=--.00014 $ Y=(l+ln u5= -- .00830 13598 u6= .00132 98820 aa=-. u4= 4.2.46 6 Integration 13161 .04165 73475 4.2.54 J JE(x)~~7X10-4 a2= .6774323 4.2.47 u3= .20800 30 a4= .12680 89 O_<x<l 10z= (1 +alx+u2x2+a3x3+u4x4t a& +w6+~7~7)2+~(~) a2= 15129 277603 .66273 088429 u3= .25439 357484 u4= .07295 173666 us=:. 01742 [(uz)n-n(az)n-1+n(n-1)(uz)n-2 + . . . +(-1>m-in!+%!] eaz 5 &=- (n-1)2"-'+laY dz@>I> S s5 (See chapters 5, 7 and 29 for other integrals involving exponential functions.) 4.3. Circular 4 The a proximations 4.2.43 to 4.S1.45 are from B. Carlson, M. E oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LIA-1943, Los Alamos, N. Mex., 1955 (with permission). 5 The approximations 4.2.46 to 4.2.47 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N.J., 1955 (with permission). Functions Definitions 111988 u6=. 00255 491796 a’=:. 00093 264267 (n20) 4.2.56 Ic(x)~<5xlO-~ al=l. Formulas eazdz earla = 4.2.55 f z”e”‘dz=s 10Z=(l+a,x+azx2+a3;c”+a4x4)2+r(x) 96 z)z’ s O<xil a1=1.14991 In a: -d y&(&-l dz 4.2.52 4.2.53 a, = - .99999 99995 u2= .49999 99206 An 64503 5270 31284 1606 03870 4116 00320 8683 00019 9919 00000 9975 00000 0415 00000 0015 For!mulas 4.2.51 -- .15953 32 .0293641 u4= A,T:(x) n=o 0001 4.2.49 le(x)/<3XlO-” u3= ID ; -: -: 3 4 85 -: 16 7 -: Differentiation e-z=l+a,x+u2x2+u3x3+u~x4+e(x) al = - .99986 84 u2= .49829 26 n A, .. 6 O<z_<l (z in the cut plrtne of Figure 4.4.) Polynomial Polynomials ei3~e-iz 4.3.1 4.3.2 sin z= 2i eiL+e--t” cos z=-2 6 The approximations 4.2.48 are from C. W. Clenshaw, Polynomial approximations to cxlementary functions, Math. Tables Aids Camp. 8, 143-147 (1954) (with permission). 72 ELEMENTARY TRANSCENDENTAL 4.3.17 tan 2=% 4.3.3 FUNCTIONS 43.18 csc 4.3.5 SW z=- 4.3.6 1 cot z=- tan 2 Periodic 4.3.7 cos (2 2 4.3.19 1 cos 2 z2 t an 2 2 (zl+z2,=“~~t~;“;i”,,“zz’ c ot 2 Half-Angle 1 Formulas 4.3.20 sin $= &(qc)’ 4.3.21 cos ;= *( 4.3.22 t,an i= &(~~>1=~=~2 Properties 2 +2/h) (k any integer) = cos 2 tan (z+&r)=tan 4.3.9 tan z,+tan 1 (zl+z2)=cAtan z1 sin z2 sin 2 sin (2+2h)=sin 4.3.8 t an q cos z2-sin (z~+~~)=c~s 1 4.3.4 2=-r-- cos 2 +=)” The ambiguity in sign may be resolved with the aid of a diagram. V , I Transforkation of Trigonometric Integrals If t,an %=z then 4.3.23 22 l-z2 sin u=--, 1+z2 Multiple-Angle 1 70 du cos u=l+zz Formulas 4.3.24 4.3.25 sinx co3 x tonx tan 22=--= 2 tan 2 l-tan22 Circular Functions 4.3.10 sin2 z+cos2 z=l 4.3.11 sec2 z-tan2 2=1 4.3.12 csc2 z-cot? 2 cot22-1 2 =-cot z-tan 4.3.28 cos 3z=-3 4.3.29 sin 42=8 cos” 2 sin 2-4 cos 2 sin 2 cos 42=8 co& z-8 Products 4.3.31 Angle sin (--)=-sin 4.3.14 cos (- z)=cos tan (--)=-tan Addition sin (zl + z2)=sin cos z+4 cosa 2 of Sines cos2 23-1 and Cosines 2 sin z1 sin zz=cos (ZI-2&-co8 (21+22) Formulas 4.3.13 4.3.16 2 cot sin 3z=3 sin z-4 sin3 2 z=l 4.3.15 sin2 z l-tan2 2 =cos2 z-sin2 z= 1+tarP 2 4.3.30 Circularfunctions. Between Negative co9 z-1=1-2 2~2 4.3.27 FIGURE 4.3. Relations cos 4.3.26 ------- sin 22=2 sin ,a cos 2=,“,“t”,“,~2 2 2 Formulas z1 cos z2+cos z1 sin 22 4.3.32 2 cos 4.3.33 2 2 sin Addition and z1 z1 cos Z:~=COS (a-z2)+cos cos z:!=sin (zl-z2)+sin Subtraction of Two Circular 4.3.34 sin zl+sin z2=2 sin(‘+) cOS(y) (21+22) (21+22) Functions 2 ELEMENTARY TRANSCENDENTAL 4.3.41 cos2 a-cos2 4.3.35 sin zl-sin z2=2 cos(F) sin(v) 73 FUNCTIONS z2=-sin (zl+z2) sin (~~-2~) 4.3.36 4.3.42 cos2 zl--sin2 z2=cos (z1+z2) cos (zl-z2) cos 21--cos z2=-2 sin(Tk) sin(y) 4.3.43 Signs of the Circular Functions in the Four Quadrants 4.3.38 z2= sin (215 z2) cos 21 cos 22 tan z,i-tan Quadrant 4.3.39 sin csc ___~- cot 21&cot z2= sin (~2fzJ . sin z1 sin z2 Relations Between Squares of ISines and Cosines :I III IV 4.3.40 sin2 q-sin2 z2=sin (zl+zJ sin (~~-4 cos set tan cot + - + J 7 - $ - 4.3.44 Functions of Angles in Any Quadrant (0 2 0 15, in Terms of Angles in the First Quadrant. k any integer) 2kufe sin-------cos-- -.---tan--_---csc------set------cot -sin cos e Fsin --SC set +sec FCSC e Ftan8 --cot Relations == sin x=u -set e ‘fc0t Between Circular = cos x=u e --OS e e e e *cot e e Fsin 8 e e e -tan0 _______ 4.3.45 8 cos 8 &tan FCSC fsin ‘fc0t - &sin +sec itan ._ a ( 1-u2)* a(l+a2)-+ Functions (l--d)+ CC (l+d)-+ u-‘(d- tan x------ a(l--a2)-* tc-‘(l--a2)* a csc x------ a-1 :1-a”)-* set xv----- (l-a2)-* 4 cot x------ a-‘(l-4)* ‘z(l--d)-* (wx~;) Illustration: set x=a -- .a-‘(cc- 1)” (1+a2)-4 a-’ a(1 +a2)-i (a”- 1)-’ (a2- 1)” a-’ a a(a”- I)-” (1 +a214 (lfa214 u(&-- 1)-t a a-‘(1+a2)* a-l (&-I)+ (a2- 1) -* a - - cot ~=a-‘(l--a2)t arcsec u=arccot (a”- 1)-i 1)’ cot x=a a-y1+a*y -1 If sin x=u, = = a-’ cos x------ e ~CSC csc x=a sin x------ - f00t e = e 8 8 e ftan ~CSC -set 8 +COS 8 e e 8 e (or Inverse Circular) tan x=a - --OS - - 74 ELEMENTARY 4.3.46 - Circular - a/12 15O is -- - _sin Functions for Certain 7d6 3o” TRANSCENDENTAL Angles *I4 45O -- FUNCTIONS *I3 60’ -- Euler’s 4.3.47 Formula ez=ez+fu=ez (cos y+i De Moivre’s 4.3.48 (cos z+i sin y) Theorem sin z)“=cos (-*<L%?;z vzfi sin ~2 <T unless Y is nn integer) 0 $ (J3-1) cos 1 JZ z cd3+ 1) 4512 112 4.3.49 sin z= --i sinh iz tan 0 2-J3 1 43 4.3.50 cos z=cosh csc co tan z= --i tanh iz 1) Jz 2&/3 4.3.51 llz<&+ 4.3.52 csc z=:i csch iz set 1 &qJ3- 1) 4 2 4.3.53 set z=:sech cot 00 2+i3 4.3.54 cot z==i coth iz &I2 l/2 &I2 Relation 1 == 4313 = = to Hyperbolic Functions (see 4.5.7 to 4.5.12) iz iz = Circular 7~112 105O n/2 30° 2~13 1200 .- - sin 1 4512 $(&i-l) 0 tan w& co csc j5<&- 1) 1 set J%&+ 1) Q1 cot a-& = cos -Jz 4 (d3- Functions in 4.3.55 sin z=sin x cash yfi cos x sinh y 4.3.56 5~/12 75O Tgazi cos z=cos x cash y--i sin 2 sinh y 4 . 3 . 57 tan Z=sin 2x-j-i sinh 2y cos 2x f cash 2y A.- -(2f43) 37r/4 135O =I JR&-1, +m+ 1) 4 . 3 . 58 Modulus and 24313 4.3.59 Phase (Argument) /sin zj= (sin2 z+sinh2 =[a -2 (cash arg sin z=arctan 4.3.61 (cos z~=(co@ =[a ’ lln/12 165’ GO0 of Circular 4.3.62 cos lj5/2 112 4 -$ z+sinh2 (cash -&/a 4% arg cos z= --arctan 22)]+ (tan x tanh y) arg tan z:=arctan 2x 4 2x (@S&$!) -1 Series tan -1 -a/3 -(2-&i) 0 csc liz 2 &m+1> set -42 - 2&/3 -&M- cot -1 4 -@+&I 1) -1 co Expansions 4.3.65 co - y)f 2y+cos ‘cash 2y-cos Itan ‘l=(,cosh 2yfcos 0 2x)]+ (cot x tanh y) 4.3.63 (Jc1) Functions y)+ 2y-cos 4.3.64 sin andImaginary 2x--i sinh cot z_sin cash 2y- cos 2x 2y -l/2 -&3 == 4243) = 5n/6 150° Real 4 1) 4.3.60 0 = of sin z=z-$+&$+ ... (14-c-1 4.3.66 cos z=l -$+2-S+. . . . .. @4<4 ELEMENTARY TRANSCENDENTAL 75 FUNCTIONS Inerlualities 4.3.67 23 225 tan z=z+~+~+~+ 1727 ... 4.3.79 22n-1$- . . +(-l).-i22.(22n-l)Bzn (2n)! ( lH<; > 4.3.80 sin 21 x_< tan 2 4.36% 12 csc 7 2=;+,+,,, 4.3.81 31 z5+ 23+15120 ... (bl<d 4.3.82 *2n-l,+ +(-l)n-12(22n-‘-l)BZn (2n) ! 4.3.69 (O<x<l) 4.3.33 +(- 4.3.84 .. l)n& 22n-- . (2n) ! .* 4.3.70 1 23 2z5 2 cot z=;-j-~-py- () lsinh y/ 5 lsin ZJ5 cash y lsinh y/ 5 Jcos 21 Icosh 4.3.85 lcsc z[ Icschlyl 4.3.86 /cos zl Scosh(z) 4.3.87 z2 5z4 61z6 set ~=l+~+~+m+. /sin zI Isinhlzl y lH<5 ... 4.3.88 22n-1,- . _ (-- w122n&n (2n) ! lcos zl< 2, /sin (l2l<O 2[<$[2( (l2l<7r) Infinite Products 4.3.71 4.3.89 2=2 4.3.90 4.3.72 l* sin cos 2=,i1 jiI (1-&) (I4<a> In cos 2=.& W-1 (-- w2n-1wn(2n) ! m, p Expansion (14<37r> (2&$ in Partial (- l)‘-‘y..;;~l- Fractions 1) Bzn 22n w<t’d where B, and E, are the Bernoulli numbers (see chapter 23). Limiting 4.3.74 4.3.75 Values 1 4.3.92 csc224 k-i--m and Euler - (z-kT)2 4.3.93 sin 2 lim ----xl z+o x lim tan=1 z-10 x . . .> (z#O,f?r,f%, Continued 4.3.94 4.3.76 > 4.3.91 4.3.73 ln tan 2 T=gI (l- tan z=c lim n sin E=x n+lim n tan 2=x ?a+- a tan tan az=-----1+ 4.3.78 lim cos z=l 22 ---3 22 22 5- 7- ... (2 #i*n*) 4.3.95 4.3.77 2 Fractions n-+- 2 (1-a’) 3f tan2 2 (4--a2) tan2 5+ (9-a”) tan2 2., . 2<g2<;, 7t ( 2 uz #+r > 76 ELEMENTARY Polynomial 4.3.96 Approximations TRANSCENDENTAL 1 FUNCTIONS o<x-$ 4.3.101 o<x<; tan x -= 2 1 +a~x2+u4x4+agx6+agx8+~,ox10 la(s)] <2x10-5 as= .33333 14036 a4= .13339 23995 u4= .00761 a3= .02456 50893 alo= .00290 05250 a6= .05337 40603 a,=-.16605 a12= .00951 68091 05x<; 4.3.102 * [E(x)(<2x10-9 a2=-. a4= as=-. 16666 66664 .00833 33315 00019 84090 4.3.98 as= .OOOOO 27526 alo= - .OOOOO00239 x cot x= I $-a2x2+a4x4+ e (2) 16(2')1<3x10-5 a2= - .332867 02x< 4.3.103 o_<xg; a2= - .33333 33410 a4= - .02222 20287 a6=-. 00211 77168 a4= .03705 O<X$ Approximations cos x=1+a~xz+a4x4+a,xe+a8x8+a,0x10+t(x) la(x)ll2X a2= - .49999 99963 a4= .04166 66418 as=-.00138 88397 alo= .00002 -. 00000 47609 02605 aa= - .00020 78504 alo= - .00002 62619 of Chebyshev Polynomials --11x51 T: (x) = cos ti, cos 8=2xsin ~~x=x *co A,T,*(,x*) 1 (see chapter 22) cos $rx=2 n-0 A,T,*(z*) A, 1.27627 8962 1 -.28526 1569 2 .00911 8016 3 - .00013 6587 n A, 0 .47200 12 16 1 - .49940 3258 .02799 2080 2 3 - .00059 6695 4 4 5 .OOOOO6704 - .ooooo 0047 0 tan 2 --l+a2x2+a4x4+t(x) 2 (a(x)(<lXlo-3 a*= .31755 Terms n o<x$ 4.3.100 in 4.3.104 10-Q aa= c (x) Ir(x)l<4Xlo-'O 16(X)1 <9x 10-h 4.3.99 ; x cot x= 1 +a2x2+a4x4+a~x6+a8x8+aIox10+ cos x=1+a,x2+a,x4++(x) a2= - .49670 a,=-.024369 a,=.20330 7 The a proximations 4.3.96 to 4.3.103 are from B. Carlson, M. e oldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N. Mex., 1955 (with permission). 5 .OOOOO1185 -.ooooo 0007 n The approximations 4.3.104 are from C. W. Clenshan, Polynomial approxinlations to elementary functions, Math. Tables Aids Camp. 8, 143-147 (1954) (with permission). *see page Ix. 8 ELEMENTARY Differentiation TRANSCENDENTAL Formulas 4.3.105 z cos z=-sin d 77 4.3.122 S d 2; sin z=cos 2 4.3.106 FUNCTIONS z dz -2 sin”=(n- 1 1) (n-2) sinnm2 2 cos 2 1) sinn-l z-(n- z 4.3.123 -$ tan z=sec2 z 4.3.107 . 4.3.108 d & csc z=-csc 4.3.109 z set z=sec z tan 2 4.3.110 d -& cot 2=-csc2 S d 2 S‘ sdz S tan z+ln Ldz=z cos2 2 cos 2 dz cosn=+1) S 2 cop-1 (La) z-(%-l) .I- S sinm z tan zdz=-ln 4.3.116 I- J csc zdz=ln 2 cos” 2 dz = sirP+’ 2 coC1 2 m+n S sin”’ cos z dz=sin ;: S 4.3.115 cm>21 4.3.127 Formulas s 4.3.114 cosn-2Z 5 S ++-2) (n-1) sin zdz=-cos 2 sin 2 4.3.112 4.3.113 (n>l) 4.3.126 4.3.111 Integration zdz zn-‘sin 4.3.124 2 cot 2 4.3.125 cosz==ln secz =- sinme 2 cosnfl 2 cosnm2 z dz 2 m+n tan :=ln (csc z-cot S sin”‘-* z)=- 1 In- 1-cosz 2 lfcosz 4.3.117 s S sin z-n zdz=zn zn cos 2 COS” 2 dz (m#-n) 4.3.128 S seczdz=ln(secz+tanz)=lntan =Inverse sinm z”Z,oP Gudermannian z=(n--1) 1 sinmS1 + Function 2 cosnel m+n-2 gd 2=2 arctan eZ-i n-l 2 dz S sinm 2 coP2 2 (n>l) S cot zdz=ln 4.3.118 4.3.119 S Psinzdz=-zz” sin z=--1n cos z+n csc z =(m-1) sin:: + z”-’ cos zdz 2 cosnml m-b-2 m-l S 2 sirF2 dz 2 cosn (m>l> 4.3.120 S 4.3.121 4.3.129 S Adz=--z sin* 2 cot z-rln sin z 4.3.130 S S tannzdz=ts-fian”-2zda cot”zdz=--- COtn-’ n-l (n#l> 2 S Cotn-2ZdZ (n#l) 2 78 ELEMENTARY TRANSCENDENTAL 4.3.131 a tan clz =------ 2 arctan (a”-- b2)t a+b sin z (a”--b”)* a tan 1 0 In (bz-a2)+ a tan 0 S S lsin2 nt dt== *cos2nt dt=% 4.3.141 S =- FUNCTIONS 0 (az>b2) 0 (n an integer, 1 z +b-(b2-a*)) S msirltmt &=; 4.3.142 =o (b2>a2) ---7-=Ttan lfsm 2 4.3.133 dz S a+b cos z 4.3.143 2 arctan (a2-b2)* (a”-b2)i (b-a) =&h-l 2 tan c- (b2-a2)+ 2 0 (b2>a?) S S dz -------=tan 1+cos 2 4.3.134 % 1-cos 2 , 0 s/2 In cos t dt=-; In 2 .O m cos mt o T+T dt=ae-” S 4.3.146 Formulas e”” sin bz dz=- In sin t dt= J i involv- 4.3.147 4.3.136 S S f -cos t2 dt=i 0 4.3.145 r/2 (b/a) (See chapters 5 and. 7 for other integrals ing circular functions.) (See [5.3] for Fourier transforms.) 2 2 dz=-cot 4.3.135 S 1S (m<O> 2 mcos at--cos bt ___- L dt=ln 6 -sin t2 dt= 4.3.144 W>W tan z+(b2-a2)+ (b-a) Jo tan i (a-b) =----- r (m=O) n =-- dz S 4.3.132 (m>O) 0 E +b+(b2-a2)t n#O) ea” (a sin bz-b a2+b2 for Solution of Plane Right Triangles cos bz) 4.3.137 S e”” cos bz dz=- em (a cos bz+b sin bz) a2fb2 4.3.138 C eazsin“-I bz S earsinn bz dz= a2+n2b2 (a sin bz-nb + eaz inne2 bz dz s 4.3.139 S e”’ cosn bz dz= cos bz) S n(n-l)b2 a2+n2b2 eaL cos”-’ bz (a cos bz+nb sin bz) a2fn2b2 + Definite n(n-W2 S eaz a2+n2b2 S S If A, B and C are the vertices (C the right angle), and a, b and c the sides opposite respectively, sin A=:=c 1 CSCA cos A=-= 6 - 1 c set A COSn-2 bz dz tan A=i=c+A Integrals versine A=vers * 4.3.140 A b A=l-cos A sin mt sin nt dt=O 0 coversine A=covers (mfn, m and n integers) * cos mt cos nt dt=O 0 haversine A=hav A=l-sin A=$ vers A exsecant A=exsec A=sec A-l A ELEMENTARY TRANSCENDENTAL 4.3.148 4.4. Formulaa for Solution of Plane 79 FUNCTIONS Inverse Circular Triangles Functions Definitions 4.4.1 * ~rcsm z= s o s l arccos z= ‘1 b cos A= c c*+ b*-a* 2bc a=b cos C-l-c cos B a+b tan $(A+B) a--b=tan #(A-B) area dt s 4.4.3 C In a triangle with angles A, B and C and sides opposite a, b and c respectively, a b sin=sin=sin S arctan z= A A/ (2=x+$ 4.4.2 0 C dt (l-t*)* ----=9-arcsin z (l-t*)* 2 2 p dt 0 1+t* The path of integration must not cross the real axis in the case of 4.4.1 and 4.4.2 and the imaginary axis in the case of 4.4.3 except possibly inside the unit circle. Each function is singlevalued and regular in the z-plane cut along the real axis from - 0~ to - 1 and + 1 to + 03in the case of 4.4.1 and 4.4.2 and along the imaginary axis from i to i 0~ and --i to --i 03in the case of 4.4.3. Inverse circular functions are also written arcsin 2= sin-’ 2, arccos z=cos-’ 2, arctan 2 =tan+ 2, . . . . When -1 _<x_<1, arcsin x and arccos x are real and 4.4.4 bc sin A =-=[s(s-a) 2 s=j(a+b+c) - $ir 5 arcsin 2 5 &r, 0 <;Irccos x,<u 4.4.5 (s- b)(s,-c)]+ arctan z+arccot R220 RZ<O z=&)s 4.4.6 4.4.7 4.3.149 Formulas for Solution of Spherical Triangles arccsc 2= urcsiu l/z arcsec z=arccos l/z 4.4.8 arccot z=arctan l/z arcsec 2 + arccsc 2= +a 4.4.9 (see 4.3.45) s C (I A 0 b m -I C sin B sm b +I I -i arcsin arccos z and z orccsc orcsec If A, B and C are the three argles and a, b and c the opposite sides, sin A y=,=-sin a 0 arctan 2 and L z orccot L sin C sm c cos a=cos b cm c+sin b sin c cos A =cos b cos (CH) CO8 e where tan B=tan b cos A FIGURE cos A=-cos B cos C+sin 13 sin C cos a 4.4. Branch cuts for jzbnctions. inverse circular 80 ELEMENTARY Fundamental TRANSCENDENTAL 4.4.23 sin t=z z=i Arcsec z= f i Arcsech 4.4.25 of the equations Arccsc 4.4.24 Property The general solutions FUNCTIONS Arccot z=:i Arccoth cos t=z Logarithmic z+ka z=arctan k is an arbitrary 4.4.13 x=-ii:Ln Arccos x=--i 4.4.28 z= k arccos z+Zkr t=Arccos where Arcsin 4.4.27 4.4.12 t=Arctan z iz Arctan x=i [(l-x”)*+ix] (~“51) Ln [x+i(l-x2)*] (x251) z= (- 1)” arcsin z+kn t=Arcsin 4.4.11 iz Representations 4.4.26 tan t=2 are respectively 4.4.10 Arccsch Interval Y Ln (x real) (9#-1) 4.4.29 arcsin x and arctan x 0 < y 5 a/2 - ?r/2 < y<O arccos x and arcsec x 0 5 y 5~12 arccot x and arccsc x 0 i y 57~12 d2<Y x=-i Ln [(x2-l)r+i] Arcsec x=--i Ln 4.4.31 containing principal value x positive x negative or zero Arccsc 4.4.30 integer. Arccot (~“2 s ?I- 1) (5 real) - ?r/2 _<y<O Addition and Subtraction of Functions Two Inverse Circular 4.4.32 Arcsin z1 f Arcsin z2 =:Arcsin [21(1-z3+~22(1-2T)t] 4.4.33 Arccos 21 f Arccos z2 =Arccos{ L-20 - FIGURE 4.5. - DlCILC OlECOl zlz2F[(l-23 (l-z$]‘j 4.4.34 I I Arctan Inverse circular junctions. Arguments arcsin (- 2) = - arcsin 2 Arcsin 4.4.15 arccos (- 2) = 7r- arccos 2 4.4.16 arctan 2 arccsc (- 2) = - arccsc 2 Arctan 4.4.18 arcsec (- 2) = 7r-arcsec 4.4.19 arccot (e) 4.4.36 4.4.17 z,=Arctan 4.4.35 4.4.14 zlfArctan Functions Relation to Inverse of Negative (- 2) = - arctan (-z)=-arccot Hyperbolic 4.6.19) 2:) =Arcsin{wdz[(l--zT) =Arccos 2 z1 &Arccot Inverse (see 4.6.14 to Circular (l-$)1’} [zz(l-2~)~~zz1(l-z~)*] z2 =Arctan 2 Functions z1f Arccos (*)=Arccot Functions Imaginary (--E?-E?) in Terms Parts of Real 4.4.37 4.4.20 Arcsin z= -i Arcsinh iz Arcsin 4.4.21 Arccos z= f i Arccosh z 4.4.38 4.4.22 Arctan z---i iz Arctanh ~=kd-(-l)~ arcsin/ f(-l)%In (zz# -1) Arccos z=2knf {arccosfi-i [cYS(d--l)q In [(~+(a~-l)~I} and ELElMENTARY TRANSCENDENTAL 4.4.39 Arctan FUNCTIONS 81 4.4.46 z=ks++ O<x<l arctan ($&) arcsin ~=T-(l--z)~(a,+a~z+a,22+a3~3 2 +; lu [$H] (zZ#-1) +u4x4+ujx5+u6x6+u,x’) k is an integer where or zero uild le(x>l<2X 10-s Cx=~[(2+1)~+?J~]~+~[(x--l)~+y~]~ uo= 1.57079 63050 4.4.40 23 1 a325 1 *3.5;:’ -arcsIn 2=2+~~+~.~ .5+2e4e6 ,7+. . . (l4<1) us= -.01708 81256 I&= Expansions CIQ= .03089 lSS10 aq= -.21459 88016 P=~[(2+1)2+y2];-3[(2--1)2f?12]; Series us= .08897 89874 a3= -.05017 43046 4.4.41 4.4.41 arcsin (l-2)+(22)’ 4.4.42 u1= 5-, “?+ . . . (/z/_<l and ~~2-1) u3= .99986 60 - aI= -.08513 30 .33029 95 uQ= .18014 10 -11&l arctan x=Continued Fractions 4.4.49 l1 arctan x in the cut plane of Figure 4.4.) (2 2 4.4.44 2 1.2.~~1.2.~~3.4~~ 3.4~~ ae22=1--r-cs-. .. in the cut plane of Figure 4.4.) (2 Polynomial ApproximaGons 4.4.45 8 O<x<l -- u2=: aq= -.21211 44 u3=- Q The approximations Univ. Press, Princeton, =l+k~u2&k+r(r) le(x)l12xlO-* u2= -.33333 u4= 14528 .19993 55085 ulo= - .07528 96400 a12= 90429096138 u14= -.01616 ua= a m= .10656 26393 57367 .00286 62257 55x10-5 a,,= 1.57072 88 ings,Jr., Approximat.ions Ojril u8= - .14208 89944 arcsin 2=~-(1-~)*(a,+am-~a?rz+asr3)+~(r) /e(x)1 1+:8x2+‘(“) Ir(x>l_<5x10-3 - z2 -4z2 -92 -1622 . 1+ 3+ 5+ 7-t 9+. . . z=z .02083 51 . .((zl>landz2#-1) 4.4.48 lo arctan 24911 le(x) 15 10-b uj= 4.4.43 a,=-.00126 -l_<s<l (121<2> =i-i+&-&+. .00667 00901 arctan x=ulx+u3x3+u~x6+u,x7+uaexQ+~(x) 1 +ks arctan z=z 2,z” 3 -tE(X) .07426 10 - .01872 93 4.4.45 to 4.4.47 are from C. Hastfor digital computers. Princeton N.J.. 1955 (with permission). 10The approximation 4.4.48 is from C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1958) (with permission). 11 The approximation 4.4.49 is from B. Carlson, M. Goldstein, Rational approximation of functions, Los Alamos Scientific Laboratory LA-1943, Los Alamos, N. Mex., 1955 (with permission). 82 ELEMENTARY Approximations in Terms 4.4.50 of Chehyshev TRANSCENDENTAL Polynomials I2 FUNCTIONS d z arccsc 2=-2 4.4.57 -l<S<l Tf(x)=cos ne, cos 0=2x- 1 (see chapter Integra-tion 22) 4.4.58 arctan x=x s g0 A,T~(zz) A, For 00000 3821 -: 01113 2925 10589 5843 -: 00138 1195 .00018 5743 -. 00002 6215 :: 3 4 5 i!t 7 ii 10 -: . 00000 0570 0086 -. 00000 0013 . 00000 0002 2 >l, use arctan s=$7r-arctan s arctan 2 dz==z arctan 2--3ln (1-j-z”) 4.4.61 arccsc 2 d2=2 arccsc 2fln [~+(2~-1)*] c O<arccsc 2<5 4.4.51 -$Ji<X<$JZ arcsin x=x -i<arccsc nTOA,Tt(2x2) arccos x=&r-x n 0 .00004 7890 .00040 6985 A,Tz(2x2) : 05494 6487 00408 0631 i ItaO A?& 1.05123 1959 ; 5 For $~<z<l, cos z=aEsiG O<arcsec 2<f n : 4 00000 5881 : 00000 0777 0107 9 [ arc- ;< arcsec 2<ir 4.4.63 S arccot 2 dz:=z arccot 2+$ In (l+z2) . : 00000 0015 0002 use arcsin x=arccos(l-x2)+, (1--x2)+. Differentiation 2<0 4.4.62 . arcsec z dz=.z arcsec zTln [2+(22-l)+] o<x<@ 4.4.52 4.4.60 s (l/z) arcsin z dz ==z arcsin z + (1- ,z2)i arccos z dz:=z arccos z-(3-z2)t A, 88137 3587 Formulas S 4.4.59 21 1 (e2-l)i 4.4.64 arcsin 2+$ (1-22)4 Formulas & arcsin z=(l-z2)-+ 4.4.65 9+1 arcsin n-j-1 S 2" arcsin 2 dz=-- 4.4.53 (n#-1) $ arccos z=-(I-z2)-f 4.4.66 4.4.54 d 1 arctan z=z 1+22 4.4.55 d -1 z arccot 2=1+22 S z arccos z dz.7 d 1 z aJJcsec 2=2 (2”- 1)t la The approximations Clenshaw, Polynomial functions, Math. Tables (with permission). 4.4.50 to 4.4.51 are from C. W. approximations to elementary Aids Comp. 8, 143-147 (1954) (l--z*)+ 4.4.67 S 2n arccos 2 dz=Ez 4.4.56 arccos z-f n,+l arccos @Z-l) , 4.4.68 1 2 arctan z dz=- (l+29 2 S arctan 2-g 2 ELEMENTARY TRANSCENDENTAL 83 FUNCTIONS L.5.8 1 arctan z---n+l p+l S z” arctan z dz=- n+l S gdz (n#-I) 4.4.70 S z arccot z dz=i nrccot z+: (14-S) cash 2=cos E.5.9 tanh z= --i’tan L.5.10 csch z=i csc iz E.5.11 4.4.69 sech z=sec iz h.5.12 coth z=i cot iz Periodic 4.4.71 S n+l arcco t z J,- Properties z (k any integer) (n#-1) 4.5. Hyperbolic iz sinh (2+2k7ri) =sinh 1.5.13 ,$a+1 zn arccot z dz=- iz Functions cash (zf2ksi) a.5.14 tanh (z+k&)=tanh 1.5.15 Definitions =cosh z Relations Between z Hyperbolic Functions 4.5.1 ez-e-2 sinh z=2 4.5.2 e”+e-” cash z=2 4.5.3 tanh z=sinh 4.5.19 cash z+sinh z=e’ 4.5.4 csch z = l/sinh 4.5.20 cash z-sinh z=e-” 4.5.5 sech z= l/cash z 4.5.6 coth z=l/tanh z cosh2 z-sinh2 tanh2 z+sech2 z= 1 4.5.18 Z/COS:I z 4.5.16 4.5.17 (z=x+iy> coth2 z-csch2 z= 1 Negative Angle z=l Formulas 4.5.21 sinh (-.z)=-sinh 4.5.22 cash ( - z) = cash z 4.5.23 tanh (-z)=-tanh z Addition 4.5.24 z z Formulas sinh (zl+z2) =sinh z1 cash 22 +cosh z1 sinh 22 4.5.25 cash (zl+ 22)=cosh ZI cash 22 +sinh 4.5.26 tanh (q+zQ=(tanh zI+tanh z1 sinh zz &)/ (1 + tanh z1 tanh 22) FIGURE 4.6. 4.5.27 Hyperbolic jusctions. coth (zl+zz)=(coth zl coth %+l)/ (coth zz+coth Relation to Circular Hyperbolic trigonometric Functions (see 4.3.49 to 4.3.34) can be derived from formulas identities by replacing z by iz Half-Angle 4.5.28 sinh E=(,,sh;-l)i 4.5.7 sinh z=--i sin i;r Formulas ZI) ELEMENTARY 84 TRANSCENDENTAL FUNCTIONS 4.5.44 cash z,-cash z2=2 s:inh (p) sinh (v) 4.5.45 zl+tanh sinh (21+z2) %=E;h L 2 cosh 2 1 2 coth z,+coth sinh (2, +2,) zz=- smh 2, sinh zz tad 4.5.46 Multiple-Angle Formulas 4.5.31 sinh 22=2 sinh z cash ~=~~‘a~~f, 4.5.32 cash 22=2 cosh2 z--l=2 Relations Between Squares of COosines 4.5.47 sinh2 zL-sinh2 zz=sinh =cosh2 4.5.48 sinh2 zl+cosh2 z2=cosh =cosh2 sinh2 z-j-1 = cosh2 z + sinh2 z Hyperbolic Sines (zl+zz) sinh (zl--q-cosh2 zz (q+zJ cash zl+sinh2 z2 and z2) (2,-z2) 4.5.33 tanh 2~=12+tt&annhhez, 4.5.34 sinh 32=3 4.5.35 cash 32=-3 4.5.36 sinh 42=4 sinh3 z cash 2+4 cosh3 z sinh z 4.5.49 sinh z=sinh x cos y+i cash x sin y 4.5.37 cash 4z=cosh4 z-j-6 sinh2 z cosh2 z+sinh’ 4.5.50 cash z=cosh x cos yfi sinh x sin y Products 4.5.38 Hyperbolic sinh 2+4 sinh3 z of Hyperbolic Sines and (Q-.zz> z and Subtraction of Two De Moivre’s (q-q) (~~-2~) 4.5.55 z2=2 sinh (‘9) cash (‘9) 4.5.42 sinh z,-sinh 4.5.54 4.5.56 4.5.57 z2=2 cash (9) (cash z+sinh and Phase Theorem z)“=cosh nz+sinh (Argument) Functions of m Hyperbolic lsinh z/ = (sinh2 x+sin2 y)* =[+(cosh 2x-cos 2y)]” Functions 4.5.41 sinh z,+sinh 4.5.53 Modulus (zI+zz) Hyperbolic Imaginary 4 . 5 .52 coth 2=sinh 2x---i sin 2y cash 2x-cos 2y 2 cash 21 cash zz=cosh (zl+zJ +sinh Addition and 4 . 5 .51 tanh 2=sinh 2x-j-i sin 2y cash 2x+cos 2y Cosines 2 sinh z1 sinh zz=cosh (zl+z2) 2 sinh z1 cash z2=sinh of Real (z:=zfiy) +cosh 4.5.40 in z;;;s cash 2+4 cosh3 z -cash 4.5.39 Functions sinh (y) 4.5.58 arg sinh z=a.rctan (coth x tan y) (cash zl= (sinh2 x+cos2 y)” =[+(cosh 2x+cos 2y)]h arg cash z=arctan (tanh x tan y) cash 2x-cos 2y * ltanh Z1=(cosh 2x+cos 2y 4.5.43 cash zl+cosh z2=2 cash (‘+) cash (y) 4.5.59 arg tanh z=arctan ~ (f:2,“,) Y ELEMENTARY Relations 4.5.60 TRANSCENDENTAL Bet ween Hyperbolic sinh x=a CO~ll x=u 85 FUNCTIONS (or Inverse Hyperbolic) tanh z=a Functions csch x=u sech x=,z = -- coth x=a sinh x----?. a (a’-. l)+ a(l-u2)-1 CL-’ a-‘(1-a2y (a’- 1) -1 cash x----o (l+u*)+ a (l-a”)+ a-‘(l+a2)t a-l a(a2- 1)-i tanh x----- a(l+a’)-4 u-‘(a2- 1)’ a (l+ay-+ (l-a’)+ a-’ csch x----- am1 (a’- 1)-’ a-‘(l-a2)+ a a(l-a2)+ (a2- 1)’ soch x----- (l+u’)-+ a-* (l-4)+ a(l+a2)-* a a-‘(a2- 1)’ coth x----- a-‘(a2+l)f u(a2-- 1)-i a-l u+a2y (1-a2)-4 a - Illustration: If sinh x=a, coth x=aB1(a2+ arcsech a=arccoth Special Values of the Hypl>rbolic 4.5.61 Functions z *. -a 2 0 2 1)’ (l--a2)-) 4.5.66 sech z=l -f+& 24-go zs+ . . . +a* 22*+ ... co ( 0 i 0 --i co 1 0 -1 0 cu coth tanh z------ 0 -i 0 -coi 1 csch 2 _______ 0~ 4 OD i 0 sech 2 _______ 1 OD -1 co 0 coth 2 _______ 0~ 0 co 0 1 > 4.5.67 z------- IM<; sinh 2 ____ cash _ __ 2=;+;-$+$f- . . . +2~22n-1+ ... (l4<*) where B, and E,, are the nth Bernoulli and Euler numbers, see chapter 23. Inhite 4.5.68 Products sinh z=z ii (l+&) -. Series Expansions 4.5.69 cash z=fi l+ k=l 23 ZfJ 4.5.62 smh ’ 4.5.63 cash z=l+$+$+;+ . . 1 (2Zzf)%2 C 2 ~ =2+9+3+~+ ... . (14-c ... (l4< m) -1 Continued 4.5.70 tanh 2=& Fraction 22 22 3+ 5f 24 7+ . . . ( Differentiation * + ... +22W2”-1)B,, 22n-l+ csch z=I ’ z--T;+360 ’ 23 II. d z sinh z=cosh 2 d cash z=sinh z 2 & tanh z=sech2 z 4.5.74 d z csch z=-csch -j&o25+... -2(22”-‘-l)B2, (2n)! page 4.5.71 4.5.73 ’’* 4.5.65 *See Formulas 4.5.72 (h)! 2 #i 2”-1 +... 2 (l4<3d z coth z ifnni > 86 ELEMENTARY d z sech z=-sech 4.5.75 TRANSCENDENTAL FUNCTIONS, 4.5.87 z tanh z S tanh” 4.5.76 -$ coth z=-csch2 Integration z s cash zdz=sinh tanh z dz=ln 4.6. Inverse cash z-n en sinh z dz=z” 4.5.84 z” cash 2 dz=z” r J 9-i arcsinh z= 4.6.2 arccosh z= 4.6.3 sinh z 4.5.83 P arctanh z= in- Functions z cash” z dz=- 1 sinh”‘+l =--& z cosh’+l S dz sinh” z coshn z=s - m+n-2 m--l sinh” z coshnm2 z dz S 4.6.4 to --i and i arccsch z = arcsinh 1/z q 4.6.5 arcsech z ==arccosh 1/z 4.6.6 arccoth z==arctanh 9 l/z iy ti -1 z coshn-l dz sinhmm2 z cash” z 1 sinh” 3 -i arcsinh z z arccsch 1 sinh”-’ n-l X (m+n#O> S S z dt - 0 1-P sinhm+ z cash” z dz =- m+n-2 + n-l iz (t2fl)t Inverse hvnerbolic functions are also written ” -. sinh-’ z, arsmh z, JS’T sinh z, etc. z sinhmel z coshn+l z m-l -m+- (z=x+iy) 4.6.3 real axis from - ~0 to - 1 and + 1 to +a +- S m+n Oz(lTt2)1 4.6.2 real axis fro.m - 03 to + 1 zn-l sinh z dz m+n S S S 4.6.1 imaginary axis from --im to iw z-n S sinh n-l 4.5.86 integrals The paths of integration must not cross the following cuts. cash z dz 4.5.85 sinh” Hyperbolic 4.6.1 (sinh z) coth zdz=ln S S (See chapters 5 and 7 for other volving hyperbolic functions.) z tanh : sech z dz=arctan 4.5.82 z dz Definitions csch zdz=ln 4.5.81 J coth” z de= -%!@.?+SCoth”-2 n-l cash z S S S 4.5.80 Wl) Wl> s 4.5.79 z dz s z s 4.5.78 tanh”-‘z+Stanh”-” n-l 4.5.88 Formulas sinh z dz=cosh 4.5.77 z dz=-- z --CD 4-I X arccosh 0 +I arctanh .z 1 arcsech t- X -I 0 iy I- z (m#l> 1 sinh”-’ dz z cosh”-2 5 cash”-’ z Wl> z FIGURE 4.7. Branch cuts for junctions. 1 arccothz inverse hyperbolic z ELESMENTARY arctanh z=arccoth 4.6.7 TRANSCENDENTAL z& $ri (see 4.5.60) Fundamental arccoth x=i 4.6.25 (according 87 FUNCTIONS as J&SO) x+1 In x-l (x2>1) Property The general solutions of the equations z=sinh t z=cosh t z=tanh t are respectively 4.6.8 t=Arcsinh ~=(-l)~ arcsinh z+kri 4.6.9 t=Arccosh 2= harccosh 4.6.10 t=Arctanh z=arctanh z+2k& z.+kti (k, integer) Functions of Negative Arguments FIGURE arcsinh (- z) = - arcsinh z 4.6.11 arccosh (- z) = ai- arccosh z *4.6.12 Addition 4.6.13 Relation Inverse hyperbolic junctiolas. 4.8. arctanh to Inverse (-2) = -arctanh Circular and Subtraction z Functions (see of Two Inverse Hyperbolic [2,(1 +z:)*d~ ~~(1 +z;>+] [ (zi- 1) (z$- l)]‘} Functions 4.4.20 to 4.6.26 4.4.25) Hyperbolic identities can be derived from trigonometric identities by replacing 2 by i2. 4.6.14 Arcsinh 4.6.15 Arccosh z= k i Arccos 4.6.16 Arctanh Arcsin iz 2=-i 2= 4.6.17 Arccsch z=i 4.6.18 Arcsech 4.6.19 Arccoth z=i -i Arcsinh z1 &Arcsinh z2 =Arcsinh 4.6.27 2 Arccosh 2, Arctrm iz f Arccosh z2 =Arccosh { 2,z2k Arccsc iz 4.6.28 2= 5 i A rcwc z Arctanh Logarithmic z,&Arctanh z2=Arctanh Arccot iz 4.6.29 Representrltions 4.6.20 arcsinh z=ln [x+(x*-+l)t] 4.6.21 arccosh x=ln [z+ (x2-- l)i] 4.6.22 1 arctanh x=4 In - +x l-x 4.6.23 arcsech s=ln [i+($- (x21) (O<X< { 2,22~:[~1+2:)(2~-1l)]‘j =Arccosh / [za(l +z:)‘& 2,(2[-- 4.6.30 Arctanh (x+0) 1>‘1 z2 =Arcsinh Wx2<1) arccsch x=ln [:+($-I-1)1] 4.6.24 Arcsinh z,fArccosh 1) 2,fArccoth z2=Arctanh =Arccoth ‘see pageII. (S> ( 2ri:i;) l)‘] ELEMENTARY 88 Series TRANSCENDENTAL Expansions FUNCTIONS 4.6.42 4.6.31 arcsinh z=z --.A- . 3 z3++-& 2 -$ arccoth. 2=(1-S)+ 25 Integration 1.3.5 -2.4.6.7 “+ ’’* 4.6.44 (lzl<l) 1.3 =In 2z+1 2 ’ 222 - 2 * 4 * 424 + 4.6.43 4.6.45 l-3.5 2.4.6.&e- '*' 4.6.46 Formulas arcsinh z dz=z s arcsinh z-(1+,9)+ arccosh zdz=z s arccosh z-(z2-1)f Sarctanhzdz=z s arctanh z++ln arccsch z dz=z arccsch arccosh z=ln 22--- 1 1.3 2 * 222 2 * 4 * 4z4 1.3.5 -2.4 . 6. 69-- 4.6.47 4.6.48 * ** arctanh z=z+$+f+g+ ... Sarcsech z dz = z arcsech Sarccoth z dz=z arccoth z arcsinh z dz=-- s (121<1) arccoth z=$-&+&~+$+ ... (z in the cut plane of Figure 4.7.) 4.6.36 arcsinh z 2 1 * 222 1 ’ 222 3 * 422 3 * 422 =1+3+5+-7+ 9+ -** 1/1+z2 s Differentiation 4.6.37 S2 arccosh 2 arccosh z=(z2-1)-f z dz:= 222- 1 arccosh z-$94 4.6.52 2”arccosh zdz=*‘- S ,,n+1 arccoshz-- 7t+1 $1 1)” p+1 (z”- l,td2 (nit-l) S 4.6.53 z arctsnh a dz--2 S 22-l arctanh z+f 4.6.54 s 2” 4.6.38 arcsinhz (n#-1) Formulas $ arcsinh z=(l +z2)-i n-i-1 4.6.51 Fractions 22 422 922 arctanh z=$- - 3- 5-- 7-. . . 4.6.35 .z++ In (~~-1) z!z2+ 1 arcsinh z-z 4 (z’+lY 4 p+1 S (I 01) Continued * 2 f arcsin z 4.6.50 zR arcsinh zdz=-- 4.6.34 * 4.6.49 (14>1) 4.6.33 z f alcsinh z (according as 9~~0) (14>1) 4.6.32 (l-9) pa+1 arctanh 2 dz:=- n+l arctanh z -&Ssdz (n#- 1) 4.6.55 4.6.39 2 arctanhz=(l-,9)-I 4.6.40 z arccsch z=T d Sz arccsch ’ z(l+S)) (according as 9?z=O) 4.6.41 *See pageII. 1 2 arcsech z= 7 dz z(l-.zy z de=< arccsh z&-f (l+zz)* (according as &z~O) 4.6.66 p+1rccsch a .z”arccsch z dz- - S n+l 2f (n# -1) * ELEMENTARY 4.6.57 s * z arcsech z dz=$ TRANSCENDENTAL arcsech tri 4.6.59 (l-z*)1 S z arccoth z dz- -2 (acc.ording as %ZO) 4.6.58 89 FUNCTIONS I 4.6.60 1 -n+1 p+1 s z” arcsech z dz =- nfl arcsech z + - S zdz (l--z*)’ z” arcc0t.h z dz=- (n#-1) of the Tables 1. Computation of Common arccoth z+- 1 n+l 2. Compute xw314 x=9.19826 to 10D usjng t,he for Table of Common Logarithms. From Table 4.1, four-point Lagrangian intierpolation gives log,, (9.19826) = .96370 56812. Then, log,, (x) = - .72277 92609=9.27722 073!$1- 10. Linear inverse interpolation in Table 4.1 yields antilog (‘i.27722) =. 18933. For 10 place ndcurncy subtnbulation with 4- oint Lngrnnginn intcrpolnnt’s produces the tn Ele N .18933 .18934 log,, 5 * 10’ A:’ A log,, N .27721 94350 2 29379 2 29366 .2’7724 23729 .2’7726 53095 .18935 x. 10’ n+l Methods -; Logarithms. To comput,e common lognrithms, the number must be expressed in the form x. log, (1 Iz<lO, - Q) 5 q _<= ). The common lognrithm of x .lOP consists of nn integral part which is called the characteristic and a deamnl part which is called the mnntissa. Table 4.1 gives the common logarithm of x. X *ll+1 1 (hrE-1) Example NOTE: In the examples given it is assumed t,hat the arguments are esact. Example arccoth z+ I Numerical 4.7. Use and Extension z*- 1 -13 By linear inverse interpolation so09836 9.836. 1O-3 5.99281:35=(--200718 15) .09836 9.836. lo-’ -i.99281,85=(-1.00718 15) Example 3. .9836 9.836.10~’ i.99281;35=(-0.00718 15) 9.836. loo 0.99281 85 9.836.10! 1.99281:35 Convert log,, x t,o In x for x=.009836. Using 4.1.23 nnd Table 4.1, In (.009836)= In 10 log,, (.009836)=2.30258 5093 (-2.00718 15) = -4.62170 62:. 9,836.10* 2.99281:35 Example 4. 9.836 98.36 983.6 x-314= .18933 05685. Interpolation in Table 4.1 between 983 nnd 984 gives -99281 85 as t,he mantissa of 9836. Note t.hat 5.99281 85=-3+.!39281 85. When p is negative the common logarithm can be expressed in the alternative forms Compute In x for x=.00278 t*o 6D. Using 4.1.7, 4.1.11 nndTable 4.2, ln (.00278)= In (.278.10-*)==ln (.278)-2 ln lo=-5.886304. between x=.002 nnd Linenr lntrrpolntion ivc ln(.00278)=-5.808. TO x= .003 would obtain 5 decimn f plncc accurncy with linenr interpolatio’n it, is neccssnry t’hnt x>.175. log,, (.009836)=?k99281 Example 5.# 85=7.99281 = -2.00718 85-10 15. The last form is convenient for conversion from common logarit.hms to natural lognrithms. The inverse of log,,x is called l;he ant,ilogarithm of x, and is written antilog x or lo -I x. The logarithm of the reciprocal of a num %er is called the cologarithm, written colog. Compute In x for x=1131.718 t,o SD. Using 4J.7, 4.1.11 and Table 4.2 1131.718 1131 In 1131.718:=hl 1131 :=ln 1’~:~‘8+1n :=ln(1.00063 1.131 +ln lo3 4836)+1x1 1.131+3 lu 10. 90 ELEMENTARY Then from 4836)-$(.00063 4836)2 1.131+3 In 10=.00063 4836-.OOOOO 0202 +.12310 2197+6.90775 Example 6. e4.80728 Example . 13489 24685 12693. Let a=x-.867. Using 4.2.1, compute successively 1.00000 00000 00000 00000 376199 a4 ---cm* 4 68. Compute ez to 18D for x=.86725 1167 693059 aa -= 3 8e.W728 10. .00048 32583 282384 a2 --=-. 2 ‘3B==e4. e’eg72* 6g=(134.28978)(1.10217 67)=148.0111. 4.2 we compute successively a= Compute e4.gg72ato 7s. 6g Using 4.2.18 and Table 4.4, Linear interpolation gives e.og72* 1.10217 67 “= with an error of 1X 10m7, Since &g- - 1.00048 32583 282384= 1+a, using and Table 9. 5279=7.03149 211. Compute In x working with 16D for x= 1.38967 12458 179231. 4.1.24 FWNCTIONS Example 4.1.24 In 1131.718=(.00063 +ln TRANSCENDENTAL a= .00025 13489 24685 12693 a2 g= 136 - 315 88140 97019 In (l-ta)= aa fg= * 2646 54842 In 1.389 = ln x= Example .00048 31415 965388 .32858 40637 722067 .32906 72053 687455. a4 ig= . 16630 e”=1.00025 7. Compute the principal value of In (f2 f 3i). 4.2 and 4.14. From 4.1.2, 4.1.3 and Tables In (2+3i)=i e.867=2.37976 08513 29496 863 from Table eae.a67=ez=2.38035 Example In (22+32)+i arctan i 13805 15472 81184 90:768 39006 4.4 089. 11. Compute eelsto 7s. =L282475+i(.982794) Let n=&. In (-2+3i)=i In 13+i 7-arctan i > =1.282475fi(2.158799) In (-2--3i)=i In 13-i-i (--?r+arctan Then exp x=exp g > 13+i -arctan ( =1.282475-G(.982794). Example Compute (.227).6gto 7D. Using 4.2.7 and Tables 4.2 and 4.4, =e--l.o~la ln lO)=exp [(n+d) In lo] =lO” exp (d In 10) 3 2> From Table 4.4 es48=exp (E In 10) ==exp (281.42282 42 In 10) =10281exp (.42282 42 In 10)=lOzsl exp .97358 87 8. (.227).8g=e. 68 (& =exp (In 10”) exp (d In 10) =1.282475-i(2.158799) In (2-33i)=iln and d=the decimal part of Go* In C.227) ,e.68(-1.48%0 [email protected]= .35g46 =10281(2.647428)=(!281)2.647428. Example 12. 5262) 60. Compute em2for x= .75 using the expansion in Chebyshev polynomials. . ELE:MENTARY Following 4.2.48 the procedure TRANSCENDENTAL in [4.3] we have from Example 15. Compute sin x and cos x for z=2.317 From 4.344 and Table 4.6 e-z=&m(Z) I * 91 FUNCTIONS where a(r) are the Chebyshev polynomials defined in chapter 22. Assuming be=kO we F;WE&IZ ba, k=7,6,5,. . . 0 from the recurrence sin (2.317)=sin (r-2.317)=sin (.82459 2654) =.73427 cos (2.317)=cos (r-2.317)= -cos be Linear interpolation error of 9X lo-*. - .ooooo 00 15 6 .ooooo 0400 - .OOOOO9560 5 4 Example .03550 4993 - .27432 74.49 .33520 2828 Example 7449) 32925 19943 !Z9577 r 08882 08665 ‘72159 62 r 48481 36811 139535 9936 r 38O=. 66322 51 r 42’=.01221 73 r 32”=.00015 51 r -38’42’32”= .675598 r. 14. Express X= 1.6789 radians in degrees, minutes and seconds to the nearest tenth of a second. From Table 1.1 giving the mathematical constants we have 1 r-l8oo --=57.29577 r 1.6789 r=96.19388’ .19388oX6O=11.633’ .633’X60=38.0” 1.6789 r=96°11‘38.0”. *See page II. 142. 951300 . . . 17. Compute sin x to 19D for x=.86725 13489 24685 12693. Let in radians to 6D. Therefore Example 12 sin 367 The method of reduction to an angle in the first quadrant which was given in Example 15 may also be used. 13. lo=.01745 1’=.00029 l”=.OOOOO 12 cos .867+cos =.29612 e-.‘6=.33520 2828-(.5)(--.27432 = .47236 6553. Express 38’42’32” 2654 gives an 16. sin (12.867)=sin sincef(z)=b,,-((2x-l)bl, Example for x=.82459 Compute sin x for x=12.867 to 8D. From 4.3.16 and Tables 4.6 and 4.8 .00018 9959 -.00300 9164 3 2 1 0 12 (.82459 2654) =-. 67885 60. b,=(4s-2)b,+l-bc+~+AI k 7 to 71). Table a=.867, /3=x-a. From 4.3.16 and 4.6 sin (a+@)=sin a co9 @+cos a sin 0 sin a=.76239 10208 07866 22598 cos a=.64711 66288 94312 75010 With the series expansions for sin B and cos fl we compute successively 1.00000 d -2=- 00000 315 * 8’ a= 00000 OOOOO 88140 97019 16630 * co9 /3= .99999 8= 99684 11859 .00025 13489 24685 12693 Ba T!=- * !c 51- * sin /3= 196li BOO25 sin a cos /3=.76239 ma a sin fl=.OOOlS sin x=.76255 2646 54842 1 13489 09967 26520 36487 22038 25351 67105 92457 57352 31308 82436 13’74 . 92 ELEMENTARY This procedure is equivalent Taylor’s formula 3.6.4. to interpolation Example TRANSCENDENTAL with ABC, a=123, B=29”16’, cos B=(123)2+(321)2 bs=az+d--2ac -2(123)(321) b=221.99934 cos 29’16’ 00 sin A-a sin B --= b (123)(-48887 50196)=.27086 221.99934 00 3gg18 ABC, c2+b2-a2=81+49-16 2bc sin A=.42591 sin B=7(*425g1 l/20= 1.52083 7931 arccot 20=;-arctan 2O=arctan a=4, b=7, 114 2-7 -9 ===.90476 c=9, Express z=3+9i 5954=25’12’31.6” 770g’, in polar form. k is an f+2rk, ~=(3~+9~)f= 9/3=arctan 770g), 3= 1.24904 58. 24. Compute arctan x for. z= l/3 to l2D. From 4.4.34 and 4.4.42 we have Bz.84106 8670 arctan s=arctan (x0+@ h =arctan 4 For k=O, exp (1.24964 58i). =48Y1’22.9” sin C=g(‘425g1 integer. qt%=9.486833 Thus 3+93=9.486833 Example 7709 4 8396. where r==($+y2)+, B=arctan 1905 .05=.04995 23. 0=arctan 19. A=.43997 arctan 20=;-arctan z=x+iy=refe, In the plane triangle find A, B, and C. cos A= 22. Example A=15’42’56.469”. Example Example Compute arctan 20 and arccot 20 to 9D. * Using 4.4.5,4.4.8, and Table 4.14 18. In the plane triangle c=321; find A, b. FUNCTIONS Ccl.86054 xo+arctan : .L+xoh+g =arcm xo+(i-$&&&+~+xJ+. 803 =106’36’5.6” .. . where the supplementary an le must be chosen for (7. As a check we get A+ 27+C=180”00’.1”. We have Example x= f = .33333 33333 33 so that h= .00033 33333 33 20. Compute cot x for x=.4589 to 6D. Since 2<.5, using Table 4.9 with interpolation 1 in (z-l-cot x), we find --cot(.4589)= .4589 .155159. Therefore cot (.4589)=2.179124.155159=2.023965. Example and, from = .32145 Table 4.1,4, arctan 05244 03. zO=arctan Since .00030 00300 03 we get arctan x= .32145 05244 03+.00030 00300 03 --.ooooo 21. Compute arcsin x for 5= .99511. For z>.95, using Table 4.14 with interpolation in the auxiliary functionf(z) we find arcsin 2=:-[2(1-x)I?f(x) =.32175 00000 09 05543 97. If x is given in the form b/a it is convenient use 4.4.34 in the form b arctan ,=arctan xO+arctan b-axe -* a+b arcsin (.99511)=;-[2(.00489)]*j(.99511) In the present example we get =1.57079 6327-(.09889 =1.47186 2100. 388252) (1.00040 7951) .333 h 1 +x&+x;= arctan jj=arctan *see page II. .333+arctan 1 -. 3333 to ELEMENTARY Example TRANSCENDENTAL arctanh .96035=+ In 1+ .96035=$ln- 1.96035 1- .96035 .03965 25. Compute arcsec 2.8 to 5D. Using 4.3.45 and Table 4.14 =a In 49.44136 191 =$(3.90078 7359) = 1.950394. (z”-- I)+ arcsec z=arcsin --y .Example 27. . [(2.8)2-l]+ arcsec 2.8=arcsm 2.8 =arcsin 93 FUNCTIONS Compute arccosh x for x=1.5368 Using Table 4.17 .93404 97735 to 6D. arccosh x=arccosh 1.5368=.852346 (x2-- 1)’ [(1.5368)2-l]* = 1.20559 or using 4.3.45 and Table 4.14 arccosh 1.5368=(.852346)(1.361754)* arcsec z=arctan arcsec 2.8=arctan (z2-l)* =(.852346)(1.166942) 2.61533 9366 = .994638. =z-arctan .38235 951564, f:rom 4.4.3 and 4.4.8 =1.570796-.365207 Example 28. Compute arccosh x for x=31.2 to 5D. Using Tables 4.2 and 4.17 with l/x=1/31.2 = .03205 128205 = 1.20559. Example 26. arccosh 31.2-m Compute arctanh 2 for x=.96035 to 6D. From 4.6.22 and Table 4.2 31.2=.692886 arccosh 31.2=.692886+3.440418=4.13330. References Texts 14.11 B. Carlson! M. Goldstein, R,ttional approximation of functions, Los Alamos Scientific Laboratory LA-1943 (Los Alamos, N. Mex., 1955). [4.2] C. W. Clenshaw, Polynomial approximations to elementary functions, Mat!l. Tables Aids Comp. 8, 143-147 (1954). of [4.3] C. W. Clenshaw, A note c’n the summation Chebyshev series, Math. ‘l’ables Aids Comp. 9, 118-120 (1955). 14.41 G. H. Hardy, A course of pure mathematics, 9th ed. (Cambridge Univ. Press, Cambridge, England, and The Macmillan Co., New York, N.Y., 1947). [4.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. F’ress, Princeton, N.J., 1955). [4.6] C. Hastings, Jr., Note 143, Math. Tables Aids Comp. 6, 68 (1953). [4.7] E. W. Hobson, A treatise on plane tri~metry, 4th ed. (Cambridge Univ. Press, ambndge, England, 1918). [4.8] H. S. Wall, Analytic theory of continued fractions (D. Van Nostrand Co., Iuc., New York, N.Y., 1948). Tables [4.9] E. P. Adams, Smithsonian mathematical formulae and tables of elliptic functions, 3d reprint (The l$~$onian Institution, Washington, D.C., [4.10] H. Andoyer, Nouvelles tables tri onometrlques fondamentales (Hermann et fils, !baris, France, 1916). [4.11] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probabilit functions, 3d ed. (Cambridge Univ. Press, 8 ambridge, England, 1951). [4.12] Chemical Rubber Corn any! Standard mathematical tables, 12th ed. ( B hemmal Rubber Publ. Co., Cleveland! Ohio, 1959). [4.13] L. J. Comne, Chambers’ six-figure mathematical tables, vol. 2 (W. R. Chambers, Ltd., London, [4.14] [4.15] Laboratory., Tables of the [4.16] Harvard. Computation function arcsin z (Harvard Univ. Press, Camz=x+iy, O<x<475, bridge, Mass., 1956). 0 5~ 1475, 6D, varying intervals. [4.17] Harvard Computation Laboratory, Tables of inverse hyperbolic functions (Harvard Univ. Press, Cambridge, Mass., 1949). arctanh x, O<z<l; arcsinh z, 0+<3.5; arccosh z, 1+<3.5; arcsinhs, arccosh x, 3.5_<~<22980, 9D, varying intervals. [4.18] National Bureau of Standards, Tables of lo”, Applied Math. Series 27 (U.S. Government Printing Office, Washington, D.C., 1953). z=O(.OOOO1)1, 10 D. Radix- table of 10n’lo-P, n=1(1)999, p=3(3)15, 15D. ELEMENTARY 94 TRANSCENDENTAL 14.191 National Bureau of Standards, Table of natural logarithms for arguments between zero and five to sixteen decimal places, 2d ed., Applied Math. Series 31 (U.S. Government Printing office, Washington, D.C., 1953). 2=0(.0001)5, 16 D. [4.20] National Bureau of Standards, Tables of the exponential function eZ, 3d ed., Applied Math. Series 14 (U.S. Government Printing Office, Washing.9999, ton, D.C., 1951). Z= -2.4999(.0001) 18D, s=l(.OOOl) 2.4999, 15D, z=2.5(.001)4.999, 15D, z=5(.01)9.99, 12D, z== -.000099(.000001) .000099, 18D, z= - 100(1)100, lQS, z= -9X 10-n(10-n)9X lo-*, n= 10, 9, 8, 7, 18D; values of , [4.21] N~t~?al%ur~~~2,5u56oD. Standards, Table of the descending exponential, 2=2.5 to z=lO, Applied Math. Series 46 (U.S. Government Printing Office, Washington, D.C., 1955). s=2.5(.001)10, ZOU. [4.22] National Bureau of Standards, Tables of sines and cosines for radian arguments, 2d ed., Applied Math. Series 43 (U.S. Government Printing Office, Washingtor, D.C., 1955). sin 5, cos z, z=O(.OO1)25.2, O(l)lOO, 8D, x=10-“(lo-“)9X lo-n, n=5,4,3, 2, 1, 15D, ~=0(.00001) 91, 12D. [4.23] xational Bureau of Standards, Tables of circular and hyperbolic sines and cosines for radian arguments, 2d ed., Applied Math. Series 36 (U.S. Government Printing O&e, Washington, D.C., 1953). sin x, cos z, sinh Z, cash 2, z=O(.OOOl) 1.9999, O(.l)lO, QD. [4,24] National Bureau of Standards, Table of circular and hyperbolic tangents and cot,angents for radian arguments, 2d printing (Columbia Univ. Press, Fey;; prk, N.Y., 1947). tan 5, cot Z, tanh z, 8D or 8S, z=O(.l)lO, ) 2=0(.0001)2, 10D. [4.25] National Bureau of Standards, Table of sines and cosines to fifteen decimal places at hundredths of a degree, Applied Math. Series 5 (U.S. Government Printing Office, Washington, D.C., 1949). sin 5, cos z, z=O”(.O1o)QOo, 15D; supplementary table 30 D. of sin Z, cos z, z=l”(lo)8Qo, [4.26] National Bureau of Standards, Table of secants and cosecants to nine significant figures at hundredths of a degree! Applied Math. Series 40 (U.S. Government Printing Office, Washington, D.C., 1954). [4.27] National Bureau of Standards, Tables of functions and of zeros of functions, Collected short tables of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954). FUNCTIONS [4.28] National Bureau of Standards, Table of arcsin z (Columbia Univ. Fress, New York, N.Y., 1945). arcsin z, s=0(.0001).989(.00001) 1, 12D; auxiliary table of f(u) =[$a-arcsin (l--)]/(2v) %, v=O(.OOOOl).OOO5, 13D. [4.29] National Bureau of Standards, Tables of arctan z, 2d ed., Applied Math. Series 26 (U.S. Government Printing Office, Washington D.C., 1953). s=0(.001,7(.01)50(.1)300(1)2000(10)10000, 12D. [4.30] National Bureau of 8tandards, Table of hyperbolic sines and cosines, x=2 to x=10, Applied Math. Series 45 (U.S. Government Printing Office, Washington, D.C., 1955). 2=2(.001)10, 9s. [4.31] B. 0. Peirce, A short table of integrals, 4th ed. (Ginn and Co., Boston, Mass., 1956). [4.32] J. Peters, Ten-place logarithm table, ~01s. 1, 2 (together with an appendix of mathematical tables) (Berlin, 1~922; rev. ed., Frederick Ungar Publ. Co., New York, N.Y., 1957). [4.33] J. Peters, Seven-place values of trigonometric functions for every thousandth of a de ree (Berlin-Friedenau., 1918; D. Van Nostrand 5 o., Inc., New York, N.Y., 1942). [4.34] L. W. Pollak, R.echentafeln zur harmonischen Analyse (Johann Ambrosius Barth, Leipzig, Germany, 1926). [4.35] A. J. Thompson, Standard table of logarithms to twenty decimal places, Tracts for Computers, No. 22 (Cambridge Univ. Press, Cambridge, England, and New York, :N.Y., 1952). [4.36] J. Todd, Table of arctangents of rational numbers, NBS Applied Math. Series 11 (U.S. Government Printing Office, ‘Washington, D.C., I D51). arctan m/n and arccot m/n, O<m<n<lOO, 12D; reductions of arctan m/n, O<m<n<lOO; reductions of arctan 1% reducible n <2089. for [4.37] U.S. Department of Commerce, Coast and Geodetic Survey, Natural sines and cosines to eight decimal places, Special Publication No. 231 (U.S. GOVernment Printing Office, Washington, D.C., 1942). [4.38] C. E. Van Ostrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences 14, 5th Memoir (U.S. Government Printing Office, Washington, D.C., 1921). [4.39] B. V. Vega, Logarithmic tables of numbers and trigonometrical functions (G. E. Stechert & Co., Ncm York, N.Y., 1905); log10 2, z=1(1)100000; logarithms of the trigonometrical functions for every ten seconds. ELEMENTARY TRANSCENDENTAL COMMON x x log10 T 100 00000 00000 101 00432 13738 102 0086001718 103 0128372247 104 0170333393 150 151 152 153 154 176119 12591 1789769473 181r34 35879 184159 14308 187152 07208 105 106 107 108 109 0211892991 0253058653 0293837777 03342 37555 0374264979 110 log10 1 95 FUNCTIONS LOGARITHMS Table 4.1 x log10 x 300 47712 12547 301 47856 64956 % 48144 69430 4800026285 log10 z 2 200 201 202 203 204 3010299957 3031960574 3053513694 3074960379 3096301674 250 251 252 253 254 3979400087 3996737215 4014005408 4031205212 4048337166 155 156 157 158 159 1903316982 205 193112 45984 206 1950996524 207 198b570870 208 2013971243 209 3117538611 3138672204 3159703455 3180633350 3201462861 255 256 257 258 259 4065401804 4082399653 40993 31233 4116197060 4132997641 305 306 307 308 309 4842998393 48572 14265 48713 83755 48855 07165 48995 84794 112 113 114 0413926852 0453229788 0492180227 0530784435 0569048513 160 161 162 163 164 204:.1 99827 210 3222192947 2061i2 58760 211 3242824553 2091il 50145 212 3263358609 212:.8 76044 213 3283796034 2140438480 214 3304137733 260 261 262 263 264 4149733480 4166405073 4183012913 41995 57485 4216039269 310 311 312 313 314 49136 16938 4927603890 4941545940 4955443375 4969296481 115 116 117 118 119 0606978404 0644579892 0681858617 0718820073 0755469614 165 166 167 168 169 217118 39442 220:.0 80880 222:'l 64711 225:bO 92817 227118 67046 215 216 217 218 219 3324384599 3344537512 3364597338 3384564936 3404441148 265 266 267 268 269 4232458739 4248816366 4265112614 4281347940 4297522800 315 316 317 318 319 49831 05538 4996870826 5010592622 5024271200 5037906831 120 121 122 123 124 0791812460 0827853703 0863598307 0899051114 0934216852 170 171 172 173 174 2304489214 232'1961104 235112 84469 2380461031 240114 92483 220 221 222 223 224 3424226808 3443922737 3463529745 3483048630 3502480183 270 271 272 273 274 4313637642 4329692909 4345689040 4361626470 4377505628 320 321 322 323 324 5051499783 5065050324 5078558717 5092025223 5105450102 125 126 127 128 129 0969100130 1003705451 1038037210 1072099696 1105897103 175 176 177 178 179 243t13 80487 225 3521825181 245f11 26678 226 3541084391 247517 32664 227 3560258572 250l.2 00023 228 3579348470 252fl5 30310 229 3598354823 275 276 277 278 279 4393326938 4409090821 4424397691 4440447959 4456042033 325 326 327 328 329 5118833610 5132176001 5145477527 5158738437 5171958979 130 131 132 133 134 1139433523 1172712957 1205739312 1238516410 1271047984 180 181 182 183 184 2552'725051 257617 85749 26Ot17 13880 2621.5 10897 264tll 78230 230 231 232 233 234 3617278360 3636119799 3654879849 3673559210 3692158574 280 281 282 283 284 4471580313 4487063199 4502491083 4517864355 4533183400 330 331 332 333 334 5185139399 5198279938 5211380837 5224442335 5237464668 135 136 137 138 139 1303337685 1335389084 1367205672 1398790864 1430148003 185 186 187 188 189 2671717284 269fsl 29442 271tb4 16065 2741578493 2761618042 235 236 237 238 239 3710678623 3729120030 3747483460 3765769571 3783979009 285 286 287 288 289 4548448600 4563660331 4578818967 4593924878 4608978428 335 336 337 338 339 5250448070 5263392774 5276299009 5289167003 5301996982 140 141 142 143 144 1461280357 1492191127 1522883444 1553360375 1583624921 190 191 192 193 194 27875 36010 281tl3 33672 283:sO 12287 2855573090 287t~O 17299 240 241 242 243 244 3802112417 3820170426 3838153660 3856062736 3873898263 290 291 292 293 294 4623979979 4638929890 4653828514 4668676204 4683473304 340 341 342 343 344 5314789170 5327543790 53402 61061 5352941200 5365584426 145 146 147 148 149 1613680022 1643528558 1673173347 1702617154 1731862684 195 196 197 198 199 29OC3 46114 2922'560714 2941.6 62262 2966651903 298E15 30764 245 246 247 248 249 3891660844 3909351071 3926969533 3944516808 3961993471 295 296 297 298 299 4698220160 4712917111 4727564493 4742162641 4756711883 345 346 347 348 349 5378190951 53907 60988 5403294748 5415792439 5428254270 150 99957 250 3979400087 300 1760912591 200 3OlCl2 (-;I6 111 [1 [(-F2 1 E log10 2: 304 48287 35836 4771212547 350 54406 80444 C-l)6 [1 For use of common logarithms see Examples 1-3. For 100<~,<135 interpolate in the range 1000<:~~<1350. Compiled from A. J. Thompson, Standard table of logarithms to twenty decimal places, Tracts for Computers, No. 22. Cambridge Univ. Press, Cambridge, England, 1952 (with permission). 96 ELEMENTARY TRANSCENDENTAL Table 4.1 2’ COMMON log10 2 .c log10 z z FUNCTIONS LOGARITHMS log10 2 2 log10 z z log10 2 350 351 352 353 354 54406 54530 54654 54777 54900 80444 71165 26635 47054 32620 400 401 402 403 404 60205 60314 60422 60530 60638 99913 43726 60531 50461 13651 450 451 452 453 454 65321 65417 65513 65609 65705 25138 65419 84348 82020 58529 500 501 502 503 504 69897 69983 70070 70156 70243 00043 77259 37171 79851 05364 550 551 552 553 554 74036 74115 74193 74272 74350 26895 15989 90777 51313 97647 355 356 357 358 359 55022 55144 55266 55388 55509 83531 99980 82161 30266 44486 405 406 407 408 409 60745 60852 60959 61066 61172 50232 60336 44092 01631 33080 455 456 457 458 459 65801 65896 65991 66086 66181 13967 48427 62001 54780 26855 505 506 507 508 509 70329 70415 70500 70586 70671 13781 05168 79593 37123 77823 555 556 557 558 559 74429 74507 74585 74663 74741 29831 47916 51952 41989 18079 360 361 362 363 364 55630 55750 55870 55990 56110 25008 72019 85705 66250 13836 410 411 412 413 414 61278 61384 61489 61595 61700 38567 18219 72160 00517 03411 460 461 462 463 464 66275 66370 66464 66558 66651 78317 09254 19756 09910 79806 510 511 512 513 514 10757 70842 70926 71011 71096 01761 09001 99610 73651 31190 560 561 562 563 564 74818 74896 74973 75050 75127 80270 28613 63156 83949 91040 365 366 367 368 369 56229 56348 56466 56504 56702 28645 10854 60643 78187 63662 415 61804, 416 61909 417 62013 418 62117 419 -62221 80967 33306 60550 62818 40230 465 ‘77; 468 469 66745 66838 66931 67024 67117 29529 59167 68806 58531 28427 515 516 517 518 519 71180 71264 71349 71432 71516 72290 97016 05431 97597 73578 565 566 567 568 569 75204 75281 75350 75434 75511 84478 64312 30589 83357 22664 370 371 372 373 374 56820 56937 57054 57170 57287 17241 39096 29399 88318 16022 420 421 422 423 424 62324 62428 62531 62634 62736 92904 20958 24510 03674 58566 470 471 472 473 474 67209 67302 67394 67486 67577 78579 09071 19986 11407 83417 520 521 522 523 524 71600 71683 71767 71850 71933 33436 77233 05030 16889 12870 570 571 572 573 574 75587 75663 75739 75815 75891 48557 61082 60288 46220 18924 375 376 377 378 379 57403 57518 57634 57749 57863 12677 78449 13502 17998 92100 425 426 427 428 429 62838 62940 63042 63144 63245 89301 95991 78750 37690 72922 475 476 477 478 479 67669 67760 67851 67942 68033 36096 69527 83790 78966 55134 525 526 527 528 529 72015 72098 72181 72263 72345 93034 57442 06152 39225 56720 575 576 577 578 579 75966 76042 76117 76192 76267 78447 24834 58132 78384 85637 380 381 382 383 384 57978 58092 58206 58319 58433 35966 49757 33629 87740 12244 430 431 432 433 434 63346 63447 63548 63648 63748 84556 72702 37468 i'8964 97295 480 481 482 483 484 68124 68214 68304 68394 68484 12374 50764 70382 71308 53616 530 531 532 533 534 72427 72509 72591 72672 72754 58696 45211 16323 72090 12570 580 581 582 583 584 76342 76417 76492 76566 76641 79936 61324 29846 85548 28471 385 386 387 388 389 58546 58658 58771 58883 58994 07295 73047 09650 17256 96013 435 436 437 438 439 63848 63948 64048 64147 64246 92570 64893 14370 41105 45202 485 486 487 488 489 68574 68663 68752 68841 68930 17386 62693 89612 98220 88591 535 536 537 538 539 72835 72916 72997 73078 73158 37820 47897 42857 22757 87652 585 586 587 588 589 76715 76789 76863 76937 77011 58661 76160 81012 73261 52948 390 391 392 393 394 59106 59217 59328 59439 59549 46070 67574 60670 25504 62218 440 441 442 443 444 64345 64443 64542 64640 64738 26765 85895 22693 37262 29701 490 491 492 493 494 69019 69108 69196 69284 69372 60800 14921 51028 69193 69489 540 541 542 543 544 73239 73319 73399 73479 73559 37598 72651 92865 98296 88997 590 591 592 593 594 77085 77158 77232 77305 77378 20116 74809 17067 46934 64450 395 396 397 398 399 59659 59769 59879 59988 60097 70956 51859 05068 30721 28957 445 446 447 448 449 64836 64933 65030 65127 65224 00110 48587 75231 80140 63410 495 496 497 t;; 69460 69548 69635 69722 69810 51989 16765 63887 93428 05456 545 546 547 548 549 73639 73719 73798 73878 73957 65023 26427 73263 05585 23445 595 596 597 598 599 77451 77524 77597 77610 77742 69657 62597 43311 11840 68224 400 60205 99913 450 65321 25138 500 69897 00043 550 74036 26895 600 77815 12504 [1 C-l)4 [93 1 [1 (73 [c-y1 ELEMENTARY TRANSCENDENTAL COMMON X log10 x 97 FUNCTIONS LOGARITHMS Table 4.1 600 601 602 603 604 77815 77887 77959 78031 78103 12504 44720 64913 73121 69386 650 651 652 653 654 81291 81358 al424 81491 81557 33566 39886 75957 31813 77483 700 701 702 703 704 log10 2 84509 80400 84571 80180 84633 71121 84695 53250 84757 26591 605 606 607 608 609 78175 78247 78318 78390 78461 53747 26242 86911 35793 72926 655 656 657 658 659 81624 81690 81756 81822 81888 13000 38394 53696 58936 54146 705 706 707 708 709 84818 84880 84941 85003 85064 91170 47011 94138 32577 62352 755 756 757 758 759 87794 87852 87909 87966 88024 69516 17955 58795 92056 17759 805 806 807 808 809 90579 90633 90687 90741 90794 58804 50418 35347 13608 85216 610 611 612 613 614 78532 78604 78675 78746 78816 98350 12102 14221 04745 83711 660 661 662 663 664 81954 82020 82085 82151 82216 39355 14595 79894 35284 80794 710 711 712 713 714 85125 85186 85247 85308 85369 83487 96007 99936 95299 82118 760 761 762 763 764 88081 88138 88195 88252 88309 35923 46568 49713 45380 33586 810 811 812 813 814 90848 90902 90955 91009 91062 50189 085412 6029:2 05456 44049 615 616 617 618 619 78887 78958 79028 79098 79169 51158 07122 51640 a4751 06490 665 666 667 668 669 82282 82347 82412 82477 82542 16453 42292 58339 64625 61178 715 716 7L7 718 719 85430 85491 85551 85612 85672 60418 30223 91557 44442 88904 765 766 767 768 769 88366 88422 88479 88536 88592 14352 87696 53639 12200 63398 815 816 817 818 819 91115 91169 91222 91275 91328 76087 01588 20565 33037 39018 620 621 622 623 624 79239 79309 79379 79448 79518 16895 16002 03847 80467 45897 670 671 672 673 674 82607 82672 82736 82801 82865 48027 25202 92731 50642 98965 720 721 722 723 724 85733 a5793 85853 85913 a5973 24964 52647 71976 82973 85662 770 771 772 773 774 88649 88705 88761 88817 88874 07252 43781 73003 94939 09607 820 821 822 823 824 91381 91434 91487 91539 91592 38524 31571 18175 98352 72117 625 626 627 628 629 79588 79657 79726 79795 79865 00173 43332 75408 96437 06454 675 676 677 678 679 82930 82994 83058 83122 83186 37728 66959 86687 96939 97743 725 726 727 728 729 86033 86093 86153 86213 86272 80066 66207 44109 13793 75283 775 776 777 778 779 88930 88986 89042 89097 a9153 17025 17213 10188 95970 74577 825 826 827 828 829 91645 91698 91750 91803 91855 39485 00473 55096 03368 45306 630 631 632 633 634 79934 80002 80071 80140 80208 05495 93592 70783 37100 92579 680 681 682 683 684 83250 83314 83378 83442 83505 89127 71119 43747 07037 61017 730 731 732 733 734 86332 86391 86451 86510 86569 28601 73770 10811 39746 60599 780 781 782 783 784 89209 89265 89320 89376 89431 46027 10339 67531 17621 60627 :zi 832 833 834 91907 91960 92012 92064 92116 80924 10238 33263 50014 60506 635 80277 37253 636 80345 711% 637 80413 94323 638 80482 06787 639 80550 08582 685 686 687 688 689 83569 83632 83695 a3758 83821 05715 41157 67371 84382 92219 735 736 737 738 739 86628 86687 86746 86805 86864 73391 78143 74879 63618 44384 785 786 787 788 789 89486 89542 89597 89652 89707 96567 25460 47324 62175 70032 835 836 837 838 839 92168 92220 92272 92324 92376 64755 62774 54580 40186 19608 640 641 642 643 644 80617 80685 80753 80821 80888 99740 80295 50281 09729 58674 690 691 692 693 694 83884 83947 84010 84073 84135 90907 80474 60945 32346 94705 740 741 742 743 744 86923 86981 87040 87098 87157 17197 82080 39053 88138 29355 790 791 792 793 794 89762 89817 89872 89927 89982 70913 64835 51816 31873 05024 840 841 842 843 844 92427 92479 92531 92582 92634 92861 59958 20915 75746 24466 645 646 647 648 649 80955 81023 81090 81157 81224 97146 25180 42807 50059 46968 695 696 697 698 699 84198 84260 84323 84385 84447 48046 92396 27781 54226 71757 745 746 747 748 749 87215 a7273 87332 87390 87448 62727 88275 06018 15979 18177 795 796 797 T2 90036 90091 90145 90200 90254 71287 30677 83214 28914 67793 845 846 847 848 849 92685 92737 92788 92839 92890 67089 036:30 34103 58523 76902 650 81291 33566 c-y 700 84509 80400 750 87506 12634 800 90308 99870 850 92941 89257 C-t,8 [1 X loglcl 2 1 [c-p 1 X 750 751 752 753 754 log10 x 87506 12634 87563 99370 87621 78406 87679 49762 a7737 13459 800 801 802 803 804 log10 2 90308 99870 90363 25161 90417 43683 90471 55453 90525 60487 2 C-47) 1 [I X [1 98 ELEMENTARY Table 4.1 F’UNCTIONS COM MON LOGARITHMS log10 X TRANSCENDENTAL % X log10 x X log10 x X log10 x 2 log10 2 00000 00000 1002 1003 1004 00043 00086 00130 00173 40775 77215 09330 37128 1050 1051 1052 1053 1054 02118 02160 02201 02242 02284 92991 27160 57398 83712 06109 33716 78923 19378 55091 86072 1005 1006 1007 1008 1009 00216 00259 00302 00346 00389 60618 79807 94706 05321 11662 1055 1056 1057 1058 1059 02325 02366 02407 02448 02489 24596 39182 49873 56677 59601 98227 98272 98317 98362 98407 12330 33877 50720 62871 70339 1010 1011 1012 1013 1014 00432 00475 00518 00560 0060:3 13738 11556 05125 94454 79550 1060 1061 1062 1063 1064 02530 02571 02612 02653 02694 58653 53839 45167 32645 16280 965 966 967 968 969 98452 98497 98542 98587 98632 73133 71264 64741 53573 37771 1015 1016 1017 1018 1019 00646 00689 00732 00774 00817 60422 37079 09529 77780 41840 1065 1066 1067 1068 1069 02734 02775 02816 02857 02897 96078 72047 44194 12527 77052 78273 96302 09211 17010 19712 970 971 972 973 974 98677 98721 98766 98811 98855 17343 92299 62649 28403 89569 1020 1021 1022 1023 1024 00860 00902 00945 00987 01029 01718 57421 08958 56337 99566 1070 1071 1072 1073 1074 02938 02978 03019 03059 03100 37777 94708 47854 97220 42814 96614 96661 96707 96754 96801 17327 09867 97341 79762 57140 975 976 977 978 979 98900 98944 98989 99033 99078 46157 98177 45637 88548 26918 1025 1026 1027 1028 1029 01072 01114 01157 01199 01241 38654 73608 04436 31147 53748 1075 1076 1077 1078 1079 03140 03181 03221 03261 03302 84643 22713 57033 87609 14447 930 931 932 933 934 96848 96894 96941 96988 97034 29486 96810 59124 16437 68762 980 981 982 983 984 99122 99166 99211 99255 99299 60757 90074 14878 35178 50984 1030 1031 1032 1033 1034 01283 72247 01325 86653 01367 96973 014:LO 03215 01452 05388 1080 1081 1082 1083 1084 03342 03382 03422 03462 03502 37555 56940 72608 84566 92822 32707 37219 36198 29658 17610 935 936 937 938 939 97081 97127 97173 97220 97266 16109 58487 95909 28384 55923 985 986 987 988 989 99343 99387 99431 99475 99519 62305 69149 71527 69446 62916 1035 1036 1037 1038 1039 01494 01535 01577 01619 01661 03498 97554 87564 73535 55476 1085 1086 1087 1088 1089 03542 03582 03622 03662 03702 97382 98253 95441 88954 78798 94939 94987 95036 95085 95133 00066 77040 48544 14589 75188 940 941 942 943 944 97312 97358 97405 97451 97497 78536 96234 09028 16927 19943 990 991 992 993 994 99563 99607 99651 99694 99738 51946 36545 16722 92485 63844 1040 1041 1042 1043 1044 01703 01745 01786 01828 01870 33393 07295 77190 43084 04987 1090 1091 1092 1093 1094 03742 03782 03822 03862 03901 64979 47506 26384 01619 73220 895 896 897 898 899 95182 95230 95279 95327 95375 30353 80097 24430 63367 96917 945 946 947 948 949 97543 97589 97634 97680 97726 18085 11364 99790 83373 62124 995 996 997 998 999 99782 99825 99869 99913 99956 30807 93384 51583 05413 54882 1045 1046 1047 1048 1049 01911 01953 01994 02036 02077 62904 16845 66817 12826 54882 1095 1096 1097 1098 1099 03941 03981 04020 04060 04099 41192 05541 66276 23401 76924 900 95424 25094 950 97772 36053 1000 00000 00000 ‘-iI 6 1050 02X18 92991 c-3815 1100 04139 26852 C-38)5 850 851 852 853 854 92941 92992 93043 93094 93145 89257 95601 95948 90312 78707 900 901 902 903 904 95424 95472 95520 95568 95616 25094 47910 65375 77503 84305 950 951 952 953 954 97772 97818 97863 97909 97954 36053 05169 69484 29006 83747 1000 ii01 855 856 857 858 859 93196 93247 93298 93348 93399 61147 37647 08219 72878 31638 905 906 907 908 909 95664 95712 95260 95808 95856 85792 81977 72871 58485 38832 955 956 957 958 959 98000 98045 98091 98136 98181 860 861 562 363 164 93449 93500 93550 93601 93651 84512 31515 72658 07957 37425 910 911 912 913 914 95904 95951 95999 96047 96094 13923 83770 48383 07775 61957 960 961 962 963 964 865 866 867 868 869 93701 93751 93801 93851 93901 61075 78920 90975 97252 97764 915 916 917 918 919 96142 96189 96236 96284 96331 10941 54737 93357 26812 55114 870 871 872 873 874 93951 94001 94051 94101 94151 92526 81550 64849 42437 14326 920 921 922 923 924 96378 96425 96473 96520 96567 875 876 877 878 879 94200 94250 94299 94349 94398 80530 41062 95934 45159 88751 925 926 927 928 929 880 881 882 a83 884 94448 94497 94546 94596 94645 26722 59084 85851 07036 22650 885 886 887 888 889 94694 94743 94792 94841 94890 890 891 892 893 894 [ c-y I [1 [1 [1 I ELEMENTARY TRANSCENDENTAL COMMON n log10 1’ log10 2 FUNCTIt 99 INS LOGARITHMS 1ogu.l T 4.1 log10 1: r x 07918 07954 07990 08026 08062 12460 30074 44677 56273 64869 1250 1251 1252 1253 1254 09691 09725 09760 09795 09829 00130 73097 43289 10710 75365 1300 1301 1302 1303 1304 11394 11427 11461 11494 11527 33523 72966 09842 44157 75914 1205 1206 1207 1208 1209 08098 08134 08170 08206 08242 70469 73078 72701 69343 63009 1255 1256 1257 1258 1259 09864 09898 09933 09968 10002 37258 96394 52777 06411 57301 1305 1306 1307 1308 1309 11561 11594 11627 11660 11693 05117 31769 55876 77440 96466 791392 22:197 61:!81 971.47 29803 1210 1211 1212 1213 1214 08278 08314 08350 08386 08421 53703 41431 26198 08009 86867 1260 1261. 1262 1263 1264 10037 10071 10105 10140 10174 05451 50866 93549 33506 70739 1310 1311 1312 1313 1314 11727 11760 11793 11826 11859 12957 26917 38350 47261 53652 06632 06669 06707 06744 06781 59;!54 851jO4 08fj60 28428 45:112 1215 1216 1217 1218 1219 08457 08493 08529 08564 08600 62779 35749 05782 72883 37056 1265 1266 1267 1268 1269 10209 10243 10277 10311 10346 05255 37057 66149 92535 16221 1315 1316 1317 1318 1319 11892 11925 11958 11991 12024 57528 58893 57750 54103 47955 1170 1171 1172 1173 1174 06818 06855 06892 06929 06966 58017 68951 76:117 80:121 80969 1220 1221 1222 1223 1224 08635 08671 08707 08742 08778 98307 56639 12059 64570 14178 1270 1271 1272 1273 1274 10380 10414 10448 10482 10516 37210 55506 71113 84037 94280 1320 1321 1322 1323 1324 12057 12090 12123 12155 12188 39312 28176 14551 98442 79851 25224 83905 39160 90996 39419 1175 1176 1177 1178 1179 07003 07040 07077 07114 07151 781i66 73?17 64h28 52905 38051 1225 1226 1227 1228 1229 08813 08849 08884 08919 08955 60887 04702 45627 83668 18829 1275 1276 1277 1278 1279 10551 10585 10619 10653 10687 01848 06744 08973 08538 05445 1325 1326 1327 1328 1329 12221 12254 12287 12319 12352 58783 35241 09229 80750 49809 05307 05346 05384 05422 05461 84435 26049 64269 99099 30546 1180 1181 1182 1183 1184 07188 07224 07261 07298 07335 201173 98976 74'165 47446 17024 1230 1231 1232 1233 1234 08990 09025 09061 09096 09131 51114 80529 07078 30766 51597 1280 1281 1282 1283 1284 10720 10754 10788 10822 10856 99696 91297 80252 66564 50237 1330 1331 1332 1333 1334 12385 12417 12450 12483 12515 16410 80555 42248 01494 58296 1135 1136 1137 1138 1139 05499 05537 05576 05614 05652 58615 83314 04647 22621 37241 1185 1186 1187 1188 1189 07371 07408 07445 07481 07518 83503 46;390 07190 64,106 18546 1235 1236 1237 1238 1239 09166 09201 09236 09272 09307 69576 84708 96996 06447 13064 1285 1286 1287 1288 1289 10890 10924 10957 10991 11025 31277 09686 85469 58630 29174 1335 1336 1337 1338 1339 12548 12580 12613 12645 12678 12657 64581 14073 61134 05770 1140 1141 1142 1143 1144 05690 05728 05766 05804 05842 48513 56444 61039 62304 60245 1190 1191 1192 1193 1194 07554 07591 07627 07664 07700 69514 171515 62354 04137 43,268 1240 1241 1242 1243 1244 09342 09377 09412 09447 09482 16852 17815 15958 11286 03804 1290 1291 1292 1293 1294 11058 11092 11126 11159 11193 97103 62423 25137 85249 42763 1340 1341 1342 1343 1344 12710 12742 12775 12807 12839 47984 87779 25158 60127 92687 1145 1146 1147 1148 1149 05880 05918 05956 05994 06032 54867 46176 34179 18881 00287 1195 1196 1197 1198 1199 07736 07773 07809 07845 07881 79,353 11797 41504 68181 91331 1245 1246 1247 1248 1249 09516 09551 09586 09621 09656 93514 80423 64535 45853 24384 1295 1296 1297 1298 1299 11226 11260 11293 11327 11360 97684 50015 99761 46925 91511 1345 1346 1347 1348 1349 12872 12904 12936 12968 13001 22843 50599 75957 98922 19497 1150 06069 78404 1200 07918 12160 1250 09691 00130 1300 11394 33523 1350 13033 1100 1101 1102 1103 1104 04139 04178 04218 04257 04296 26852 73190 15945 55124 90734 1105 1106 1107 1108 1109 04336 04375 04414 04453 04493 1110 1111 1112 1113 1114 2 1150 1151 1152 1153 1154 06069 06107 06145 06182 06220 781104 53:!36 24.791 93073 58088 1200 1201 1202 1203 1204 22780 51270 76209 97604 15461 1155 1156 1157 1158 1159 06258 06295 06333 06370 06408 19842 78341 331590 851594 34360 04532 04571 04610 04649 04688 29788 40589 47872 51643 51908 1160 1161 1162 1163 1164 06445 06483 06520 06557 06595 1115 1116 1117 1118 1119 04727 04766 04805 04844 04883 48674 41946 31731 18036 00865 1165 1166 1167 1168 1169 1120 1121 1122 1123 1124 04921 04960 04999 05037 05076 80227 56126 28569 97563 63112 1125 1126 1127 1128 1129 05115 05153 05192 05230 05269 1130 1131 1132 1133 1134 c1 c-3815 [1 c-3814 log10 Table [I c-3814 [I c-3813 2 37685 [I C-38)3 100 ELEMENTARY NATURAL Table 4.2 3 0.000 0: 001 -6.90775 TRANSCENDENTAL In x -m FUNCTIONS LOGARITHMS In 2 X In x X 821371 221917 140274 622464 0.050 0.051 0.052 0.053 0.054 -2.99573 -2.97592 -2.95651 -2.93746 -2.91877 22735 96462 15604 33654 12324 539910 578113 007097 300152 178627 0.100 0.101 0.102 0.103 0.104 -2.30258 -2.29263 -2.28278 -2.27302 -2.26336 50929 47621 24656 62907 43798 940457 408776 978660 525013 407644 0.002 0.003 0.004 -6.21460 -5.80914 -5.52146 52789 80984 29903 09178 0.005 3.006 0.007 0.008 0.009 -5.29831 -5.11599 -4.96184 -4.82831 -4.71053 73665 58097 51299 37373 07016 480367 540821 268237 023011 459177 0.055 0.056 0.057 0.058 oIo59 -2.90042 -2.88240 -2.86470 -2.84731 -2.83021 20937 35882 40111 22684 78350 496661 469878 475869 357177 764176 0.105 0.106 0.107 0.108 0.109 -2.25379 -2.24431 -2.23492 -2.22562 -2.21640 49288 61848 64445 40518 73967 246137 700699 202309. 579174 529934 0.010 0.011 0.012 0.013 0.014 -4.60517 -4.50986 -4.42284 -4.34280 -4.26869 01859 00061 86291 59215 79493 880914 837665 941367 206003 668784 0.060 0,061 0.062 0.063 0.064 -2.81341 -2.79688 -2.78062 -2.76462 -2.74887 07167 14148 08939 05525 21956 600364 088258 370455 906044 224652 0.110 0.111 0.112 0.113 0.114 -2.20727 -2.19822 -2.18925 -2.18036 -2.17155 49131 50776 64076 74602 68305 897208 698029 870425 697965 876416 0.015 0.016 OiO17 0.018 0.019 -4.19970 -4.13516 -4.07454 -4.01738 -3.96331 50778 65567 19349 35210 62998 799270 423558 259210 859724 156966 0.065 0.066 0.067 0.068 0.069 -2.73336 -2.71810 -2.70306 -2.68824 -2.67364 80090 05369 26595 75738 87743 864999 557115 911710 060304 848777 0.115 0.116 0.117 0.118 0.119 -2.16282 -2.15416 -2.14558 -2.13707 -2.12863 31506 50878 13441 06545 17858 188870 757724 843809 164723 706077 0.020 0.021 0.022 0.023 0.024 -3.91202 -3.86323 -3.81671 -3.77226 -3.72970 30054 28412 28256 10630 14486 281461 587141 238212 529874 341914 0.070 0.071 0.072 0.073 0.074 -2.65926 -2.64507 -2.63108 -2.61729 -2.60369 00369 54019 91599 58378 01857 327781 408216 660817 337459 779673 0.120 0.121 0.122 0.123 0.124 -2.12026 -2.11196 -2.10373 -2.09557 -2.08747 35362 47333 42342 09236 37133 000911 853960 488805 097196 771002 0.025 0.026 0.027 0.028 0.029 -3.68887 -3.64965 -3.61191 -3.57555 -3.54045 94541 87409 84129 07688 94489 139363 606550 778080 069331 956630 0.075 0.076 0.077 0.078 0.079 -2.59026 -2.57702 -2.56394 -2.55104 -2.53830 71654 19386 98571 64522 74265 458266 958060 284532 925453 151156 0.125 0.126 0.127 0.128 0.129 -2.07944 -2.07147 -2.06356 -2.05572 -2.04794 15416 33720 81925 50150 28746 798359 306591 235458 625199 204649 0.030 0.031 0.032 0.033 0.034 -3.50655 -3.47376 -3.44201 -3.41124 -3.38139 78973 80744 93761 77175 47543 199817 969908 824105 156568 659757 0.080 0.081 0.082 0.083 0.084 -2.52572 -2.51330 -2.50103 -2.48891 -2.47693 86443 61243 60317 46711 84801 082554 096983 178839 855391 388234 0.130 0.131 0.132 0.133 0.134 -2.04022 -2.03255 -2.02495 -2.01740 -2.00991 08285 79557 33563 61507 54790 265546 809855 957662 603833 312257 0.035 0.036 0.037 0.038 0.039 -3.35240 -3.32423 -3.29683 -3.27016 -3.24419 72174 63405 73663 91192 36328 927234 260271 379126 557513 524906 0.085 0.086 0.087 0.088 0.089 -2.46510 -2.45340 -2.44184 -2.43041 -2.41911 40224 79827 71603 84645 89092 918206 286293 275533 039306 499972 0.135 0.136 0.137 0.138 0.139 -2.00248 -1.99510 -1.98777 -1.98050 -1.97328 05005 03932 43531 15938 13458 437076 460850 540121 249324 514453 0.040 0.041 0.042 0.043 0.044 -3.21887 -3.19418 -3.17008 -3.14655 -3.12356 58248 32122 56606 51632 56450 682007 778292 987687 885746 638759 0.090 0.091 0.092 0.093 0.094 -2.40794 -2.39689 -2.38596 -2.37515 -2.36446 56086 57724 67019 57858 04967 518720 652870 330967 288811 121332 0.140 0.141 0.142 0.143 0.144 -1.96611 -1.95899 -1.95192 -1.94491 -1.93794 28563 53886 82213 06487 19794 728328 039688 808763 222298 061364 0.045 0.046 0.047 0.048 0.049 -3.10109 -3.07911 -3.05760 -3.03655 -3.01593 27892 38824 76772 42680 49808 118173 930421 720785 742461 715104 0.095 0.096 0.097 0.098 0.099 -2.35387 -2.34340 -2.33304 -2.32278 -2.31263 83873 70875 43004 78003 54288 815962 143008 787542 115651 475471 0.145 0.146 0.147 0.148 0.149 -1.93102 -1.92414 -1.91732 -1.91054 -1.90380 15365 86572 26922 30052 89730 615627 738006 034008 180220 366779 0.050 -2.99573 22735 539910 0.100 -2.30258 50929 940457 0.150 -1.89711 99848 858813 For use of natural logarithms see Examples 4-7. In 10 = 2.30258 50929 940457 ELEMJCNTARY TRANSCENDENTAL NATURAL x 11 47 87 31 80 X 101 FUNCTIONS LOGARITHMS Table 4.2 X In x In 2 99848 54421 47581 73575 26765 858813 672127 358607 897016 685079 0.200 0.201 0.202 0.203 0.204 -1.60943 -1.60445 -1.59948 -1.59454 -1.58963 79124 03709 75815 92999 52851 341004 230613 809323 403497 379207 0.250 0.251 0.252 0.253 0.254 -1.38629 -1.38230 -1.37832 -1.37436 -1.37042 43611 23398 61914 57902 10119 In 2 198906 503532 707137 546168 636005 0.150 0.151 0.152 0.153 0.154 -1.897 -1.890 -1.883 -1.877 -1.870 0.155 0.156 0.157 0.158 0.159 -1.86433 -1.85789 -1.85150 -1.84516 -1.83885 01620 92717 94736 02459 10767 628904 326000 338290 551702 619055 0.205 0.206 0.207 0.208 0.209 -1.58474 -1.57987 -1.57503 -1.57021 -1.56542 52998 91101 64857 71992 10270 437289 925560 167680 808191 173260 0.255 0.256 0.257 0.258 0.259 -1.36649 -1.36257 -1.35867 -1.35479 -1.35092 17338 78345 91940 56940 72172 237109 025746 869173 605196 825993 0.160 0.161 0.162 0.163 0.164 -1.83258 -1.82635 -1.82015 -1.81400 -1.80788 14637 09139 89437 50781 88511 483101 976741 497530 753747 579386 0.210 0.211 0.212 0.213 0.214 -1.56064 -1.55589 -1.55116 -1.54646 -1.54177 77482 71455 90043 31132 92639 646684 060706 101246 727119 602856 0.260 0.261 0.262 0.263 0.264 -1.34707 -1.34323 -1.33941 -1.33560 -1.33180 36479 48716 07752 12468 61758 666093 594436 210402 043725 358209 0.165 0.166 0.167 0.168 0.169 -1.80180 -1.79576 -1.78976 -1.78379 -1.77785 98050 74906 14665 12995 65640 815564 255938 653819 788781 590636 0.215 0.216 0.217 0.218 0.219 -1.53711 -1.53247 -1.52785 -1.52326 -1.51868 72508 68712 79254 02161 35491 544743 979720 416775 930480 656362 0.265 0.266 0.267 0.268 0.269 -1.32802 -1.32425 -1.32050 -1.31676 -1.31304 54529 89702 66205 82984 38993 959148 004380 818875 712804 802979 0.170 0.171 0.172 0.173 0.174 -1.77195 -1.76609 -1.76026 -1.75446 -1.74869 68419 17224 08021 36844 99797 318753 794772 686840 843581 676080 0.220 0.221 0.222 0.223 0.224 -1.51412 -1.50959 -1.50507 -1.50058 -1.49610 77326 25774 78971 35075 92271 297755 643842 098576 220183 270972 0.270 0.271 0.272 0.273 0.274 -1.30933 -1.30563 -1.30195 -1.29828 -1.29462 33199 64581 32126 34837 71725 837623 024362 861397 971773 940668 0.175 0.176 0.177 0.178 0.179 -1.74296 -1.73727 -1.73160 -1.72597 -1.72036 93050 12839 55464 17286 94731 586230 439853 083079 900519 413821 0.225 0.226 0.227 0.228 0.229 -1.49165 -1.48722 -1.48280 -1.47840 -1.47403 48767 02797 52615 96500 32754 777169 098512 007344 276963 278974 0.275 0.276 0.277 0.278 0.279 -1.29098 -1.28735 -1.28373 -1.28013 -1.27654 41813 44132 77727 41652 34971 155658 649871 947986 915000 607714 0.180 0.181 0.182 0.183 0.184 -1.71479 -1.70925 -1.70374 -1.69826 -1.69281 84280 82477 85919 91261 95213 919267 163113 053417 407161 731514 0.230 Oi231 0.232 Oi2j3 0.234 -1.46967 -1.46533 -1.46101 -1i45671 -1.45243 59700 75684 79073 68254 41636 589417 603435 158271 164365 244356 0.280 0.281 0.282 0.283 0.284 -1.27296 -1.26940 -1.26584 -1.26230 -1.25878 56758 06096 82080 83813 10408 128874 483913 440235 388994 209310 0.185 0.186 0.187 0.188 0.189 -1.68739 -1.68200 -1.67664 -1.67131 -1.66600 94539 86052 66621 33161 82639 038122 689358 275504 521878 224947 3-22;: 0:237 0.238 0.239 -1.44392 -1.44816 -1.43969 -1.43548 -1.43129 34739 97648 51378 46053 17270 565270 379781 470059 106624 506264 0.285 0.286 0.287 0.288 0.289 -1.25526 -1.25176 -1.24827 -1.24479 -1.24132 60987 34681 30632 47988 85908 134865 622845 225159 461911 697049 0.190 0.191 0.192 0.193 0.194 -1.66073 -1.65548 -1.65025 -1.64506 -1.63989 12068 18509 99069 50900 71199 216509 355072 543555 772515 188089 0.240 0.241 0.242 0.243 0.244 -1.42711 -1.42295 -1.41881 -1.41469 -1.41058 63556 83454 75528 38356 70536 401457 914821 254507 415886 889352 0.290 0.291 0.292 0.293 0.294 -1.23787 -1.23443 -1.23100 -1.22758 -1.22417 43560 20118 14767 26699 55116 016173 106445 138553 650697 434554 0.195 0.196 0.197 0.198 0.199 -1.63475 -1.62964 -1.62455 -1.61948 -1.61445 57204 06197 15502 82482 04542 183903 516198 441485 876018 576447 0.245 0.246 0.247 0.248 0.249 -1.40649 -1.40242 -1.39836 -1.39432 -1.39030 70684 37430 69423 65328 23825 374101 497742 541599 171549 174294 0.295 0.296 0.297 0.298 0.299 -1.22077 -1.21739 -1.21402 -1.21066 -1.20731 99226 58246 31401 17924 17055 423172 580767 794374 767326 914506 0.200 -1.60943 79124 341004 0.250 -1.38629 43611 198906 0.300 -1.20397 28043 259360 C-7612 [1 In 10 = 2.30258 50929 940457 102 ELEMENTARY Table 4.2 TRANSCENDENTAL NATURAL .I’ In $7’ 2’ FUNCTIONS LOGARITHMS :c In .f’ In .I’ ;.;N; 0:302 0.303 0.304 -1.20064 -1.20397 -1.19732 -1.19402 -1.19072 28043 50142 82616 24734 75775 332613 259360 072674 727679 759154 0.350 0.351 0.352 0.353 0.354 -1.04982 -1.04696 -1.04412 -1.04128 -1.03845 21244 90555 41033 72220 83658 986777 162712 840400 488403 483626 0.400 0.401 0.402 0.403 0.404 -0.91629 -0.91379 -0.91130 -0.90881 -0.90634 07318 38516 31903 87170 04010 741551 755679 631160 354541 209870 0.305 0.306 0.307 0.308 0.309 -1.18744 -1.18417 -1.18090 -1.17765 -1.17441 35023 01770 75313 54960 40020 741254 297563 949399 085626 843916 0.355 0.356 0.357 0.358 0.359 -1.03563 -1.03282 -1:OSOOl -1.02722 -1.02443 74895 45481 94972 22925 28904 067213 301066 024980 814367 938582 0.405 0.406 0.407 0.408 0.409 -0.90386 -0.90140 -0.89894 -0.89648 -0.89404 82118 21193 20935 81045 01229 755979 804044 395421 779754 393353 0.310 0.311 0.312 0.313 0.314 -1.17118 -1.16796 -1.16475 -1.16155 -1i15836 29815 23668 20911 20884 22930 029451 029029 726547 419838 738837 0.360 0.361 0.362 0.363 0.364 -1.02165 -1.01887 -1.01611 -1.01335 -1.01060 12475 73206 10671 24447 14113 319814 492561 563660 172863 453964 0.410 0.411 0.412 0.413 0.414 -0.89159 -0.88916 -0.88673 -0.88430 -0.88188 81192 20644 19296 76860 93051 837836 859024 326107 211043 568227 0.315 0.316 Oi317 0.318 0.319 -1.15518 -1.15201 -1i14885 -1.14570 -1.14256 26401 30653 35051 38962 41761 565040 952249 048564 019602 972925 0.365 0.366 0.367 0.368 0.369 -1.00785 -1.00512 -1.00239 -0.99967 -0.99695 79253 19455 34309 23408 86349 996455 807708 275668 132061 416099 0.415 0.416 0.417 0.418 0.419 -0.87947 -0.87707 -0.87466 -0.87227 -0.86988 67587 00187 90571 38464 43590 514388 208738 833356 573807 599993 0.320 0.321 0.322 0.323 0.324 -1.13943 -1.13631 -1.13320 -1.13010 -1.12701 42831 41558 37334 29557 17631 883648 521212 377287 594805 898077 0.370 0.371 0.372 0.373 0.374 -0.99425 -0.99155 -0.98886 -0.98617 -0.98349 22733 32163 14247 68593 94815 438669 747019 089905 383215 676051 0.420 0.421 0.422 0.423 0.424 -0.86750 -0.86512 -0.86274 -0.86038 -0.85802 05677 24452 99649 30999 18237 047231 997556 461252 358591 501793 0.325 0.326 0.327 0.328 0.329 -1.12393 -1.12085 -1.11779 -1.11474 -1.11169 00966 78976 51080 16705 75282 523996 154294 848837 979933 167652 0.375 0.376 0.377 0.378 0.379 -0.98082 -0.97816 -0.97551 -0.97286 -0.97021 92530 61355 00915 10833 90738 117262 922425 341263 625494 997107 0.425 0.426 0.427 0.428 0.429 -0.85566 -0.85331 -0.85097 -0.84863 -0.84629 61100 59327 12657 20834 83600 577202 127666 535125 003403 541201 0.330 0.331 0.332 0.333 0.334 -1.10866 -1.10563 -1.10262 -1.09961 -1.09661 26245 69036 03100 27890 42860 216111 050742 656485 016932 054366 0.380 0.381 0.382 0.383 0.384 -0.96758 -0.96495 -0.96233 -0.95972 -0.95711 40262 59038 46703 02898 27263 617056 554361 755619 014911 944102 0.430 0.431 0.432 0.433 0.434 -0.84397 -0.84164 -0.83932 -0.83701 -0.83471 00702 71888 96907 75509 07448 945289 783893 380267 796472 817322 0.335 0.336 0: s>i 0.338 0.339 -1.09362 -1.09064 -ii08767 -1.08470 -1.08175 47471 41190 23486 93834 51716 570706 189328 297753 991183 016868 0.385 0.386 0.387 0.388 0.389 -0.95451 -0.95191 -0.94933 -0.94674 -0.94417 19446 79095 05859 99393 59353 943528 173062 523552 588636 636908 0.435 OI4% 0.431 0.438 0.439 -0.83240 -0.83011 -0.82782 -0i82553 -0.82325 92478 30356 20838 63686 58659 934530 331027 865469 056909 069657 0.340 0.341 0.342 0.343 0.344 -1.07880 -1.07587 -1.07294 -1.07002 -1.06711 96613 28016 45419 48318 36216 719300 986203 195319 161971 087387 0.390 0.391 0.392 0.393 0.394 -0.94160 -0.93904 -0.93649 -0.93394 -0.93140 85398 77189 34391 56671 43696 584449 967713 916745 128758 842032 0.440 0.441 0.442 0.443 0.444 -0.82098 -0.81871 -0.81644 -0.81418 -0.81193 05520 04035 53969 55089 07165 698302 352911 044389 370014 499123 0.345 Oi346 0.347 0.348 0.349 -1.06421 -1.06131 -1.05843 -1.05555 -1.05268 08619 65039 04990 27992 33567 507773 244128 352779 076627 797099 0.395 0.396 0.397 0.398 0.399 -0.92886 -0.92634 -0.92381 -0.92130 -0.91879 95140 10677 89982 32736 38620 810152 276565 949466 976993 922736 0.445 0.446 0.447 0.448 0.449 -0.80968 -0.80743 -0.80519 -0.80296 -0.80073 09968 63269 66843 20465 23912 158968 620730 685682 671519 398828 0.350 -1.04982 21244 986777 0.400 -0.91629 07318 741551 0.450 -0.79850 [‘-,6’1] L ’ J In 10 = 2.30258 50929 940457 *see page n. * (76962 177716 -77)8 [I ELEMEiNTARY TRANSCENDENTAL NATURAL In X LOGARITHMS In X 2 103 FUNCTIONS x Table In X 4.2 x 0.450 0.451 0.452 0.453 0.454 -0.79850 -0.79628 -0.79407 -0.79186 -0.78965 76962 79394 30991 31534 80809 177716 794587 499059 991030 407891 0.500 0.501 0.502 0.503 0.504 -0.69314 -0.69114 -0.68915 -0.68716 -0.68517 71805 91778 51592 51088 90109 599453 972723 904079 823978 107684 0.550 0.551 0.552 0.553 0.554 -0.59783 -0.59602 -0.59420 -0.59239 -0.59059 70007 04698 72327 72774 05922 556204 292226 050417 598023 348532 0.455 0.456 0.457 0.458 0.459 -0.78745 -0.78526 -0.78307 -0.78088 -0.77870 78600 24694 18880 60948 50689 311866 677510 879324 679521 215919 0.505 0.506 0.507 0.508 0.509 -0.68319 -0.68121 -0.67924 -0.67727 -0.67530 68497 86096 42753 38314 72624 067772 946715 909539 036552 316143 0.555 0.556 0.557 0.558 0.559 -0.58878 -0.58698 -0.58519 -0.58339 -0.58160 71652 69847 00390 63166 58058 357025 315547 548530 008261 270379 0.460 0.461 0.462 0.463 0.464 -0.77652 -0.77435 -0.77219 -0.77002 -0.76787 87894 72359 03879 82248 07267 989964 854885 003982 959030 558818 0.510 0.511 0.512 0.513 0.514 -0.67334 -0.67138 -0.66943 -0.66747 -0.66553 45532 56887 06539 94338 20135 637656 784326 426293 113675 269719 0.560 0.561 0.562 0.563 0.564 -0.57981 -0.57803 -0.57625 -0.57447 -0.57270 84952 43734 34290 56508 10274 529421 594407 884460 424467 840782 0.465 0.466 0.467 0.468 0.469 -0.76571 -0.76356 -0.76142 -0.75928 -0.75715 78733 96448 60213 69830 25105 947807 564912 132397 644903 358577 0.515 0.516 0.517 0.518 0.519 -0.66358 -0.66164 -0.65971 -0.65778 -0.65585 83783 85135 24044 00367 13958 184009 005743 737079 226540 162484 0.565 0.566 0.567 0.568 0.569 -0.57092 -0.56916 -0.56739 -0.56563 -0.56387 95478 12007 59752 38602 48448 356961 789541 543850 609857 558061 0.470 0.471 0.472 0.473 0.474 -0.75502 -0.75289 -0.75077 -0.74865 -0.74654 25842 71849 62933 98904 79572 780328 657193 965817 902041 870606 0.520 0.521 0.522 0.523 0.524 -0.65392 -0.65200 -0.65008 -0.64817 -0.64626 64674 52372 76910 38149 35946 066640 287701 994983 172142 610949 0.570 0.571 0.572 0.573 0.574 -0.56211 -0.56036 -0.55861 -0.55686 -0.55512 89181. 60693 62876 95622 58826 535412 261268 023392 673975 625706 0.475 0.476 0.477 0.478 0.479 -0.74444 -0.74233 -0.74023 -0.73814 -0.73605 04749 74247 87880 45464 46815 474958 507170 937958 906811 712218 0.525 0.526 0.527 0.528 0.529 -0.64435 -0.64245 -0.64055 -0.63865 -0.63676 70163 40662 47304 89952 68471 905133 444272 407747 758756 238377 0.575 0.576 0.577 0.578 0.579 -0.55338 -0.55164 -0.54991 -0.54818 -0.54645 52381 76182 30124 14103 28014 847866 862458 740375 097596 091418 0.480 0.481 0.482 0.483 0.484 -0.73396 -0.73188 -0.72981 -0.72773 -0.72567 91750 80088 11649 86253 03722 802004 763759 315367 295644 655053 0.530 0.531 0.532 0.533 0.534 -0.63487 -0.63299 -0.63111 -0.62923 -0.62735 82724 32577 17896 38548 94400 359695 401982 404927 162925 219422 0.580 O;iiSl 0.582 0.583 0.584 -0.54472 -0i54300 -0.54128 -0.53956 -0.53785 71754 45221 48312 80926 42961 416720 302258 506992 316447 539100 0.485 0.486 0.487 0.488 0.489 -0.72360 -0.72154 -0.71949 -0.71743 -0.71539 63880 66550 11558 98731 27895 446539 816433 995473 289899 072650 0.535 0.536 0.537 0.538 0.539 -0.62548 -0.62362 -0.62175 -0.61989 -0.61803 85320 11179 71844 67188 97080 861305 113351 732724 203526 731399 0.585 0.586 0.587 0.588 0.589 -0.53614 -0.53443 -0.53273 -0.53102 -0.52932 34317 54894 04591 83310 90953 502806 051244 540406 835101 305503 0.490 0.491 0.492 0.493 0.494 -0.71334 -0.71131 -0.70927 -0.70724 -0.70521 98878 11511 65624 61049 97617 774648 876165 898289 394469 942145 0.540 0.541 0.542 0.543 0.544 -0.61618 -0.61433 -0.61248 -0.61064 -0.60880 61394 60001 92775 59590 60321 238170 356555 424908 482016 261944 0.590 ;A;; 0:593 0.594 -0.52763 -0.52593 -0.52424 -0.52256 -0.52087 27420 92615 86440 08799 59596 823719 760389 981314 844116 194921 0.495 0.496 0.497 0.498 0.499 -0.70319 -0.70117 -0.69916 -0.69715 -0.69514 75164 93522 52528 52019 91832 134468 572096 855083 574841 306184 0.545 0.546 0.547 0.548 0.549 -0.60696 -0.60513 -0.60330 -0.60147 -0.59965 94843 63032 64765 99920 68374 188930 372320 601558 341215 726064 0.595 0.596 0.597 0.598 0.599 -0.51919 -0.51751 -0.51583 -0.51416 -0.51249 38734 46119 81655 45250 36808 365073 167873 895350 315053 666877 0.500 -0.69314 71805 599453 0.550 -0.59783 70007 556204 0.600 -0.51082 56237 659907 [I C-76 [1 95 ln lo= 2.30258 50929 940457 104 ELEMENTARY TRANSCENDENTAL Table 4.2 NATURAL In z 2 56231 659907 469295 733160 549516 473221 0.650 0.651 0.652 0.653 0.654 -0.43078 -0.42924 -0.42771 -0.42617 -0.42464 -0.50252 -0.50087 -0.49922 -0.49758 -0.49593 68209 52929 64879 70112 512956 128226 226388 159700 722400 0.655 0.656 0.657 0.658 0.659 0.610 0.611 0.612 0.613 0.614 -0.49429 -0.49265 -0.49102 -0.48939 -0.48776 63218 83198 29964 03430 03508 147801 105417 698110 459257 349946 0.660 0.661 0.662 0.663 0.664 -0.41400 -0.41248 -0.41098 -0.40947 0.615 0.616 0.617 0.618 0.619 -0.48613 -0.48450 30111 83154 756192 486173 -0.48126 -0.47965 68215 00062 244463 0.665 0.666 0.667 0.668 0.669 0.620 0.621 0.622 0.623 0.624 -0.47803 -0.47642 -0.47481 -0.47320 -0.47160 58009 41970 51862 87601 49106 429998 486583 429576 946839 127094 0.670 0.671 0.625 -0.47000 36292 457356 0.626 -0.46840 -0.46680 -0.46521 49078 87383 820385 492164 51125 40222 139384 816965 0.675 0.676 0.677 0.678 0.679 54595 94164 58848 965587 409239 352796 0.680 -0.51082 -0.50916 -0.50749 -0.50583 -0.50418 0.605 0.606 0.607 0.608 0.609 :* 2 0: 629 0.630 03970 -0.48288 62550 767492 -0.46362 -0.46203 -0.46044 -0.45886 -0.45728 975409 0.672 0.673 0.674 0.681 0.682 924543 56367 735678 07170 81497 79275 554841 057060 249384 -0.42312 -0.42159 00433 44900 468851 380480 -0.42007 -0.41855 -0.41703 12604 975265 03476 17444 568199 796298 54439 14391 97230 02887 31295 616658 304508 451288 962745 057032 -0.40796 -0.40646 -0.40496 -0.40346 -0.40197 82383 56084 52330 71054 262829 417479 665133 454913 x 0.700 0.701 0.702 0.703 0.704 -0.35667 -0.35524 -0.35382 -0.35239 -0.35097 49439 73919 18749 83871 69228 387324 475470 563259 714721 0.705 0.706 0.707 0.708 0.709 -0.34955 -0.34814 -0.34672 -0.34531 -0.34389 74761 00414 46130 11852 97524 698684 888950 855643 884173 500096 0.710 0.711 0.712 -0.34249 -0.34108 03089 28491 788962 -0.33967 73675 0.713 701613 -0.33827 -0.33687 38585 678411 -0.33547 -0.33407 27362 0.716 51120 881294 214914 12188 539086 ao@ . 5:i 0.719 -0.33128 -0.33267 -Or32989 5 4383 97099 39212 339129 825167 610904 -0.40047 -0.39898 -0.39749 -0.39600 -0.39452 75665 61420 69384 99493 51680 971253 104553 589875 374092 698300 0.720 0.721 0.722 0.723 0.724 -0.32850 -0.32711 -0.32573 -0.32434 -0.32296 40669 61416 01400 60568 38865 720361 971880 893108 233724 964207 -0.39304 -0i39i56 -0.39008 25881 22029 40060 096072 391730 698621 0.7i!5 0.726 0.727 -0.32158 -0.32020 -0.31882 -0.31745 36241 52641 88014 42307 274623 573410 486177 854511 -0.41551 -0Ij8860 79910 417415 0.714 0.71!i 0.728 240947 467759 23166 425527 -0.38713 41514 234409 0.729 -0.31608 15469 734789 -0.38566 -0.38419 -0.38272 24808 29728 56211 04194 73613 119847 326247 0.7'30 -0.31471 07448 -0.31334 -0.31197 18192 323585 386750 a. 731 0.732 113470 595866 0.733 -0.31060 47650 95770 0.134 -0.30924 62503 208255 954856 676215 -0.45570 48568 63245 379609 0.683 -0.38126 449111 0.684 -0.37979 0.635 0.636 0.637 0.638 0.639 -0.45413 -0.45255 -0.45098 -0.44941 -0.44785 02800 67156 56234 69956 08246 894454 420149 099737 373472 046022 0.685 0.686 -0.37833 -0.37687 0.681 0.688 -0.37542 -0.37396 0.689 0.640 0.641 0.642 0.643 0.644 -0.44628 -0.44472 -0.44316 -0.44161 -0.44005 71026 58220 05547 65528 284195 614670 921759 445177 777834 0.645 0.646 0.647 0.649 -0.43850 -0.43695 -0.43540 -0.43386 -0.43232 49621 57751 89844 45826 25622 0.650 -0.43078 29160 0.648 In 2 29160 i* t;: 0:633 0.634 69752 LOGARITHMS In 2 X 03444 78336 80822 10810 0.600 OibOl 0.602 0.603 0.604 FUNCTIONS 64407 199118 76512 09867 562518 597877 0.735 0.736 0.137 -0.30788 -0.30652 -0.30516 -0.37251 64410 40079 487934 684785 0.738 0.739 0.690 0.691 0.692 0.693 0.694 -0.37106 -0.36961 -0.36816 -0.36672 -0.36528 36813 54552 93233 52797 33184 908320 144672 644675 922330 753326 863646 995352 812365 298624 780471 0.695 0.696 0.697 0.698 0.699 -0.36384 -0.36240 -0.36096 -0.35953 -0.35810 34334 56186 98682 61762 45367 216132 197646 483268 924543 0.700 -0.35667 49439 381324 173449 477174 In lO= 2.30258 50929 940457 397002 47797 693004 -0.30381 51602 73867 14543 532608 928004 816646 -0: 30245 73580 339353 0.740 0.741 -0.30110 -0.29975 50927 839216 0.742 -0.29840 46536 60358 0.743 0.744 -0.29705 -0.29571 92342 42441 0.745 0.746 0.747 0.748 -0.29437 -0.29302 -0i29169 -0.29035 10606 96787 00938 23010 0.749 -0.28901 62954 076598 649176 0.750 -0.28768 20724 517809 860502 147566 643779 490452 025775 783762 493197 ELEMENTARY TRANSCENDENTAL NATURAL In X x LOGARITHMS In T X 105 FUNCTIONS Table 4.2 In X x 0.750 0.751 0.752 0.753 0.754 -0.28768 -0.28634 -0.28501 -0.28369 -0.28236 20724 96272 89550 00511 29109 517809 180023 322973 822435 741810 0.800 0.801 0.802 0.803 0.804 -0.22314 -0.22189 -0.22064 -0.21940 -0.21815 35513 43319 66711 05650 60098 142098 137778 156226 353754 031707 0.850 0.851 0.852 0.853 0.854 -0.16251 -0.16134 -0.16016 -0.15899 -0.15782 89294 31504 87521 57314 40851 977749 087629 528213 904579 935672 0.755 0.756 0.757 0.758 0.759 -0.28103 -0.27971 -0.27839 -0.27707 -0.27575 75297 39028 20255 18933 35015 331123 026041 446883 397654 865071 0.805 0.806 0.807 0.808 0.809 -0.21691 -0.21567 -0.21443 -0.21319 -0.21195 30015 15364 16107 32204 63619 635737 755088 121883 610417 236454 0.855 0.856 0.857 0.858 0.859 -0.15665 -0.15548 -0.15431 -0.15315 -0.15198 38100 49028 73603 11794 63569 453768 403950 843573 941748 978817 0.760 0.761 0.762 0.763 0.764 -0.27443 -0i27312 -0.27180 -0.27049 -0.26918 68457 19211 87232 72476 74898 017603 204512 954908 976800 156166 0.810 0.811 0.812 0.813 0.814 -0.21072 -0.20948 -0.20825 -0.20702 -0.20579 10313 72248 49388 41694 49129 156526 667241 204591 343265 795968 0.860 0.861 0.862 0.863 0.864 -0.15082 -0.14966 -0.14850 -0.14734 -0.14618 28897 07745 00083 05878 25101 345836 544063 184440 987091 780814 0.765 0.766 0.767 0.768 0.769 -0.26787 -0.26657 -0.26526 -0.26396 -0.26266 94451 31092 84776 55458 43094 556012 415458 148809 344649 764931 0.815 0.816 0.817 0.818 0.819 -0.20456 -0.20334 -0.20211 -0.20089 -0.19967 71657 09240 61841 29423 11951 412743 180300 221342 793900 290676 0.865 0.866 0.867 0.868 0.869 -0.14502 -0.14387 -0.14271 -0.14156 -0.14041 57720 03704 63022 35643 21537 502577 197019 015952 217869 167450 0.770 0.771 0.772 0.773 0.774 -0.26136 -0.26006 -0.25877 -0.25747 -0.25618 47641 69054 07289 62303 34053 344075 188076 573609 947151 924099 0.820 0.821 0.822 0.823 0.824 -0.19845 -0.19723 -0.19601 -0.19479 -0.19358 09387 21695 48839 90783 47490 238383 297088 259571 050672 726654 0.870 0.871 0.872 0.873 0.874 -0.13926 -0.13811 -0.13696 -0.13581 -0.13467 20673 33021 58550 97231 49033 335076 296343 731574 425348 266016 0.775 0.776 0.777 0.778 0.779 -0.25489 -0.25360 -0.25231 -0.25102 -0.24974 22496 27587 49286 87548 42331 287901 989183 144896 037454 113888 0.825 0.826 0.827 0.828 0.829 -0.19237 -0.19116 -0.18995 -0.18874 -0.18753 18926 05054 05839 21245 51238 474561 611590 584457 968774 468421 0.875 0.876 0.877 0.878 0.879 -0.13353 -0.13238 -0.13124 -0.13010 -0.12897 13926 91880 82866 86853 03812 245226 457456 099540 470204 969601 0.780 0.781 0.782 0.783 0.784 -0.24846 -0.24718 -0.24590 -0.24462 -0.24334 13592 01291 05384 25829 62586 984996 424511 368260 913340 317292 0.830 0.831 0.832 0.833 0.834 -0.18632 -0.18512 -0.18392 -0.18272 -0.18152 95781 54841 28381 16368 18766 914934 266889 609285 152944 233903 0.880 0.881 0.882 0.883 0.884 -0.12783 -0.12669 -0.12556 -0.12443 -0.12329 33715 76530 32229 00783 82163 098849, 459575 753457 781770 444936 0.785 0.786 0.787 0.788 0.789 -0.24207 -0.24079 -0.23952 -0.23825 -0.23698 15611 84865 70305 71891 89581 997286 529305 647338 242579 362628 0.835 0.836 0.837 0.838 0.839 -0.18032 -0.17912 -0.17793 -0.17673 -0.17554 35541 66658 12084 71785 45725 312816 974354 926617 000540 149309 0.885 0.886 0.887 0.888 0.889 -0.12216 -0.12103 -0.11991 -0.11878 -0.11765 76339 83283 02966 35359 80434 742075 770561 725576 899670 682325 0.790 Oi791 0.792 0.793 0.794 -0.23572 -0i23445 -0.23319 -0.23193 -0.23067 23335 73112 38871 20573 18177 210699 144832 677112 472891 350013 0.840 0.841 0.842 0.843 0.844 -0.17435 -0.17316 -0.17197 -0.17078 -0.16960 33871 36190 52647 83209 27843 447778 091890 398103 802816 861799 0.890 0.891 0.892 0.893 0.894 -0.11653 -0.11541 -0.11428 -0.11316 -0.11204 38162 08515 91464 86981 95038 559515 113277 021277 056380 086229 0.795 0.796 0.797 0.798 0.799 -0.22941 -0.22815 -0.22690 -0.22564 -0.22439 31643 60931 06001 66815 43332 278052 377540 919220 323283 158624 0.845 0.846 0.847 ;.;&I; . -0.16841 -0.16723 -0.16605 -0.16487 -0.16369 86516 59193 45843 46431 60926 249632 759138 300827 902340 707897 0.895 0.896 0.897 ;.;N: . -0.11093 -0.10981 -0.10869 -0.10758 -0.10647 15607 48660 94169 52106 22445 072817 072066 233409 799374 105168 0.800 -0.22314 35513 142098 0.850 -0.16251 89294 977749 0.900 -0.10536 05156 578263 [1 (72 [1 C-67)2 In lO= 2.30258 50929 940457 [1 (72 106 ELEMENTARY Table 4.2 NATURAL In z x TRANSCENDENTAL FUNCTIONS LOGARITHMS In X In x X 2 0.900 0.901 0.902 0.903 0.904 -0.10536 -0.10425 -0.10314 -0.10203 -0.10092 05156 00213 07589 27255 59185 578263 737991 195134 651516 899606 0.950 0.951 0.952 0.953 0.954 -0.05129 -0.05024 -0.04919 -0.04814 -0.04709 32943 12164 02441 03753 16075 875505 367467 907717 279349 338505 1.000 0.00000 1.0011 0.00099 1 OOZ! 0.00199 1:003 0.00299 1.004 0.00399 00000 000000 95003 80026 55089 20212 330835 626731 797985 695375 0.905 0.906 0.907 0.908 0.909 -0.09982 -0.09871 -0.09761 -0.09651 -0.09541 03352 59729 28288 09003 01848 822109 391577 670004 808438 046582 0.955 0.956 0.957 0.958 0.959 -0.04604 -0.04499 -0.04395 -0.04290 -0.04186 39385 73659 18875 75010 42040 014068 307358 291828 112765 986988 1.005 1.006 1.007 1.008 1.009 0.00498 0.00598 0.00697 0.00796 0.00895 75415 20716 56137 81696 97413 110391 775475 364252 491769 714719 0.910 0.911 0.912 0.913 0.914 -0.09431 -0.09321 -0.09211 -0.09101 -0.08992 06794 23817 52889 93983 47075 712413 221787 078057 871686 279870 0.960 0.961 0.962 0.963 0.964 -0.04082 -0.03978 -0.03874 -0.03770 -0.03666 19945 08700 08283 18671 39843 202551 118446 164306 840115 715914 1.010 1.011. 1 OK! 1:013 1.014 0.00995 0.01093 0.01192 0.01291 0.01390 03308 99400 85708 62252 29051 531681 383344 652738 665463 689914 0.915 0.916 0.917 0.918 0.919 -0.08883 12137 -0iO8773 89143 -0.08664 78067 -0iO8555 78883 -0.08446 91566 066157 080068 256722 616466 264500 0.965 Oi966 0.967 0.968 0.969 -0.03562 -0iO3459 -0.03355 -0.03252 -0.03149 71776 14447 67835 31917 06670 431511 696191 288427 055600 913708 1.015 1.016 1.017 l.OlEl 1.019 0.01488 OiOi587 0.01685 0.01783 0.01882 86124 33491 71170 99181 17542 937507 562901 664229 283310 405878 0.920 Oi921 0.922 Oi923 0.924 -0.08338 -0.08229 -0.08121 -0.08012 -0.07904 16089 52427 00554 60444 32073 390511 268302 255432 792849 404529 0.970 0.971 0.972 0.973 0.974 -0.03045 92074 847085 -0iO2942 88106 908121 -0.02839 94745 216980 -0.02737 11967 961320 -0.02634 39753 396020 1.020 1.021. l.O2i! 1.023 1.024 0.01980 OiO2078 0.02176 OiO2273 0.02371 26272 25391 14917 94869 65266 961797 825285 815127 694894 173160 0.925 0.926 0.927 0.928 0.929 -0.07796 -0.07688 -0.07580 -0.07472 -0.07364 15414 10443 17134 35461 65401 697119 359577 162819 959365 682985 0.975 0.976 0.977 0.978 0.979 -0.02531 -0.02429 -0.02326 -0.02224 -0.02122 78079 26925 86269 56089 36364 842899 690446 393543 473197 516267 :- i:: 1: 02i: 1.028 1.029 0.02469 0.02566 0.02664 0.02761 0.02858 26125 77467 19309 51670 74568 903715 485778 464212 329734 519126 '0.930 0.931 0.932 0.933 0.934 -II.07257 -6.07149 -0.07042 -0.06935 -0.06827 06928 60017 24642 00781 88407 348354 050700 965459 347932 532944 0.980 0.981 0.982 0.983 0.984 -0.02020 -0.01918 -0.01816 -0.01714 -0.01612 27073 28194 39706 61588 93819 175194 167740 276712 349705 298836 1.030 1.031. 1 03i! 11033 1.034 0.02955 0.03052 0.03149 0.03246 0.03343 88022 92050 86670 71901 47760 415444 348229 593710 375015 862374 0.935 0.936 0.937 0.938 0.939 -0.06720 -0.06613 -0.06507 -0.06400 -0.06293 87496 98025 19967 53299 97997 934501 045450 437149 759124 738741 0.985 0.986 0.987 0.988 0.989 -0.01511 -0.01409 -0.01308 -0.01207 -0.01106 36378 89243 52395 25812 09473 100482 795016 486555 342692 594249 1.035 1.036 1.037 1.038 1.039 0.03440 0.03536 0.03633 0.03729 0.03825 14267 71438 19292 57847 87121 173324 372913 473903 436969 170903 0.940 0.941 0.942 0.943 0.944 -0.06187 -0.06081 -0.05975 -0.05868 -0.05762 54037 21393 00044 89963 91128 180875 967574 057740 486796 366364 0.990 o1991 0.992 0.993 0.994 -0.01005 -0~00904 -0.00803 -0.00702 -0.00601 03358 07446 21716 46149 80723 535014 521491 972643 369645 255630 1.040 1.041. 1 042 1:043 1.044 0.03922 OiO4018 0.04114 0.04210 0.04305 07131 17896 19433 11760 94894 532813 328318 311752 186354 604470 0.945 0.946 0.947 0.948 0.949 -0.05657 -0.05551 -0.05445 -0iO5340 -0.05234 03514 27099 61857 07767 64803 883943 302588 960588 271152 722092 0.995 0.996 0.997 0.998 0.999 -0.00501 -0.00400 -0.00300 -0.00200 -0.00100 25418 80213 45090 20026 05003 235443 975388 202987 706731 335835 1.045 1.046 1.047 1.048 1.049 0.04401 0.04497 0.04592 0.04688 0.04783 68854 33656 89318 35858 73294 167743 427312 883998 988504 141601 0.950 -0.05129 32943 875505 1.000 0.00000 00000 000000 1.050 0.04879 01641 694320 [1 C-67)2 [ In 10=2.30258 (-iI 1 I 50929 940457 1 [C-67) 1 ELEMENTARY TRANSCENDENTAL NATURAL In 2 X 107 FUNCTIONS LOGARITHMS In X Table 4.2 In X x x 1.050 1.051 1.052 1.053 1.054 0.04879 OiO4974 0.05069 0.05164 0.05259 01641 20918 31143 32331 24501 694320 948141 155181 518384 191706 1.100 1.101 1.102 1.103 1.104 0.09531 0.09621 0.09712 0.09803 0.09893 01798 88577 67107 37402 99478 043249 405429 307227 713654 549036 1.150 1.151 1.152 1.153 1.154 0.13976 0.14063 0.14149 0.14236 0.14323 19423 11297 95622 72412 41680 751587 397456 736995 869220 859078 1.055 1.056 1.057 1.058 1.059 0.05354 0.05448 0.05543 0.05638 0.05732 07669 81852 47068 03334 50666 280298 840697 881006 361076 192694 1.105 1.106 1.107 1.108 1.109 0.09984 0.10074 0.10165 0.10255 0.10345 53349 99031 36537 65883 87083 697161 001431 264998 250921 682300 1.155 1.156 1.157 1.158 1.159 0.14410 0.14496 0.14583 0.14669 0.14755 03439 57702 04482 43791 75643 737569 501857 115395 508035 576147 1.060 1.061 1.062 1.063 1.064 0.05826 0.05921 0.06015 0.06109 0.06203 89081 18596 39228 50993 53909 239758 318461 197471 598109 194526 1.110 1.111 1.112 1.113 1.114 0.10436 0.10526 0.10616 0.10705 0.10795 00153 05106 01958 90722 71415 242428 574929 283906 934078 050923 1.160 1.161 1.162 1.163 1.164 0.14842 0.14928 0.15014 0.15100 0.15186 00051 17027 26584 28735 23493 182733 157544 297195 365274 092461 1.065 1.066 1.067 1.068 1.069 0.06297 0.06391 0.06485 0.06578 0.06672 47991 33257 09723 77405 36320 613884 436528 196163 380031 429082 1.115 1.116 1.117 1.118 1.119 0.10885 0.10975 0.11064 0.11154 0.11243 44049 08639 65200 13747 54293 120821 591192 870637 329074 297882 1.165 1.166 1.167 1.168 1.169 0.15272 0.15357 0.15443 0.15529 0.15614 10870 90879 63533 28844 86824 176639 283006 044189 060353 899314 1.070 1.071 1.072 1.073 1.074 0.06765 0.06859 0.06952 OiO7045 0.07138 86484 27914 60626 84636 99960 738148 656117 486102 485614 866729 1.120 1.121 1.122 1.123 1.124 0.11332 0.11422 0.11511 0.11600 0.11689 86853 11440 28071 36757 37514 070032 900229 005046 563061 714993 1.170 1.171 1.172 1.173 1.174 0.15700 0.15785 0.15871 0.15956 0.16041 37488 80846 16911 45696 67214 096648 155803 548209 713384 059047 1.075 1.076 1.077 1.078 1.079 0.07232 0.07325 0.07417 0.07510 0.07603 06615 04617 93981 74724 46862 796261 395927 742515 868054 759976 1.125 1.126 1.127 1.128 1.129 0.11778 0.11867 0.11955 0.12044 0.12133 30356 15297 92350 61530 22851 563835 174986 576392 758672 675250 1.175 1.176 1.177 1.178 1.179 0.16126 0.16211 0.16296 0.16381 0.16466 81475 88494 88282 80852 66215 961223 764352 781397 293950 552339 1.080 1.081 1.082 1.083 1.084 0.07696 0.07788 0.07881 0.07973 0.08065 10411 65386 11804 49680 79030 361283 570712 242898 188536 174545 1.130 1.131 1.132 1.133 1.134 0.12221 0.12310 0.12398 0.12486 0.12575 76327 21971 59797 89820 12053 242492 339834 809912 458693 055603 1.180 1.181 1.182 1.183 1.184 0.16551 0.16636 0.16720 0.16805 0.16889 44384 15372 79189 35849 85364 775734 152253 839065 962497 618139 1.085 1.086 1.087 1.088 1.089 0.08157 0.08250 0.08342 0.08434 0.08525 99869 12215 16081 11484 98439 924229 117437 390724 337509 508234 1.135 1.136 1.137 1.138 1.139 0.12663 0.12751 0.12839 0.12927 0.13015 26509 33202 32147 23357 06844 333660 989596 683990 041392 650451 1.185 1.186 1.187 1.188 1.189 0.16974 0.17058 0.17142 0.17227 0.17311 27745 63005 91156 12209 26177 870945 755337 275310 404532 086448 1.090 1.091 1.092 1.093 1.094 0.08617 0.08709 0.08801 0.08892 0.08984 76962 47068 08773 62091 07039 410523 509338 227133 944015 997895 1.140 1.141 1.142 1.143 1.144 0.13102 0.13190 0.13278 0.13365 0.13453 82624 50708 11112 63848 08929 064041 799386 338185 126736 576062 1.190 1.191 1.192 1.193 1.194 0.17395 0.17479 0.17563 0.17647 0.17730 33071 32903 25686 11431 90149 234380 731631 431580 157791 704103 1.095 1.096 1.097 1.098 1.099 0.09075 0.09166 0.09257 0.09349 0.09440 43632 71885 91812 03430 06754 684641 258238 930932 873389 214843 1.145 1.146 1.147 1.148 1.149 0.13540 0.13627 0.13714 0.13802 0.13889 46370 76182 98381 12978 19988 062030 925478 472336 973747 666186 1.195 1.196 1.197 1.198 1.199 0.17814 0.17898 0.17981 0.18065 Oil8148 61853 26555 84265 34996 78760 834740 284400 758361 932576 453772 1.100 0.09531 01798 043249 6;) 1 [ I 1.150 0.13976 19423 751587 1.200 0.18232 15567 939546 In 10 = 2.30258 50929 940457 [c-y1 108 ELEMENTARY TRANSCENDENTAL NATURAL Table 4.2 In 2 X FUNCTIONS LOGARITHMS In X x In X x 1.200 1.201 1.202 1.203 1.204 0.18232 0.18315 0.18398 0.18481 0.18564 15567 45430 68361 84369 93468 939546 978465 130158 925418 866293 1.250 1.251 1.252 1.253 1.254 0.22314 0.22394 0.22474 0.22554 0.22633 35513 32314 22726 06759 84422 142098 847741 779068 139312 107290 1.300 1.301 1.302 1.303 1.304 0.26236 0.26313 0.26390 0.26466 0.26543 42644 31995 15437 92981 64635 674911 303682 863775 427081 044612 1.205 1.206 1.207 1.208 1.209 0.18647 0.18730 0.18813 0.18896 0.18979 95669 90983 79421 60995 35716 426183 049937 153944 126232 326556 1.255 1.256 1.257 1.258 1.259 0.22713 0.22793 0.22872 0.22952 0.23031 55725 20680 79296 31582 77550 837472 460069 081104 782488 622101 1.305 1.306' 1.307 1.308 1.309 0.26620 0.26696 0.26773 0.26849 0.26926 30407 90308 44346 92530 34869 746567 542393 420849 350070 277629 1.210 1.211 1.212 1.213 1.214 0.19062 0.19144 0.19227 0.19309 0.19392 03596 64645 18876 66299 06926 086497 709552 471227 619131 373065 1.260 1.261 1.262 1.263 1.264 0.23111 0.23190 0.23269 0.23348 0.23428 17209 50569 77641 98433 12957 633866 827825 190214 683541 246657 1.310 1.311 1.312 1.313 1. 314 0.27002 0.27079 0.27155 0.27231 0.27307 71372 02047 26905 45953 59200 130602 815628 218973 206591 624188 1.215 1.216 1.217 1.218 1.219 0.19474 0.19556 0.19638 0.19721 0.19803 40767 67835 88140 01692 08504 925118 439753 053901 877053 991345 1.265 1.266 1.267 1.268 1.269 0.23507 0.23586 0.23665 0.23744 0.23822 21221 23237 19013 08560 91887 794836 219844 390020 150342 322506 1. 31'5 1.316 1.317 1.318 1.319 0.27383 Oi27459 0.27535 0.27611 0.27687 66656 68329 64227 54360 38737 297279 031255 611440 803155 351775 1.220 1.221 1.222 1.223 1.224 0.19885 0.19967 0.20048 0.20130 0.20212 08587 01951 88607 68567 41840 451652 285676 494036 050353 901343 1.270 1.271 1.272 1.273 1.274 0.23901 0.23980 0.24059 0.24137 0.24216 69004 39922 04649 63195 15571 704999 073170 179304 752695 499716 0.27763 :-3322: 0.27838 1:322 0.27914 1.32:3 1.3214 17365 90255 57414 0.27990 18851 0.28065 74575 982795 401883 294945 328186 148165 1.225 1.226 1.227 1.228 1.229 0.20294 0.20375 0.20457 0.20538 0.20620 08439 68375 21657 68297 08305 966903 140197 287744 249507 838978 1.275 1.276 1.277 1.278 1.279 0.24294 0.24373 0.24451 0.24529 0.24607 61786 01849 35770 63559 85225 103895 225981 504022 553431 967056 1.325 1.326 1.3i!7 1.328 1.329 0.28141 0.28216 0.28292 0.28367 0.28442 24594 68917 07553 40510 67797 381855 636708 500705 542421 311083 1.230 1.231 1.232 1.233 1.234 0.20701 0.20782 0.20863 0.20945 0.21026 41693 68472 88651 02241 09254 843261 023165 113280 822072 831961 1.280 1.281 1.282 1.283 1.284 0.24686 0.24764 0.24842 0.24920 0.24998 00779 10229 13584 10856 02052 315258 145972 984783 334994 677694 1.330 1.331 1.332 1.333 1.334 0.28517 0.28593 0.28668 0.28743 0.28818 89422 05394 15721 20411 19474 336624 129746 181974 965716 934320 1.235 1.236 1.237 1.238 1.239 0.21107 0.21188 0.21268 0.21349 0.21430 09700 03590 90934 71742 46026 799405 354990 103508 624044 470054 1.285 1.286 1.287 1.288 1.289 0.25075 0.25153 0.25231 0.25309 0.25386 87183 66258 39286 06276 67239 471831 154276 139896 821619 570503 1.335 1.336 1. 337 1.338 1.339 0.28893 0.28968 0.29042 0.29117 Oi29192 12918 00751 82981 59617 30667 522129 144540 198061 060367 090355 1.240 1.241 1.242 1.243 1.244 0.21511 0.21591 0.21672 0.21752 0.21833 13796 75062 29835 78125 19943 169455 224702 112870 285741 169877 1.290 1.291 1.292 1.293 1.294 0.25464 0.25541 0.25619 0.25696 0.25773 22183 71118 14053 50997 81960 735807 645054 604101 897204 787088 1.340 1.341 1.342 1.343 1.344 0.29266 0.29341 0.29416 0.29490 0.29565 96139 56042 10385 59175 02421 628200 995415 494901 411005 009578 1.245 1.246 1.247 1.248 1.249 6.21913 0.21993 0.22074 0.22154 0.22234 55299 84203 06666 22699 32311 166709 652614 978994 472359 434406 1.295 1.296 1.297 1.298 1.299 0.25851 0.25928 0.26005 0.26082 0.26159 06951 25979 39053 46182 47376 515011 300830 343068 818983 884625 1.1345 1.346 1.347 1.348 1.349 0.29639 0.29713 0.29787 0.29862 0.29936 40130 72312 98974 20124 35772 538024 225361 282269 901153 256188 1.250 0.22314 35513 142098 1.300 0.26236 42644 674911 1.350 0.30010 45924 503381 cI c -p In 10 = 2.30258 50929 940457 [ c-y I ELEMENTARY TRANSCENDENTAL NATURAL In x x 109 FUNCTIONS LOGARITHMS In X Table 4.2 In X x x 1.350 1.351 1.352 1.353 1.354 0.30010 0.30084 0.30158 0.30232 0.30306 45924 50589 49776 43491 31744 503381 780618 207723 886510 900833 1.400 1.401 1.402 1.403 1.404 0.33647 0.33718 0.33789 0.33861 0.33932 22366 62673 97886 28011 53056 212129 548700 123983 203239 036194 1.450 1.451 1.452 1.453 1.454 0.37156 0.37225 0.37294 P.37363 0.37431 35564 29739 19164 03845 83791 324830 020508 026043 881459 113276 1.355 1.356 1.357 1.358 1.359 0.30380 0.30453 Oi30527 0.30601 0.30674 14543 91895 63808 30291 91351 316642 182038 527321 365044 690067 1.405 1.406 1.407 1.408 1.409 0.34003 0.34074 0.34145 0.34217 0.34288 73027 87933 97781 02577 02329 857091 884732 322520 358507 165432 1.455 1.456 1.457 1.458 1.459 0.37500 0.37569 0.37637 0.37706 0.37775 59006 29497 95272 56335 12695 234558 744942 130678 864664 406486 1.360 1.361 1.362 1.363 1.364 0.30748 0.30821 0.30895 0.30968 0.31042 46997 97236 42077 81527 15594 479606 693290 273206 143956 212704 1.410 1.411 1.412 1.413 1.414 0.34358 0.34429 0.34500 0.34571 0.34642 97043 86728 71390 51037 25674 900769 706770 710503 023904 743810 1.460 1.461 1.462 1.463 1.464 0.37843 0.37912 0.37980 0.38048 0.38117 64357 11327 53613 91220 24155 202451 685624 275868 379873 391198 1.365 1.366 1.367 1.368 1.369 0.31115 0.31188 0.31261 0.31334 0.31408 44286 67611 85577 98192 05463 369231 485983 418125 003587 063118 1.415 1.416 1.417 1.418 1.419 0.34712 0.34783 0.34854 0.34924 0.34995 95310 59952 19607 74281 23981 952009 715280 085434 099358 779056 1.465 1.466 1.467 1.468 1.469 0.38185 Or38253 0.38321 0.38390 0.38458 52424 76034 94991 09301 18971 690306 644597 608447 923238 917403 1.370 :* z: 1:373 1.374 0.31481 0.31554 0.31626 0.31699 0.31772 07398 04005 95293 81267 61938 400335 801773 036935 858340 001576 1.420 1.421 1.422 1.423 1.424 0.35065 0.35136 0.35206 0.35276 0.35346 68716 08491 43313 73191 98129 131694 149636 810491 077153 897840 1.470 1.471 1.472 1.473 1.474 0.38526 0.38594 0.38662 0.38730 0.38797 24007 24416 20203 11374 97937 906449 193005 066845 804932 671449 1.375 1.376 1.377 1.378 1.379 0.31845 0.31918 0.31990 0.32063 0.32135 37311 07395 72197 31725 85988 185346 111519 465178 914668 111648 1.425 1.426 1.427 1.428 1.429 0.35417 0.35487 0.35557 0.35627 0.35697 18137 33219 43384 48639 48989 206138 921042 946994 173926 477304 1.475 1.476 1.477 1.478 1.479 0.38865 0.38933 0.39001 0.39068 0.39136 79897 57261 30035 98225 61837 917831 782808 492427 260100 286627 1.380 1.381 1.382 1.383 1.384 0.32208 0.32280 0.32353 0.32425 0.32497 34991 78744 17253 50526 78571 691133 271551 454782 826212 954778 1.430 1.431 1.432 1.433 1.434 0.35767 0.35837 0.35907 0.35977 0.36046 44442 35005 20685 01488 77421 718159 743139 384539 460348 774286 1.480 1.481 1.482 1.483 1.484 0.39204 0.39271 0.39339 0.39406 0.39474 20877 75352 25268 70631 11447 760237 856617 738951 557950 451887 1.385 1.386 1.387 1.388 1.389 0.32570 0.32642 0.32714 0.32786 0.32858 01396 19007 31413 38620 40637 393018 677115 326945 846128 722067 1.435 1.436 1.437 1.438 1.439 0.36116 0.36186 0.36255 0.36325 0.36394 48492 14706 76070 32592 84279 115844 260324 968879 988549 052308 1.485 1.486 1.487 1.488 1.489 0.39541 0.39608 0.39676 0.39743 0.39810 47722 79462 06674 29364 47537 546629 955674 780180 109001 018719 1.390 1.391 1.392 1.393 1.394 0.32930 0.33002 0.33074 0.33145 0.33217 37471 29129 15619 96947 73123 426004 413059 122279 976686 383321 1.440 1.441 1.442 1.443 1.444 0.36464 0.36533 0.36603 0.36672 0.36741 31135 73170 10388 42797 70404 879093 173850 627573 917338 706345 1.490 1.491 1.492 1.493 1.494 0.39877 0.39944 0.40011 0.40078 0.40145 61199 70357 75017 75185 70867 573678 826014 815691 570533 106256 1.395 11. z: 1:398 1.399 0.33289 0.33361 0.33432 0.33504 0.33575 44152 10043 70802 26438 76956 733290 401807 748248 116185 833441 1.445 1.446 1.447 1.448 1.449 0.36810 0.36880 0.36949 0.37018 0.37087 93215 11237 24476 32939 36633 643955 365729 493468 635246 385453 1.495 1.496 1.497 1.498 1.499 0.40212 0.40279 0.40346 0.40413 0.40479 62068 48795 31054 08850 82191 426497 522855 374913 950277 204607 1.400 0.33647 22366 212129 1.450 0.37156 35564 324830 (-fV 1.500 0.40546 51081 081644 [ c-y I [1 In lo= 2.30258 50929 940457 110 ELEMENTARY TRANSCENDENTAL NATURAL Table 4.2 In x X FUNCTIONS LOGARITHMS In z 5 In X 1.500 1.501 1.502 1.503 1.504 0.40546 0.40613 0.40679 0.40746 0.40812 51081 15526 75533 31107 82255 081644 513249 419430 708374 276481 1.550 1.551 1.552 1.553 1.554 0.43825 0.43889 0.43954 0.44018 0.44083 49309 98841 44217 85441 22519 311553 944018 610270 665500 454557 1.600 1.505 1.506 1.507 1.508 1.509 0.40879 0.40945 0.41012 0.41078 0.41144 28982 71293 09196 42695 71797 008391 777018 443584 857643 857118 1.555 1.556 1.557 1.558 1.559 0.44147 0.44211 0.44276 0.44340 0.44404 55456 84257 08928 29474 45900 1.510 1: 511 1.512 1.513 1.514 0.41210 0141277 0.41343 Oi41409 0.41475 96508 16832 32777 44348 51550 268330 906025 573413 062189 152570 1.560 1.561 1.562 1.563 1.564 0.44468 0.44532 0.44596 0.44660 0.44724 1.515 1.516 1.517 1.518 1.519 0.41541 0.41607 0.41673 0.41739 0.41805 54389 52872 47003 36789 22236 613325 201799 663952 734382 136358 1.565 1.566 1.567 1.568 1.569 1.520 1.521 1.522 1.523 1.524 0.41871 0.41936 0.42002 0.42068 0.42133 03348 80132 52594 20739 84572 581850 771558 394941 130248 644545 1.525 1.526 1.527 1.528 1.529 0.42199 0.42264 0.42330 0.42395 0.42461 44100 99328 50262 96907 39269 1.530 1.531 1.532 1.533 1.534 0.42526 0.42592 0.42657 0.42722 0.42787 1.535 1.536 1.537 1.538 1.539 x :- El: 1:603 1.604 0.47000 0.47062 0.47125 0.47187 0.47250 36292 84340 28486 68736 05094 457356 145776 461675 274159 443228 311975 561999 518613 485565 756395 1.605 1.606 1.607 1.608 1.609 0.47312 0.47374 0.47436 0.47499 0.47561 37565 66155 90867 11707 28680 819792 245699 553755 567746 102462 58212 66415 70514 70514 66421 614457 332950 174942 393396 231193 1.610 1.611 1.612 1.613 1.614 0.47623 Oi47685 0.47747 Oi47809 0.47871 41789 51041 56440 57991 55698 963716 948373 844365 430718 477571 0.44788 0.44852 0.44916 0.44980 0.45043 58239 45975 29633 09219 84737 921165 686114 738838 282161 508955 1.615 1.616 1.617 1.618 1.619 0.47933 0.47995 0.48057 0.48119 0.48180 49566 39600 25805 08186 86746 746199 989036 949698 362999 954981 1.570 1.571 1.572 1.573 1.574 0.45107 0.45171 0.45234 0.45298 0.45362 56193 23592 86940 46240 01499 602167 734841 070148 761408 952115 1.620 1.621 1.622 1.623 1.624 0.48242 0.48304 0.48365 0.48427 0.48489 61492 32427 99556 62885 22417 442927 535391 932212 324542 394862 593749 622653 364954 443287 469252 1.575 1.576 1.577 1.578 1.579 0.45425 0.45488 0.45552 0.45615 0.45679 52722 99914 43079 82224 17352 775964 356874 809013 236825 735050 1.625 1.626 1.627 1.628 1.629 0.48550 0.48612 0.48673 0.48735 0.48796 78157 30111 78282 22675 63296 817008 256188 369007 803486 199081 77354 11166 40713 65998 87029 043441 755467 183996 896771 450644 1.580 1.581 1.582 1.583 1.584 0.45742 0.45805 0.45868 0.45932 0.45995 48470 75582 98693 17808 32933 388754 273350 454621 988751 922341 1.630 1.631 1.632 1.633 1.634 0.48858 0.48919 0.48980 0.49041 0.49103 00148 33236 62565 88139 09964 186710 388768 419153 883281 378111 0.42853 0.42918 0.42983 0.43048 0.43113 03810 16347 24645 28710 28548 391605 254804 564588 834522 567422 1.585 1.586 1.587 1.588 1.589 0.46058 0.46121 0.46184 0.46247 0.46310 44073 51232 54415 53628 48875 292439 126562 442720 249440 545789 1.635 1.636 1.637 1.638 1.639 0.49164 0.49225 0.49286 0.49347 0.49408 28043 42381 52983 59854 62997 492167 805553 889979 308777 616926 1.540 1.541 1.542 1.543 1.544 0.43178 0.43243 0.43308 0.43372 0.43437 24164 15563 02751 85733 64516 255378 379787 411377 810238 025844 1.590 1.591 1.592 1.593 1.594 0.46373 0.46436 0.46499 0.46561 0.46624 40162 27493 10874 90309 65803 321402 556498 221913 279115 680233 1.640 1.641 1.642 1.643 1.644 0.49469 0.49530 0.49591 0.49652 0.49713 62418 58121 50110 38390 22966 361071 079538 302365 551310 339882 1.545 1.546 1.547 1.548 1.549 0.43502 0.43567 0.43631 0.43696 0.43760 39103 09501 75715 37751 95614 497088 652302 909291 675354 347316 1.595 1.596 1.597 1.598 1.599 0.46687 0.46750 0.46812 0.46875 0.46937 37362 04990 68692 28473 84338 368079 276170 328754 440829 518172 1.645 1.646 1.647 1.648 1.649 0.49774 0.49834 0.49895 0.49956 0.50016 03842 81022 54511 24314 90435 173352 548781 955033 872800 774619 1.550 0.43825 49309 311553 C-58)6 1.600 0.47000 36292 457356 1.650 0.50077 52879 124892 E1 5 [(-;I1 In 10 = 2.30258 50929 940457 c-p I: 3 ELEMXNTARY TRANSCENDENTAL NATURAL X LOGARITHMS 2 In x 111 FUNCTIONS In 2 Table 4.2 In x X 1.650 1.651 1.652 1.653 1.654 0.50077 0.50138 0.50198 0.50259 0.50319 52879 11649 66750 18188 65966 124892 379910 987863 388871 014996 1.700 1.701 1.702 1.703 1.704 0.53062 0.53121 0.53180 0.53239 0.53297 82510 63134 40301 14016 84284 621704 137247 511824 805512 071240 1.750 1.751 1.752 1.753 1.754 0.55961 0.56018 0.56075 {Or56132 0.56189 57879 70533 79925 86059 88939 354227 037148 141997 390974 499913 1.655 1.656 1.657 1.658 1.659 0.50380 0.50440 0.50500 0.50561 0.50621 10088 50559 87384 20567 50112 290262 630679 444259 131032 083074 1.705 1.706 1.707 1.708 1.709 0.53356 0.53415 0.53473 0.53532 0.53590 51107 14490 74438 30953 84041 354801 694874 123036 663781 334538 1.755 1.756 1.757 1.758 1.759 0.56246 0.56303 0.56360 0.56417 0.56474 88569 84952 78092 67992 54657 178291 129249 049601 629853 554211 1.660 1.661 1.662 1.663 1.664 0.50681 0.50741 0.50802 0.50862 0.50922 76023 98306 16964 32002 43423 684519 311578 332564 107906 990168 1.710 1.711 1.712 1.713 1.714 0.53649 0.53707 0.53766 0.53824 0.53882 33705 79949 22777 62193 98201 145685 100564 195504 419829 755880 1.760 1.761 1.762 1.763 1.764 0.56531 0.56588 0.56644 0.56701 0.56758 38090 18295 95275 69034 39575 500604 140691 139878 157332 845996 1.665 1.666 1.667 1.668 1.669 0.50982 0.51042 0.51102 0.51162 0.51222 51234 55437 56037 53039 46446 324071 446509 686569 365550 796980 1.715 1.716 1.717 1.718 1.719 0.53941 0.53999 0.54057 0.54116 0.54174 30806 60010 85819 08235 27264 179032 657705 153385 620636 007122 1.765 1.766 1.767 1.768 1.769 0.56815 0.56871 0.56928 0.56984 0.57041 06903 71021 31933 89642 44151 852601 817683 375593 154517 776482 1.670 1.671 1.672 1.673 1.674 0.51282 0.51342 0.51402 0.51461 0.51521 36264 22496 05146 84220 59720 286637 132567 625099 046869 672836 1.720 1.721 1.722 1.723 1.724 0.54232 0.54290 0.54348 0.54406 0.54464 42908 55172 64060 69575 71722 253617 294024 055391 457926 415014 1.770 1.771 1.772 1.773 1.774 0.57097 0.57154 0.57210 0.57267 0.57323 95465 43588 88521 30270 68838 857378 006965 828892 920708 873877 1.675 1.676 1.677 1.678 1.679 0.51581 0.51641 0.51700 0.51760 0.51819 31652 00020 64828 26080 83780 770298 598913 410718 450144 954038 1.725 1.726 1.727 1.728 1.729 0.54522 0.54580 0.54638 0.54696 0.54754 70504 65926 57991 46703 32067 833231 612362 645415 818639 011534 1.775 1.776 1.777 1.778 1.779 0.57380 0.57436 0.57492 0.57548 0.57605 04229 36445 65491 91370 14086 273791 699783 725143 917128 836981 1.680 1.681 1.682 1.683 1.684 0.51879 0.51938 0.51998 0.52057 0.52117 37934 88544 35615 79152 19158 151676 264786 507563 086690 201350 1.730 1.731 1.732 1.733 1.734 0.54812 0.54869 0.54927 0.54985 0.55043 14085 92761 68101 40107 08783 096876 940722 402434 334690 583501 1.780 1.781 1.782 1.783 1.784 0.57661 0.57717 0.57773 0.57829 0.57885 33643 50043 63290 73388 80341 039938 075246 486176 810034 578176 1.685 1.686 1.687 1.688 1.689 0.52176 0.52235 0.52295 0.52354 0.52413 55638 88595 18035 43961 66378 043250 796637 638312 737654 256630 1.735 1.736 1.737 1.738 1.739 0.55100 0.55158 0.55215 0.55273 0.55331 74133 36162 94872 50268 02353 988225 381584 589679 432003 721460 1.785 1.786 1.787 1.788 1.789 0.57941 0.57997 0.58053 0.58109 0.58165 84152 84824 82361 76767 68045 316024 543073 772910 513224 265821 1.690 1.691 1.692 1.693 1.694 0.52472 0.52532 0.52591 0.52650 0.52709 85289 00699 12611 21031 25962 349821 164432 840315 509983 298627 1.740 1.741 1.742 1.743 1.744 0.55388 0.55445 0.55503 0.55560 0.55618 51132 96607 38784 77665 13254 264377 860520 303111 378839 867879 1.790 1.791 1.792 1.793 1.794 0.58221 0.58277 0.58333 0.58389 0.58444 56198 41230 23145 01946 77636 526636 785747 527387 229958 366044 1.695 1.696 1.697 1.698 1.699 0.52768 0.52827 0.52886 0.52945 0.53003 27408 25373 19862 10878 98426 324136 697113 520893 891556 897950 1.745 1.746 1.747 1.748 1.749 0.55675 0.55732 0.55790 0.55847 0.55904 45556 74574 00311 22772 41960 543905 174105 519195 333437 364650 1.795 1.796 1.797 1.798 1.799 0.58500 Oi58556 0.58611 Oi58667 0.58723 50219 19698 86078 49360 09549 402422 800079 014220 494285 683961 1.700 0.53062 82510 621704 1.750 0.55961 57879 354227 1.800 0.58778 66649 021190 C-58)5 [1 [I C-5814 In 10 = 2.30258 50929 940457 [C-5814I 112 ELEMENTARY NATURAL Table 4.2 FUNCTIONS LOGARITHMS x In x X TRANSCENDENTAL In x In x 2 1.800 1.801 1.802 1.803 1.804 0.58778 0.58834 0.58889 0.58945 0.59000 66649 20661 71591 19442 64216 021190 938190 861462 211802 404319 1.850 1.851 1.852 1.853 1.854 0.61518 0.61572 0.61626 0.61680 0.61734 56390 60335 61362 59473 54671 902335 913605 239876 032227 436634 1.900 1.901 1.902 1.903 1.904 0.64185 0.64238 0.64290 0.64343 0.64395 38861 00635 59641 15883 69363 723948 062921 231986 140124 691736 1.805 1.806 1.807 1.808 1.809 0.59056 0.59111 0.59166 0.59222 0.59277 05917 44549 80116 12619 42064 848442 947937 100914 699848 131581 1.855 1.856 1.857 1.858 1.859 0.61788 0.61842 0.61896 0.61950 0.62003 46960 36343 22823 06403 87087 593985 640088 705687 916468 393070 1.905 1.906 1.907 1.908' 1.909 0.64448 0.64500 0.64553 0.64605 0.64657 20085 68052 13266 55730 95437 786643 320104 182820 260948 436106 1.810 1.811 1.812 1.813 1.814 0.59332 0.59387 0.59443 0.59498 0.59553 68452 91789 12076 29317 43516 777344 012763 207876 727140 929449 1.860 1.861 1.862 1.863 1.864 0.62057 0.62111 0.62165 0.62218 0.62272 64877 39776 11788 80916 47162 251099 601137 548753 194514 633994 1.910 1.911. 1.912 1.913 1.914 0.64710 0.64762 0.64814 0.64867 0.64919 32420 66652 98146 26904 52930 585385 581360 292095 581158 307625 1.815 1.816 1.817 1.818 1.819 0.59608 0.59663 0.59718 0.59773 0.59828 54677 62801 67894 69957 68995 168141 791016 140341 552871 359852 1.865 1.866 1.867 1.868 1.869 0.62326 0.62379 0.62433 0.62486 0.62540 10530 71024 28645 83398 35284 957789 251521 595856 066509 734258 1.915 1.916 1.917 1.918 1.919 0.64971 0.65023 0.65076 0.65128 0.65180 76226 96795 14640 29764 42170 326093 486688 635074 612465 255629 1.820 0.59883 65010 887040 0.59938 58007 454709 ;.;;;' 1:823 1.824 0.60048 47988 965260 0.59993 34956 377666 0.60103 18916 521396 1.870 1.871 1.872 1.873 1.874 0.62593 0.62647 0.62700 0.62754 0.62807 84308 30472 73780 14234 51838 664953 919526 554003 619515 162304 1.92:O 1. 9i!l 1.9;!2 1.9:!3 1.924 0.65232 0.65284 0.65336 0.65388 0.65440 51860 58837 63105 64666 63522 396902 864196 481007 066427 435147 1.825 1.826 1.827 1.828 1.829 0.60157 0.60212 0.60267 0.60322 Oi60376 99870 77821 52773 24730 93694 344548 727767 958697 319583 087286 1.875 1.876 1.877 1.878 1.879 0.62860 0.62914 0.62967 0.63020 0.63073 86594 18505 47576 73807 97204 223741 840329 043718 860712 313283 1.925 1.926 1.927 1.928 1,929 0.65492 0.65544 0.65596 0.65648 0.65700 59677 53133 43894 31961 17339 397475 759338 322293 883539 235920 1.830 1.831 1.832 1.833 1.834 0.60431 0.60486 0.60540 0.60595 0.60649 59668 22656 82662 39688 93738 533296 923737 519385 575680 342731 1.880 1.881 1.882 1.883 1.884 0.63127 0.63180 0.63233 0.63286 0.63339 17768 35503 50411 62496 71761 418578 188933 631879 750154 541713 1.930 1.931 1.932 1.933 1.934 0.65752 0.65803 0.65855 0.65907 0.65959 00029 80034 57357 32002 03970 167942 463774 903263 261938 311026 1.835 1.836 1.837 1.838 1.839 0.60704 0.60758 0.60813 0.60867 0.60922 44815 92921 38062 80239 19456 065336 982987 329886 334953 221840 1.885 1.886 1.887 1.888 1.889 0.63392 0.63445 0.63498 0.63551 0.63604 78208 81842 82663 80677 75885 999741 112658 864132 233089 193725 1.935 1.936 1.937 1.938 1.'939 0.66010 0.66062 0.66114 0.66165 0.66217 73264 39888 03844 65134 23762 817451 543853 248588 685745 605148 1.840 1.841 1.842 1.843 1.844 0.60976 0.61030 0.61085 0.61139 0.61193 55716 89022 19378 46786 71251 208943 509408 331151 876862 344021 1.890 1.891 1.892 1.893 1.894 0.63657 0.63710 0.63763 0.63816 0.63869 68290 57896 44706 28722 09947 715510 763204 296865 271858 638865 1.940 1.941 :* zt; 1:944 0.66268 0.66320 0.66371 0.66423 0.66474 79730 33041 83698 31703 77060 752368 868732 691332 953030 382473 1.845 1.846 1.847 1.848 1.849 0.61247 0.61302 0.61356 0.61410 Or61464 92774 11360 27012 39732 49524 924905 806604 171029 194924 049878 1.895 1.896 1.897 1.898 1.899 0.63921 0.63974 0.64027 0.64080 0.64132 88385 64038 36909 07001 74318 343897 328301 528772 877361 301488 1.945 1.946 1.947 1.948 1.949 0.66526 0.66577 0.66628 0.66680 0.66731 19770 59837 97263 32052 64205 704096 638133 900626 203434 254238 1.850 0.61518 56390 902335 C-58)4 1.900 0.64185 38861 723948 ‘-;8’4 L950 0.66782 93725 756554 [1 [I In 10 = 2.30258 50929 940457 [3 C-i)3 ELEIKENTARY TRANSCENDENTAL NATURAL In x X LOGARITHMS In X 113 FUNCTIONS 2 Table 4.2 In x X 1.950 1.951 1.952 1.953 1.954 0.66782 Oi66834 0.66885 Oi66936 0.66987 93725 20616 44879 66518 85536 756554 409742 909007 945419 205910 2.000 2.001 2.002 2.003 2.004 0.69314 0.69364 0.69414 0.69464 0.69514 71805 70556 66808 60566 51832 599453 015964 930288 836812 226184 2.050 2.051 2.052 2.053 2.054 0.71783 0.71832 0.71881 0.71930 0.71978 97931 74790 49273 21380 91115 503168 902436 085231 367965 063665 1.955 1.956 1.957 1.958 1.959 0.67039 0.67090 0.67141 0.67192 0.67243 01934 15716 26884 35441 41389 373291 126256 139392 083186 624037 2.005 2.006 %*00:: 2:009 0.69564 0.69614 0.69664 0.69713 0.69763 40607 26895 10698 92018 70858 585325 397438 142011 294828 327974 2.055 2.056 2.057 2.058 2.059 0.72027 0.72076 0.72124 0.72173 0.72222 58479 23475 86106 46374 04280 481979 929187 708201 118579 456524 1.960 1.961 1.962 1.963 1.964 0.67294 Oi67345 0.67396 0.67447 0.67498 44732 45472 43611 39152 32099 424259 142092 431713 943240 322741 2.010 2.011 2.012 2.013 2.014 0.69813 0.69863 0.69912 0.69962 0.70012 47220 21107 92522 61466 27942 709844 905150 374928 576544 963706 2.060 2.061 2.062 2.063 2.064 0.72270 0.72319 0.72367 0.72416 0.72464 59828 13019 63855 12340 58476 014897 083220 947682 891148 193163 1.965 1.966 1.967 1.968 1.969 0.67549 0.67600 0.67650 0.67701 0.67752 22453 10217 95394 77986 57996 212246 249748 069220 300617 569885 I* 00:: 2:017 2.018 2.019 0.70061 0.70111 0.70161 0.70210 0.70260 91953 53502 12589 69219 23393 986463 091222 720747 314172 307004 2.065 2.066 2.067 2.068 2.069 0.72513 0.72561 0.72609 0.72658 0.72706 02264 43706 82806 19566 53987 129961 974468 996312 461827 634060 1.970 11971 1.972 11973 1.974 0.67803 Oi678ii 0.67904 Oi67955 0.68006 35427 ib281 82561 52270 19410 498971 7Oii83i 804437 404783 112898 2.020 :- 8;: 2:023 2.024 0.70309 0.70359 0.70408 0.70458 0.70507 75114 24384 71205 15581 57514 131134 214840 982797 856084 252191 2.070 2.071 2.072 2.073 2.074 0.72754 0.72803 0.72851 0.72899 0.72947 86072 15824 43243 68334 91098 772777 134471 972366 536425 073356 1.975 1.976 1.977 1.978 1.979 0.68056 0.68107 0.68158 0.68208 0.68259 83983 45993 05441 62332 16666 530852 256761 884799 005204 204287 2.025 27026 2.027 z*. 82298 0.70556 0.70606 0.70655 0.70705 0.70754 97005 34058 68674 30608 00857 585025 264916 698630 436777 289367 2.075 2.076 2.077 2.078 2.079 0.72996 0.73044 0.73092 0.73140 0.73188 11536 29653 45448 58926 70088 826616 036422 939753 770357 758759 1.980 1.981 1.982 1.983 1.984 0.68309 68447 0.6836$'?7677 0.68410 64359 0.68461 08495 0.68511 50088 064439 164139 077962 376589 626811 2.030 2.031 2.032 2.033 2.034 0.70803 0.70852 0.70902 0.70951 0.71000 57930 82825 05297 25346 42976 536960 982476 162355 462096 263682 2.080 2.081 2.082 2.083 2.084 0.73236 0.73284 0.73332 0.73380 0.73428 78937 85474 89701 91622 91238 132266 114974 927771 788349 911205 1.985 1.986 1.987 1.988 1.989 0.68561 0.68612 0.68662 0.68712 0.68763 89141 25656 59635 91082 19998 391537 229808 696798 343823 718351 2.035 2:036 2.037 2.038 2.039 0.71049 0.71098 0.71147 0.71196 0.71245 58188 70986 81372 89348 94915 945583 882763 446688 005331 923181 2.085 2.086 2.087 2.088 2.089 0.73476 0.73524 0.73572 0.73620 0.73668 88552 83565 76280 66700 54825 507648 785807 950637 203923 744287 1.990 11991 1.992 1.993 1.994 0.68813 Or68863 0.68913 0.68964 0.69014 46387 70250 91591 10412 26715 364010 820592 624065 306577 396466 0.71294 %- 0.71343 FA! 0.71392 2:042 2.043 2.044 98078 98838 97197 0.71441 93158 0.71490 86723 561250 277077 424738 354850 414580 2.090 2.091 2.092 2.093 2.094 0.73716 0.73764 0.73812 0.73859 0.73907 40659 24204 05462 84434 61124 767196 464965 026765 638627 483451 1.995 i;99i 1.997 1.998 1.999 0.69064 Oi69114 0.69164 0.69214 0.69264 40503 51778 60544 66802 70555 418268 892722 336782 263618 182630 2.045 2iO46 2.047 2.048 2.049 0.71539 0.71588 0.71637 0.71686 0.71735 947651 294347 791525 772614 567627 2.095 2.096 2.097 2.098 2.099 0.73955 0.74003 0.74050 0.74098 0.74146 35533 07664 77519 45099 10408 741011 587957 197829 741054 384959 2.000 0.69314 71805 599453 2.050 0.71783 97931 503168 2.100 0.74193 73447 293773 r (91 L5J For x>2.1 see Example 5. 77894 66675 53066 37071 18692 [1 In 10 = 2.30258 50929 940457 C-i)3 [C-i)3 1 114 Table ELEMENTARY 4.3 RADIX TRANSCENDENTAL TABLE OF 10 10 10 10 10 10 10 10 10 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 ; 9 ; 0.00000 0.00000 0.00000 00019 00009 99999 99950 00000 99800 00029 99999 99550 00000 00049 00039 99999 98750 00000 99200 9 0.00000 00059 99999 98200 00001 ; 9 0.00000 0.00000 00079 00069 99999 97550 00001 96800 00002 00089 99999 95950 00002 8" 0.00000 00199 00099 99999 95000 00027 80000 00003 8 8 0.00000 0.00000 00299 99999 55000 00090 00399 99999 20000 00213 ii 8" 0.00000 0.00000 00499 00599 99998 75000 00417 20000 00720 00799 99996 80000 01707 00699 99997 55000 01143 8 0.00000 00899 99995 95000 02430 7 0.00000 00999 99995 00000 03333 s : 0.00000 0.00000 01999 99980 00000 26667 02999 99955 90000 04999 99920 00004 13333 03999 99875 00002 16667 7 0.00000 05999 99820 00007 20000 ; 7 0.00000 0.00000 6 6 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 6" i 6 6" For n>lO,ln (1+x10-n, 99999 99999 99999 99998 99999 99995 99999 99992 99999 99987 99999 99982 99999 99975 99999 99968 99999 99959 NATURAL In 00000 00001 00002 00003 00004 00005 00006 00007 00008 n FUNCTIONS LOGARITHMS 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -In (I-210Wn) 0000~1 00000 00000 OOOCl200000 00002 oooCl3 00000 00004 00004 00000 00008 oooCl5 00000 00012 00006 00000 00018 00007 00000 00024 00008 00000 00032 00009 00000 00040 50000 00000 50000 00000 50000 00000 50000 00000 50000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 00010 ooo;!o 00030 00040 00050 00060 00070 00080 00090 00000 00000 00000 00000 00000 00000 00000 00000 00000 00050 00200 00450 00800 01250 01800 02450 03200 04050 00000 00000 00000 00000 00000 00001 00001 00002 00002 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 00100 00200 00300 00400 00500 00600 007OO 00800 00900 00000 00000 00000 00000 00001 00001 00002 00003 00004 05000 20000 45000 80000 25000 80000 45000 20000 05000 00003 00027 00090 00213 00417 00720 01143 01707 02430 07999 99680 00017 06666 06999 99755 00011 43333 08999 99595 00024 29998 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 OlOOO 00005 02000 00020 03000 00045 040'00 00080 05000 00125 06000 00180 07000 00245 08000 00320 09000 00405 00000 00000 00000 00002 00004 00007 00011 00017 00024 03333 26667 90000 13333 16667 20000 43334 06668 30002 09999 19999 29999 39999 49999 59999 69999 79999 89999 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 10000 20000 30000 40000 50000 60000 70000 80000 90000 00033 00266 00900 02133 04166 07200 11433 17066 24300 33336 66707 00203 33973 68229 03240 39336 76907 16403 99500 98000 95500 92000 87500 82000 75500 68000 59500 00033 00266 00899 02133 04166 07199 11433 17066 24299 (l~~10-~)=f~10-~-~~210-2~to 50000 00000 50000 00000 50000 00000 50000 00000 50000 33331 66627 99798 32693 65104 96760 27331 56427 83598 25D. 00500 02000 04500 08000 12500 18000 24500 32000 40500 ELEMENTARY RADIX OF NATURAL FUNCTIONS Table -In 21 : 0.00001 0.00000 99998 00002 66662 66673 99999 !jOOOO 33333 08334 ; 5 5 s 0.00003 0.00002 0.00004 99992 150008 99979 75049 99995 130021 33269 33538 99987 50041 66510 42292 ; 9" z 5 0.00005 0.00006 0.00008 0.00007 99975 50114 32733 11695 99982 130071 99676 01555 99959 50242 65642 73220 99968 130170 98359 86809 : 3 4 2 1 4 4 4 0.00019 0.00009 0.00029 0.00039 0.00059 0.00049 99800 99950 99550 99200 98200 98750 ; 9 : t 0.00079 0.00069 0.00089 96801 70564 33215 90059 97551 14273 34192 77369 95952 42836 09300 94948 13 2 ; ; 0.00099 0.00199 0.00299 0.00399 95003 33083 53316 68094 80026 62673 05601 82538 20212 69537 47881 16106 55089 79798 45299 90751 5 ; 0.00498 75415 11039 07361 21022 ; 8 9 3 ; 3 0.00697 0.00598 0.00796 0.00895 20716 77547 46378 20189 56137 36425 24209 95222 81696 49176 87351 07973 97413 71471 90444 31465 ; 2 0.01980 0.00995 26272 96179 08284 60291 03308 53168 71302 82154 ; 5 ; 2 2 0.02955 0.03922 0.04879 07131 53281 29626 92009 88022 41544 40273 26194 01641 69432 00306 53744 ; ; 0.06765 0.05826 86484 73814 80526 84159 89081 23975 77552 57184 9" 2 0.08617 0.07696 76962 41052 32498 42170 10411 36128 33234 13335 ; : 0.18232 0.09531 15567 93954 62621 39521 01798 04324 86004 17180 3 4 i 0.26236 0.33647 42644 67491 05203 54960 22366 21212 93050 45934 2 1 0.47000 0.40546 36292 08164 38197 09370 51081 45735 55365 80131 ii 9 1' 11 0.58778 0.53062 0.64185 66649 02119 00818 97311 82510 62170 39623 15432 38861 72394 77599 10360 1 0 0.69314 71805 59945 30941 72321 00333 32666 D8997 21326 71967 41651 115 LOGARITHMS In (1+x10-") n X ‘TABLE TRANSCENDENTAL 26673 30833 97548 93538 61554 04791 53332 06560 58785 06509 40636 42280 4.3 (l-x10-n) 0.00001 0.00002 0.00003 0.00004 0.00005 0.00006 0.00007 0.00008 0.00009 00000 00002 00004 00008 00012 00018 00024 00032 00040 50000 00002 50009 00021 50041 00072 50114 00170 50243 33333 66670 00020 33397 66822 00324 33933 67690 01640 58334 66673 25049 33538 92292 01555 61695 73221 36811 0.00010 0.00020 0.00030 0.00040 0.00050 0.00060 0.00070 0.00080 0.00090 00050 00200 00450 00800 01250 01800 02451 03201 04052 00333 02667 09002 21339 41682 72032 14393 70769 43164 35833 06673 02548 73538 29791 41555 39196 13224 14318 53335 06773 61215 20162 92719 97800 69533 63873 66419 0.00100 0.00200 0.00300 0.00400 0.00501 0.00601 0.00702 0.00803 0.00904 05003 20026 45090 80213 25418 80723 46149 21716 07446 33583 70673 20298 97538 23544 25563 36964 97264 52149 53350 07735 72181 81834 28204 01620 45987 25903 06220 01430 16511 32509 87927 30937 19350 41123 86494 55241 0.01005 03358 53501 44118 35489 0.02020 27073 17519 44840 80453 0.03045 92074 84708 54591 92613 0.04082 19945 20255 12955 45771 0.05129 32943 87550 53342 61961 0.06187 54037 18087 47179 78001 0.07257 06928 34835 43071 15733 0.08338 16089 39051 05839 47658 0.09431 06794 71241 32687 71427 0.10536 0.22314 0.35667 0.51082 0.69314 0.91629 1.20397 1.60943 2.30258 05156 35513 49439 56237 71805 07318 28043 79124 50929 57826 14209 38732 65990 59945 74155 25935 34100 94045 30122 75576 37891 68320 30941 06518 99262 37460 68401 75010 62951 26387 55141 72321 35272 27462 07593 79915 ELEMENTARY Table 4.4 TRANSCENDENTAL EXPONENTIAL FTJNCTIONS FUNCTION e-z N 0.001 0.002 0.003 0.004 0.000 1.00000 1.00100 1.00200 iI 1.00400 00000 05001 20013 45045 80106 00000 66708 34000 03377 77341 000 342 267 026 872 1.00000 0.99900 0.99800 0.99700 0.99600 00000 04998 19986 44955 79893 00000 33374 67333 03372 43991 000 992 067 976 472 0. 003 0.006 0.007 0.008 0.009 1.00501 1.00601 1.00702 1.00803 1.00904 25208 80360 45572 20855 06217 59401 54064 66848 04273 73867 063 865 555 431 814 0.99501 0.99401 0.99302 0.99203 0.99104 24791 79640 44429 19148 03787 92682 53935 33235 37060 72883 313 265 105 630 662 0.010 0.011 0.012 0.013 0.014 1.01005 1.01106 1.01207 1.01308 1.01409 01670 07224 22888 48673 84589 84168 44719 66077 59809 38492 058 556 754 158 345 0.99004 0.98906 0.98807 0.98708 0.98609 ,d337 02787 17128 41350 75442 49168 75368 61930 20287 62861 054 698 540 583 903 0.015 0.016 0.017 0.018 0.019 1.01511 1.01612 1.01714 1.01816 1.01918 30646 86854 53223 29763 16486 15718 06094 25240 89793 17408 979 822 748 761 011 0.98511 0.98412 0.98314 0.98216 0.9811'7 19396 73200 36846 10323 93622 03062 55285 34909 58300 42806 661 115 635 718 006 0.020 0.021 0.022 0.023 0.024 1.02020 1.02122 1.02224 1.02326 1.02429 13400 20516 37844 65395 03178 26755 37528 70438 47217 90621 810 653 235 475 534 0.98019 0.97921 0.97824 0.97726 0.97628 86733 89645 02350 24837 57097 06755 69459 51210 73277 57909 302 588 045 073 314 0.025 0.026 0.027 0.028 0.029 1.02531 1.02634 1.02736 1.02839 1.02942 51205 09484 78027 56844 45944 24428 73442 63489 21425 75130 841 115 392 045 820 0.97530 0.97433 0.97336 0.97238 0.97141 99120 50896 12415 83668 64644 28332 08749 24336 01246 66604 669 328 791 891 825 0.030 0.031 0.032 0.033 0.034 1.03045 1.03148 1.03251 1.03355 1.03458 45339 55038 75053 05392 46067 53516 86522 05118 41305 28117 856 716 420 472 894 0.97044 0.96947 0.96850 0.96753 0.96657 55335 55730 65820 85595 15046 48508 76025 79197 89032 37506 177 948 585 009 651 0.035 0.036 0.037 0.038 0.039 1.03561 1.03665 1.03769 1.03873 1.03977 97087 58464 30208 12328 04836 99623 90923 38157 78497 50157 260 727 074 733 831 0.96560 0.96464 0.96367 0.96271 0.96175 54162 02934 61353 29408 07091 57566 83123 49053 91199 46366 478 030 452 529 723 0.040 0.041 0.042 0.043 0.044 1.04081 1.04185 1.04289 1.04393 1.04498 07741 21055 44787 78948 23548 92388 45479 50763 50612 88443 227 549 238 586 779 0.96078 0.95982 0.95886 0.95791 0.95695 94391 91299 97805 13900 39574 52323 47798 72484 67030 73046 209 914 552 669 678 0.045 0.046 0.047 0.048 0.049 1.04602 1.04707 1.04812 1.04917 liO5022 78599 44109 20090 06553 03507 08716 56937 79655 24470 40028 943 184 638 516 148 0.95599 0.95504 0.95408 0.95313 0.95218 74818 19621 73975 37870 11296 33099 90714 90371 77504 98504 907 635 141 745 853 0.050 1.05127 10963 76024 040 For use and extension of the table see Examples 8-11. 2 -23 SW Table 7.1 for values of -e and Table 26.1 for & & 0. 95122 94245 00714 009 -- 23 e 2. ELEMENTARY TRANSCENDENTAL EXPONENTIAL Table 4.4 FUNCTION ez X 117 F’UNCTIONS e-z 0.050 0.051 0.052 0.053 0.054 1.05127 1.03232 1.05337 1.05442 1.05548 10963 28932 57425 96451 46021 76024 83203 13364 19355 55080 040 913 763 907 041 0.95122 0.95027 0.94932 0.94838 0.94743 94245 86705 88668 00124 21065 00714 32426 42889 82298 01798 009 935 583 184 300 0.055 0.056 0.057 0.058 0.059 1.05654 1.05759 1.05865 1.05971 1.015077 06146 76837 58103 49957 52407 75494 36611 95500 10287 40159 286 252 087 540 012 0.94648 0.94553 0.94459 0.94364 0.94270 51479 91358 40693 99474 67691 53483 90396 66523 36798 57099 869 267 349 514 754 0.060 0.061 0.062 0.063 0.064 l.Otj183 1.015289 1.015396 1.015502 1.015609 65465 89141 23447 68392 23987 45359 87195 28033 31305 61505 622 264 669 464 244 0.94176 0.94082 0.93988 0.93894 0.93800 45335 32397 28867 34736 49995 84248 76009 91088 89133 30729 710 730 928 241 488 0.065 0.066 0.067 0.068 0.069 l.Oli715 1.015822 1.0~5929 1.07036 1.07143 90243 67171 54781 53084 62091 84192 65993 74600 78774 48346 625 321 202 366 205 0.93706 Oi93613 0.93519 0.93426 0.93332 74633 08642 52013 04735 66800 77403 91618 36776 77213 78201 433 844 558 542 958 0.070 0.071 0.072 0.073 0.074 1.0'7250 1.0'7358 1.0'7465 1.0'7573 1.0'7680 81812 12258 53440 05369 68054 54216 68357 63813 14703 96219 479 383 620 476 891 0.93239 0.93146 0.93053 0.92960 0.92867 38199 18921 08958 08300 16938 05948 27592 11205 25792 41287 229 106 732 713 187 0.075 0.076 0.077 0.078 0.079 1.0'7788 1.0'7896 1.013004 1.0:3112 1.0;3220 41508 25741 20763 26586 43220 84631 57281, 92600 70083 70314 536 889 313 133 717 0.92774 0.92681 0.92588 0.92496 0.92403 34863 62065 98536 44265 99244 28552 59382 06495 43539 45086 892 237 377 280 807 0.080 0.081 0.082 0.083 0.084 1.013328 1.013437 1.0~3545 1.013654 1.013762 70676 08965 58098 18085 88938 74958 66760 29549 48238 08826 554 341 059 061 156 0.92311 0.92219 0.92127 0.92035 0.91943 63463 36914 19586 11472 12560 86635 44608 96348 20124 95124 783 072 654 706 674 0.085 0.086 0.087 0.088 0.089 1.0;3871 1.013980 1.0'3089 1.0'3198 1.0'3308 70666 63283 66797 81220 06563 98398 05128 18277 28197 26330 696 660 747 460 201 0.91851 Oi91759 0.91667 0.91576 0.91484 22844 42312 70956 08767 55735 01457 20150 33152 23325 74452 356 982 295 631 003 0.090 0.091 0.092 0.093 0.094 1.0'3417 1.0'3526 1.0'3636 1.0'3746 1.0'3855 42837 90052 48220 17352 97459 05210 58465 80816 68081 17173 358 401 975 994 736 0.91393 0:91301 0.91210 0.91119 0.91028 11852 77108 51495 35002 27622 71228 99265 45090 96140 40766 187 803 403 557 940 0.095 0.096 0.097 0.098 0.099 1.0'3965 1.10075 1.113186 1.11296 1.10406 88551 90639 03736 27851 62995 26102 93978 21010 08507 58881 942 912 606 743 902 0.90937 0.90846 0.90755 0.90664 0.90574 29344 40160 60061 89037 27080 68231 68706 33272 53920 23548 420 150 654 921 496 0.100 1.13517 09180 75647 625 [c-J)1 1 0.90483 74180 35959 573 [(-;I1 1 118 ELEMENTARY Table TRANSCENDENTAL 4.4 EXPONENTIAL FUNCTIONS FUNCTION e-z ez X 0.101 0.102 0.103 0.104 0.100 1.10517 1.10627 1.10738 1.10849 1.10960 09180 75647 625 66417 63423 521 34717 27933 371 14090 76007 230 04549 15582 540 0.90483 0.90393 0.90302 0.90212 0.90122 74180 35959 573 30328 85864 089 95516 68876 819 69734 81516 470 52974 21204 780 0.105 0.106 0.107 0.108 0.109 1.11071 1.11182 1.11293 1.11404 1.11516 06103 55705 232 18765 06530 839 42544 79325 605 77453 86467 594 23503 41447 807 0.90032 0.89942 0.89852 0.89762 0.89673 45225 86265 613 46480 75924 059 56729 90305 534 75964 30434 876 04174 98235 450 0.110 0.111 0.112 0.113 0.114 1.11627 1.11739 1.11851 1.11963 1.12075 80704 58871 292 49068 54458 258 28606 45045 196 19329 48585 987 21248 84153 031 0.89583 0.89493 0.89404 0.89315 0.89225 41352 96528 251 87489 29031 000 42575 00357 257 06601 16015 519 79558 82408 325 0.115 0.116 0.117 0.118 0.119 1.12187 1.12299 1.12411 1.12524 1.12636 34375 71938 354 58721 33254 738 94296 90536 839 41113 67342 307 99182 88352 913 0.89136 0.89047 0.88958 0.88869 0.88780 61439 06831 368 52232 97472 599 51931 63411 334 60526 14617 364 78007 61950 067 0.120 0.121 0.122 0.123 0.124 1.12749 1.12862 1.12975 1.13088 1.13201 68515 79375 671 49123 67343 967 41017 80318 682 44209 47489 324 58709 99175 153 0.88692 0.88603 0.88514 0.88426 0.88337 04367 17157 516 39595 92875 591 83685 02627 096 36625 60820 866 98408 82750 886 0.125 0.126 0.127 0.128 0.129 1.13314 1.13428 1.13541 1.13655 1.13769 84530 66826 317 21682 83024 976 70177 81486 442 30026 97060 307 01241 65731 582 0.88249 0.88161 0.88073 0.87985 0.87897 69025 84595 403 48467 83416 046 36725 97156 940 33791 44643 827 39655 45583 178 0.130 0.131 0.132 0.133 0.134 1.13882 1.13996 1.14110 1.14224 1.14339 83833 24621 831 77813 11990 306 83192 67235 091 99983 30894 235 28196 44646 898 0.87809 0.87721 0.87634 0.87546 0.87459 54309 20561 324 77143 91043 564 09950 79373 297 50921 08771 138 00646 03334 043 0.135 0.136 0.137 0.138 0.139 1.14453 1.14568 1.14682 1.14797 1.14912 67843 51314 488 18935 94861 807 81485 20398 195 55502 74178 672 41000 03605 088 0.87371 0.87284 0.87197 0.87109 0.87022 59116 88034 434 26324 88719 322 02261 32109 436 86917 45798 347 80284 58251 595 0.140 0.141 0.142 0.143 0.144 1.15027 1.15142 1.15257 1.15372 1.15488 37988 57227 268 46479 84744 161 66485 37004 992 98016 66010 407 41085 24913 632 0.86935 0.86848 0.86762 0.86675 0.86588 82353 98805 820 93116 97667 890 12564 85914 032 40688 95488 962 77480 59205 017 0.145 0.146 0.147 0.148 0.149 1.15603 1.15719 1.15835 1.15951 1.16067 95702 68021 623 61880 50796 218 39630 29855 297 28963 62973 936 29892 09085 563 0.86502 22931 10741 288 0.86415 77031 84642 755 0.86329 39774 16319 421 0.86243 11149 42045 443 1 0.86156 91148 98958 277 0.150 1.16183 42427 28283 123 0.86070 79764 25057 807 [c-p1 1 [(-l’l1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL X FUNCTIONS 119 FUNCTION Table 4.4 e-1 ez 0.150 0.151 0.152 0.153 0.154 1.16183 1.16299 1.16416 1.16532 1.16649 42427 66580 02364 49789 08867 28283 81820 32112 42737 78439 123 230 335 886 490 0.86070 0.85984 0.85898 0.85812 0.85727 79764 76986 82807 97218 20210 25057 59205 41123 11393 11457 807 488 482 800 440 0.155 0.156 0.157 0.158 0.159 1.16765 1.16882 1.16999 1.17116 1.17233 79611 62030 56139 61947 79466 05125 89869 00913 07669 80717 080 080 572 465 662 0.85641 0.85555 0.85470 0.85384 0.85299 51774 91903 40588 97819 63589 83613 71018 17685 68481 69131 531 473 083 735 511 0.160 0.161 0.162 0.163 0.164 1.17351 1.17468 1.17586 1.17703 1.17821 08709 49688 02413 66896 43150 91810 13871 20999 88467 92722 235 592 654 025 171 0.85214 0.85129 0.85044 0.84959 0.84874 37889 20711 12045 11884 20218 66211 07151 40232 14590 80206 338 144 998 263 741 0.165 0.166 0.167 0.168 0.169 1.17939 iIi8Oi7 1.18175 ii18293 1.18412 31187 31017 42653 66106 01389 11390 23276 08361 47810 23969 594 011 533 843 378 0.84789 0.84704 0.84619 0.84535 Oi84450 37040 62341 96113 38346 89033 87915 89399 37188 84658 86034 828 660 270 733 326 0.170 0.171 0.172 0.173 0.174 1.18530 1.18649 1.18767 1.18886 1.19005 48513 07490 78332 61050 55658 20365 21711 13905 84032 20362 514 746 874 188 660 0.84366 0.84282 0.84197 0.84113 0.84029 48165 15734 91731 76148 68976 96383 71619 68499 44623 58431 682 939 904 201 438 0.175 0.176 0.177 0.178 0.179 1.19124 1.19243 1.19363 1.19482 1.19602 62166 80586 10931 53212 07441 12358 50669 27138 34800 67883 122 468 834 796 563 0.83945 0.83861 0.83777 0.83694 Oi83610 70207 79833 97845 24234 58993 69207 37074 22993 88768 97035 358 003 869 073 511 0.180 0.181 0.182 0.183 0.184 1.19721 ii19841 1.19961 1.20081 1.20201 73631 51792 41938 44080 58230 21810 93199 79868 80830 96301 165 657 311 812 462 0.83527 0.83443 0.83360 0.83276 0.83193 02114 53586 13404 81557 58038 11272 95789 15735 37090 26671 021 549 309 951 728 0.185 0.186 0.187 0.188 0.189 1.20321 1.20442 1.20562 1.20683 1.20804 84401 22603 72850 35153 09524 27695 77629 49924 49605 82901 376 686 742 317 811 0.83110 0.83027 0.82944 0.82861 0.82778 42838 35949 37363 47072 65066 52125 81932 85403 32680 94733 659 701 915 634 637 0.190 0.191 0.192 0.193 0.194 1.2'0924 iIi1045 1.21167 1.2'1288 1.2'1409 95976 94520 05169 27935 62829 57251 81299 64900 19119 56233 458 533 562 527 085 0.82695 0.82613 0.82530 0.82448 0.82365 91339 25881 68684 19741 79042 43362 51193 91682 39108 68576 318 854 387 186 832 0.195 0.196 0.197 0.198 0.199 1.2'1531 1.211652 1.211774 1.2'1896 1.212018 09864 69053 40407 23938 19658 89730 34316 05908 21642 99872 774 229 396 747 499 0.82283 ii82201 0.82119 0.82036 0.81954 46580 22346 06333 98531 98933 56018 78186 12657 37831 32925 384 562 919 021 626 0.200 1.;!2140 27581 60169 834 [ (-Y1 0.81873 07530 77981 859 [c-y1 120 ELEMENTARY TRANSCENDENTAL EXPONENTIAL Table 4.4 FUNCTIONS FUNCTION e-z X 0.200 0.201 0.202 0.203 0.204 1.22140 1.22262 1.22384 1.22507 1.22629 27581 47718 80081 24682 81534 60169 23327 11358 47499 56210 834 112 099 185 607 0.81873 0.81791 0.81709 0.81627 Oi81546 07530 24315 49279 82414 23711 77981 53859 42236 25609 87292 859 397 649 934 668 0.205 0.206 0.207 0.208 0.209 1.22752 1.22875 1.22998 1.23121 1.23244 50649 32039 25717 31695 49985 63177 95312 80752 48867 30254 678 005 723 721 869 0.81464 0.81383 0.81301 0.81220 0.81139 73164 30762 96499 70367 52356 11414 82920 87570 11939 43411 545 720 998 015 427 0.210 0.211 0.212 0.213 0.214 1.23367 1.23491 1.23614 1.23738 1.23862 80599 23550 78850 46512 26547 56743 61394 78503 43600 93452 251 396 512 719 285 0.81058 0.80977 0.80896 0.80815 0.80734 42459 40668 46975 61372 83850 70187 81276 66499 16488 22681 100 291 845 379 475 0.215 0.216 0.217 0.218 0.219 1.23986 1.24110 1.24234 1.24358 1.24483 18969 23790 41021 70676 12766 66061 00671 37764 19061 87531 862 728 020 978 187 0.80654 0.80573 0.80492 0.80412 0.80332 14401 53018 99693 54416 17181 77326 73479 05001 66559 53626 874 662 467 655 521 0.220 0.221 0.222 0.223 0.224 1.24607 1.24732 1.24857 1.24982 1.25107 67305 34305 13778 05737 10194 87380 64064 64283 35983 28362 820 879 447 926 294 0.80251 0.80171 0.80091 0.80011 0.79931 87979 66802 53643 48492 51343 62478 90195 34659 94554 69365 483 284 186 165 114 0.225 0.226 0.227 0.228 0.229 1.25232 l.25357 1.25482 1.25608 1.25734 27161 56652 98679 53254 20390 91864 78186 40279 32344 09839 345 948 295 151 113 0.79851 0.79771 0.79692 0.79612 0.79532 62187 81016 07822 42598 85335 59377 65674 90139 35453 05093 043 '274 647 721 973 0.230 0.231 0.232 0.233 0.234 1.25860 1.25985 1.26111 1.26238 1.26364 00099 92394 97288 14793 44922 29477 4v31 28329 27261 07777 863 426 426 349 797 0.79453 0.79373 0.79294 0.79215 0.79136 36025 94660 61233 35735 18158 03334 35242 06683 24314 95583 008 758 687 003 855 0.235 0.236 0.237 0.238 0.239 1.26490 1.26617 1.26744 1.26870 1.26997 87687 43101 11177 91928 85365 32891 66879 75283 24910 83836 756 857 640 818 547 0.79057 0.78978 0.78899 0.78820 0.78741 08496 06739 12880 26910 48823 28735 32802 17609 93770 72687 550 754 706 426 922 0.240 0.241 0.242 0.243 0.244 1.27124 1.27252 1.27379 1.27506 1.27634 91503 10353 41928 86241 43304 21404 08229 16194 18459 89454 692 095 849 570 665 0.78662 0.78584 0.78505 0.78427 0.78348 78610 16263 61775 15137 76342 66553 88345 51829 71556 62862 409 515 496 451 532 0.245 0.246 0.247 0.248 0.249 1.27762 1.27889 1.28017 1.28145 1.28274 13132 95735 91127 99321 20330 04886 41738 78269 94021 69811 611 230 966 162 341 0.78270 0.78192 0.78114 0.78035 0.77957 45382 22249 06935 99432 99733 41868 25477 31376 78034 84700 168 270 458 273 396 0.250 1.28402 54166 87741 484 C-i)2 [ 1 0.77880 07830 71404 868 [c-y1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL FUNCTION Table ez X 121 FUNCTIONS 4.4 e-2 0.250 0.251 0.252 0.253 0.254 1.28402 1.26531 1.28659 1.28788 ii28917 54166 00843 60372 32768 18042 87741 31195 84840 34630 67804 484 317 591 366 299 0.77880 0.77802 0.77724 0.77646 0.77569 07830 23715 47380 78818 18020 71404 58957 68946 23737 46476 868 312 150 828 034 0.255 0.256 0.257 0.258 0.259 1.29046 1.29175 1.23304 1.29433 1.29563 16208 27279 51267 88186 38048 72889 39703 59353 24237 28048 931 974 603 745 373 0.77491 0.77414 0.77336 0.77259 0.77182 64979 19687 82137 52321 30230 61080 92248 65449 06928 43703 928 360 096 045 483 0.260 0.261 0.262 0.263 0.264 1.29693 1.29822 1.29952 1.30082 1.30212 00866 76654 65424 67189 81963 65771 33689 29381 51724 00894 798 967 755 266 131 0.77105 0.77028 0.76951 0.76874 0.76797 15858 09196 10237 18973 35396 03566 15079 07575 11160 56706 284 142 806 303 173 0.265 0.266 0.267 0.268 0.269 1.30343 1.30473 1.30604 1.30734 1.30865 09757 50586 04463 71400 51410 78368 86927 30654 14936 46466 808 883 372 028 646 0.76720 0.76643 0.76567 0.76490 0.76414 59499 91275 30714 77811 32556 75855 01019 65373 02864 48198 698 133 938 015 937 0.270 0.271 0.272 0.273 0.274 1.30996 1.31127 1.31258 1.31390 1.31521 44507 50703 70013 02448 48022 33247 84587 11108 24739 38724 364 979 252 218 500 0.76337 0.76261 0.76185 0.76109 0.76033 94943 64964 42610 27876 20752 36853 05065 89837 28934 60882 186 386 543 278 066 0.275 0.276 0.277 0.278 0.279 1.31653 ii31784 1.31916 1.32048 1.32180 06748 78640 63710 61972 73438 67621 27303 34958 09095 69539 623 324 873 387 151 0.75957 0.75881 0.75805 0.75729 0.75653 21232 29307 44971 68215 99032 24968 61241 10508 14335 15047 476 409 337 547 380 0.280 0.281 0.282 0.283 0.284 1.32312 li32445 1.32577 1.32710 1.32843 98123 36039 87199 51618 29307 37436 35257 86792 17157 52794 936 318 007 164 731 0.75578 0.75502 0.75427 0.75351 0.75276 37414 83354 36845 97878 66447 55725 80208 33088 59717 06196 472 002 932 250 222 0.285 0.286 0.287 0.288 0.289 1.32976 ii33109 1.33242 1.33375 1.33509 20281 24552 42134 73041 17285 21473 52291 75675 23384 28508 753 710 843 488 403 0.75201 0.75126 0.75051 0.74976 0.74901 42543 26159 17288 15922 22054 19382 46886 37067 39041 02669 630 026 974 301 348 0.290 0.291 0.292 0.293 0.294 1.33642 1.33776 1.33910 1.34044 1.34178 74880 45839 30176 27904 39036 25472 50035 39293 31681 66971 103 199 724 481 373 0.74826 0.74751 0.74676 0.74602 0.74527 35675 56780 85359 21406 64914 78565 18091 73357 97221 43288 215 016 128 444 626 0.295 0.296 0.297 0.298 0.299 1.34312 1.34447 1.34581 1.34716 1.34850 63586 01568 52994 17878 96234 86276 32052 48097 79554 72910 747 735 594 052 654 0.74453 0.74378 0.74304 0.74230 0.74155 15874 74280 40123 13397 94094 65909 20179 61939 47774 35010 357 599 843 369 502 0.300 1.34985 88075 76003 104 0.74081 82206 [c-p1 866 [1 1 C-l)2 81717 122 ELEMENTARY Table TRANSCENDENTAL EXPONENTIAL 4.4 FUNCTIONS FUNCTION e-2 ez X 0.300 0.301 0.302 0.303 0.304 1.34985 1.35120 1.35256 1.35391 1.35526 88075 76003 104 93415 38015 618 12267 09482 272 44644 42288 348 90560 89671 692 0.74081 0.74007 0.73933 0.73859 0.73786 82206 81717 866 77727 46707 647 80648 89531 848 90963 70482 549 08664 50591 171 0.305 0.306 0.307 0.308 0.309 1.35662 1.35798 1.35934 1.36070 1.36206 50030 06224 066 23065 47892 497 09680 71980 642 09889 37150 137 23705 03421 961 0.73712 0.73638 0.73565 0.73491 0.73418 33743 91627 732 66194 56100 112 06009 07253 313 53180 09068 726 07700 26263 391 0.310 0.311 0.312 0.313 0.314 1.36342 1.36478 1.36615 1.36752 1.36888 51141 32177 794 92211 86161 378 46930 29479 880 15310 27605 258 97365 47375 624 0.73344 0.73271 0.73198 0.73124 0.73051 69562 24289 264 38758 69332 482 15282 28312 628 99125 68882 001 90281 59424 881 0.315 0.316 0.317 0.318 0.319 1.37025 1.37163 1.37300 1.37437 1.37575 93109 56996 611 02556 26042 743 25719 25458 804 62612 27561 208 13249 06039 370 0.72978 0.72905 0.72833 0.72760 0.72687 88742 69056 797 94501 67623 797 07551 25701 720 27884 14595 463 55493 06338 254 0.320 0.321 0.322 0.323 0.324 1.37712 1.37850 1.37988 1.38126 1.38264 77643 35957 085 55808 93753 895 47759 57246 476 53509 05630 003 73071 19479 542 0.72614 0.72542 0.72469 0.72397 0.72325 90370 73690 925 32509 90141 181 81903 29902 880 38543 67915 300 02423 79842 419 0.325 0.326 0.327 0.328 0.329 1.38403 1.38541 1.38680 1.38818 1.38957 06459 80751 421 53688 72784 617 14771 80302 136 89722 89412 403 78555 87610 642 0.72252 73536 42072 189 Oi72180 51874 31715 812 0.72108 37430 26607 016 0.72036 30197 05301 338 0.71964 30167 47075 395 0.330 0.331 0.332 0.333 0.334 1.39096 1.39235 1.39375 1.39514 1.39654 81284 63780 266 97923 08194 268 28485 12516 609 72984 69803 608 31435 74505 339 0.71892 0.71820 0.71748 0.71677 0.71605 37334 31926 170 51690 40570 286 73228 54443 294 01941 55698 947 37822 27208 486 0.335 0.336 0.337 0.338 0.339 1.39794 1.39933 1.40073 1.40214 1.40354 03852 22467 023 90248 10930 424 90637 38535 249 05034 05320 540 33452 12726 081 0.71533 0.71462 0.71390 0.71319 0.71248 80863 52559 924 31058 16057 326 88399 02720 095 52878 98282 260 24490 89191 756 0.340 0.341 0.342 0.343 0.344 1.40494 1.40635 1.40776 1.40916 1.41057 75905 63593 797 32408 62169 155 02975 14102 572 87619 26450 817 86355 07678 418 0.71177 0.71105 0.71034 0.70963 0.70892 03227 62609 715 89082 06409 751 82047 09177 248 82115 60208 649 89280 49510 748 0.345 0.346 0.347 0.348 0.349 1.41198 1.41340 1.41481 1.41623 1.41764 99196 67659 075 26158 17677 066 67253 70428 658 22497 40023 522 91903 41986 146 0.70822 0.70751 0.70680 0.70609 0.70539 03534 67799 973 24871 06501 685 53282 57749 463 88762 14384 398 31302 69954 390 0.350 1.41906 75485 93257 248 [(-;I2 1 0.70468 80897 18713 434 [ C-29 1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL FUNCTIONS FUNCTION Table e= X 1.23 4.4 e-= 0.350 0.351 0.352 0.353 0.354 1.41906 1.42048 1.42190 1.42333 1.42475 75485 73259 85237 11434 51864 93257 12195 18577 33601 79888 248 200 438 886 380 0.70468 0.70398 0.70328 0.70257 0.70187 80897 37538 01219 71933 49673 18713 55620 76340 77241 55394 434 921 929 521 037 0.355 0.356 0.357 0.358 0.359 1.42618 1.42760 1.42903 1.43046 1.43189 06542 75482 58698 56204 68015 81480 63844 53876 79897 71658 082 915 979 983 672 0.70117 0.70047 0.69977 0.69907 0.69837 34432 26202 24977 30750 43513 08572 35252 34611 06525 51573 398 !399 008 666 587 0.360 0.361 0.362 0.363 0.364 1.43332 1.43476 1.43619 1.43763 1.43907 94145 34608 89419 58592 42141 60340 78555 60351 41209 58046 258 848 880 556 276 0.69767 0.69697 0.69628 0.69558 0.69489 63260 89984 23678 64334 11947 71031 66872 41770 99095 42910 057 738 967 062 621 0.365 0.366 0.367 0.368 0.369 1.44051 1.44195 1.44339 1.44484 1.44628 40081 52426 79191 20389 76036 49217 54516 15177 73879 74739 078 071 881 090 677 0.69419 0.69350 0.69280 0.69211 0.69142 66508 28012 96450 71816 54104 77978 09755 44391 88730 50308 831 768 707 425 508 0.370 0.371 0.372 0.373 0.374 1.44773 1.44918 1.45063 1.45208 1.45353 46146 30733 29812 43398 71504 63324 86644 93158 32775 56852 462 554 799 223 487 0.69073 0.69004 0.68935 0.68866 0.68797 43306 39415 42425 52328 69118 37354 58789 24222 43955 28979 660 010 423 806 422 0.375 0.376 0.377 0.378 0.379 1.45499 1.45644 1.45790 1.45936 1.46082 14146 71337 43093 29428 30357 18201 71086 71225 75796 43431 336 052 910 632 842 0.68728 0.68660 0.68591 0.68523 0.68454 92787 23330 60738 05006 56126 90972 42301 96020 65870 66278 199 040 141 297 222 0.380 0.381 0.382 0.383 0.384 1.46228 1.46374 1.46521 1.46667 1.46814 45894 76054 20851 80300 54416 34224 09728 32959 68398 81989 532 512 881 485 380 0.68386 0.68317 0.68249 0.68181 0.68113 14092 78896 50532 28992 14271 12355 19899 05390 85990 79547 858 696 084 553 125 0.385 0.386 0.387 0.388 0.389 1.46361 1.47108 1.47255 1.47402 1.47550 43214 46708 64912 97842 45513 41144 14743 73135 88141 33054 302 133 370 592 939 0.68045 0.67977 0.67909 0.67841 0.67773 06362 05256 10949 23432 42700 04587 80321 26636 64104 13971 638 060 810 077 142 0.390 0.391 0.392 0.393 0.394 1.47698 1.47845 1.47993 1.48141 1.48290 07938 85134 77114 83893 05486 82642 13147 02288 29264 74753 577 180 401 352 084 0.67705 0.67638 0.67570 0.67502 0.67435 68744 01560 41139 87475 40562 98164 39289 60626 86133 40444 700 177 058 209 198 0.395 0.396 0.397 0.398 0.399 1.48438 1.48586 1.48735 1.48884 1.49033 41909 93175 59300 40299 36186 20914 51389 51306 07277 07402 066 667 642 615 565 0.67368 0.67300 0.67233 0.67166 0.67099 00392 66959 40256 20276 07013 48867 37386 32657 62009 53445 624 438 274 771 901 0.400 1.49182 46976 41270 318 0.67032 00460.35639 C-l)9 [1 c-y 11 301 124 ELEMENTARY TRANSCENDENTAL Table 4.4 EXPONENTIAL FUNCTIONS FUNCTION ez X e+ 0.400 0.401 0.402 0.403 0.404 1.49182 1.49331 1.49481 1.49630 1.49780 46976 72684 13326 68916 39469 41270 99960 76042 63582 58138 318 030 686 585 840 0.67032 0.66965 0.66898 0.66831 0.66764 00460 00610 07456 20993 41212 35639 37934 90346 23560 68928 301 596 733 309 902 0.405 0.406 0.407 0.408 0.409 1.49930 1.50080 1.50230 1.50380 1.50531 25000 25524 41056 71611 17204 56766 58019 61950 70111 85559 870 898 452 860 754 0.66697 0.66631 0.66564 0.66497 0.66431 68108 01674 41903 88788 42323 58474 24886 01521 22401 22216 400 338 227 888 786 0.410 0.411 0.412 0.413 0.414 1.50681 1.50832 1.50983 1.51134 1.51285 77851 53565 44363 50259 71268 12853 58058 28744 33993 84394 578 082 838 742 526 0.66365 0.66298 0.66232 0.66166 0.66100 02501 69316 42760 22828 09512 36319 00727 52122 27848 65912 366 386 256 372 454 0.415 0.416 0.417 0.418 0.419 1.51437 1.51588 1.51740 1.51892 1.52044 07406 58688 25129 06744 03547 92048 70568 35084 02239 90196 265 894 718 927 115 0.66034 0.65968 0.65902 0.65836 0.65770 02807 02704 09199 22284 41952 04982 84389 44120 24827 67816 886 050 673 158 932 0,420 0,421 0.422 0.423 0.424 1.52196 1.52348 1.52500 1.52653 1.52806 15556 42784 85246 42959 15937 18633 08753 83279 66456 84057 796 926 422 685 126 0.65'704 0.65639 0.65573 0.65507 0.65442 68198 01014 40393 86331 38819 15056 09171 93441 11806 08858 782 201 728 293 560 0.425 0.426 0.427 0.428 0.429 1.52959 1.53112 1.53265 1.53418 1.53572 04196 07751 26617 60809 10343 63378 33247 24018 67579 97349 690 382 802 666 347 0.65376 0.65311 0.65246 0.65181 0.65115 97851 63421 35522 14147 99291 29847 20675 27900 98731 81032 271 593 462 930 515 0.430 0.431 0.432 0.433 0.434 1.53725 1.53879 1.54033 1.54187 l.54341 75235 55499 51151 62207 88681 48281 56865 61127 00632 16487 402 110 008 428 038 0.65050 0.64985 0.64920 0.64856 0.64791 90947 89107 93766 04918 22554 23316 74749 85147 04976 85350 545 506 398 075 604 0.435 0.436 0.437 0.438 0.439 1.54496 1.54650 1.54805 1.54960 1.55115 30589 87947 60770 49074 52873 51338 49377 56340 19508 87714 384 427 096 826 108 0.64726 0.64661 0.64597 0.64532 0.64468 46670 77259 14314 57828 07796 78034 35439 10624 57294 29801 611 635 479 565 285 0.440 0.441 0.442 0.443 0.444 1.55270 1.55426 1.55581 1.55737 1.55893 72185-11336 07023 42305 57404 34107 23343 41779 04856 21915 042 879 580 367 277 0.64403 0.64339 0.64274 0.64210 0.64146 64210 27065 96354 72070 54208 83141 72956 55531 87795 27319 359 185 200 233 863 0.445 0.446 0.447 0.448 0.449 1.56049 1.56205 1.56361 1.56517 1.56674 01958 14665 42992 86956 46571 719 035 055 663 356 0.64082 0.64018 0.63954 0.63890 0.63826 42760 37720 39082 46840 60986 32318 61647 74800 31916 93768 776 123 880 208 809 0.450 1.56831 21854 90168 811 0.63762 81516 21773 293 32666 33744 86418 53521 99451 11 t-p Ic-y1 I ELEMENTARY TRANSCENDENTAL EXPONENTIAL 5 125 FUNCX’IONS FUNCTION Table ez 4.4 ecz 0.450 0.451 0.452 0.453 0.454 1.56831 1.56988 1.57145 1.57302 1.57459 21854 90168 811 12820 93202 449 19485 77649 003 41865 14175 089 79974 75018 775 0.63762 0.63699 0.63635 0.63571 0.63508 81516 21773 293 08421 77982 535 41697 25087 037 81336 26414 293 27332 45928 153 0.455 0.456 0.457 0.458 0.459 1.57617 1.57775 1.57932 1.53090 1.53249 33830 33991 152 03447 66477 911 88842 49440 916 90030 61419 781 07027 82533 449 0.63444 0.63381 0.63318 0.63254 0.63191 79679 48228 182 38370 98549 030 03400 62759 794 74762 07363 387 52448 99495 898 0.460 0.461 0.462 0.463 0.464 1.53407 39849 94481 775 1.55565 88512 80547 101 1.5~372453032 25595 846 1.5'3883 33424 16080 087 1.5'3042 29704 40039 147 0.63128 0.63065 0.63002 0.62939 0.62876 36455 06925 969 26773 98054 154 23399 41912 291 26325 08162 872 35544 67098 411 0.465 0.466 0.467 0.468 0.469 1.5'3201 41888 87101 182 1.5'3360 69993 48484 772 1.5'3520 14034 17000 511 1.5'3679 74026 87052 601 1.59839 49987 54640 444 0.62813 0.62750 0.62688 0.62625 0.62562 51051 89640 814 72840 47340 750 00904 12377 027 35236 57555 956 75831 56310 730 0.470 0.471 0.472 0.473 0.474 1.5'3999 41932 17360 241 1.61)159 49876 74406 589 1.60319 73837 26574 077 1.61148013829 76258 891 1.60640 69870 27460 416 0.62500 0.62437 0.62375 0.62313 0.62250 22682 82700 796 75784 11411 229 35129 17752 104 00711 77657 876 72525 67686 754 0.475 0.476 0.477 0.478 0.479 1.60801 41974 85782 835 1.60962 30159 58436 741 1.6X123 34440 54240 740 1.6:!284 54833 83623 064 1.6:1445 91355 58623 174 0.62188 0.62126 0.62064 0.62002 0.61940 50564 65020 075 34822 47461 685 25292 93437 314 21969 81993 957 24846 92799 250 0.480 0.481 0.482 0.483 0.484 l-6:.607 1.6:.769 1.6:.930 1.6:!092 1.6:!255 44021 92893 382 12849 01700 456 97853 01927 238 99050 12074 265 16456 52261 382 0.61878 0.61816 0.61754 0.61692 0.61631 33918 06140 853 49177 02925 827 70617 64680 018 98233 73547 436 32019 12289 639 0.485 0.486 0.487 0.488 0.489 1.6:!417 1.6;!579 1.6;!742 1.6;!905 1.6:,068 50088 44229 364 99962 11341 538 66093 78585 406 48499 72574 272 47196 21548 865 0.61569 0.61508 0.61446 0.61385 0.61323 71967 64285 113 18073 13528 659 70329 44630 776 28730 42817 043 93269 93927 508 0.490 0.491 0.492 0.493 0.494 1.6:1231 62199 55378 970 1.63394 93526 05565 057 1.63558 41192 05239 912 1.6:,722 05213 89170 270 1.6:)885 85607 93758 453 0.61262 0.61201 0.61140 0.61079 0.61018 63941 84416 069 40740 01349 867 23658 32408 668 12690 65884 251 07830 90679 799 0.495 0.496 0.497 0.498 0.499 1.64049 1.64213 1.64378 1.64542 1.64707 82390 57044 002 95578 18705 315 25187 20061 292 71234 04072 971 33735 15345 173 0.60957 09072 96309 287 Oi60896 16410 72896 868 0.60825 29838 11176 269 Oi60774 49349 02490 178 0.60713 74937 38789 634 0.500 1.61187212707 00128 147 0.60653 06597 12633 424 172 1 ( [ c-y1 ELEMENTARY Table TRANSCENDENTAL 4.4 EXPONENTIAL FUNCTIONS FUNCTION e2 X e-x 0.500 0.501 0.502 0.503 0.504 1.64872 1.65037 1.65202 1.65367 1.65532 12707 08166 20128 48611 93631 00128 06319 83464 82760 57054 147 214 418 175 920 0.6065:3 0.60592 0.60531 0.60471 0.60410 06597 44322 88106 37943 93828 12633 17187 46224 94122 55864 424 470 228 075 709 0.505 0.506 0.507 0.508 0.509 1.65698 1.65864 1.66030 1.66196 li66362 55204 33347 28076 39409 67361 60850 50305 83232 19105 19058 766 156 516 918 736 0.603510 0.602910 0.60229 0.6016'3 0.60109 55754 23715 97704 77717 63747 27040 03842 83065 62109 38975 541 093 390 362 237 0.510 0.511 0.512 0.513 0.514 1.66529 1.66695 1.66862 1.67029 1.67196 11949 73190 51101 45698 56998 45886 64047 39666 40535 36112 308 601 871 333 826 0.60049 0.59989 0.5992'3 0.5986'3 0.5980'3 55788 53833 57878 67916 83940 12265 81185 45538 05729 62761 943 502 434 153 369 0.515 0.516 0.517 0.518 0.519 1.67363 1.67531 1.67698 1.67866 1.68034 85017 29773 91283 69562 64627 97529 97587 10762 13205 82744 486 414 348 342 439 0.59750 0.596910 0.596310 0.59571 0.59511 05946 33926 67876 07789 53658 18237 74358 33920 00321 77550 489 019 965 238 053 0.520 0.521 0.522 0.523 0.524 1.68202 1.68371 1.68539 1.68708 1.68876 76496 05186 50712 13093 92344 98886 42818 97408 47211 78463 347 123 851 326 738 0.59452 0.59392 0.59333 0.5927.3 0.59214 05479 63245 26951 96589 72156 70194 83436 23052 95412 07481 339 138 015 460 294 0.525 0.526 0.527 0.528 0.529 1.69045 1.69215 1.69384 1.69553 1.69723 88483 01527 31492 78396 42254 79091 38708 48618 01819 93001 359 232 855 881 803 0.5915s 0.59096 0.5903'7 0.58978 0.58919 53643 41046 34359 33576 38690 66815 81562 60463 12850 48643 082 533 912 450 749 0.530 0.531 0.532 0.533 oI534 1.69893 1.70063 1.70233 1.70403 1.70574 23086 20906 35733 67583 16474 18550 76549 66781 90727 51574 654 702 146 817 883 0.58860 0.5880:1 0.58742 0.58684 0.5862!5 49696 66589 89361 18008 52523 78355 13085 64523 44946 67219 196 372 463 670 626 0.535 0.536 0.537 0.538 0.539 1.70744 1.70915 1.71086 1.71257 1.71429 82422 65445 65559 82781 17130 54211 05232 12940 87347 40175 545 748 887 510 036 0.58566 0.58508 0.5844'3 0.58391 0.58333 92901 39135 91221 49151 12920 44793 91706 22582 52628 97638 803 932 409 716 836 0.540 0.541 0.542 0.543 0.544 1.71600 1.71772 1.71944 1.72116 1.72288 68621 37273 23102 26125 46360 84858 36547 12106 30118 10887 460 069 159 747 296 0.58274 0.58216 0.581513 0.581013 0.58042 82523 57953 39205 26273 19151 73989 98641 89137 63601 40742 665 430 107 839 351 0.545 0.546 0.547 0.548 0.549 1.72460 1.72633 1.72806 1.72978 1.73152 83823 38533 10506 99760 06311 76435 50509 58581 27847 87233 429 656 095 197 477 0.57984 0.57926 0.57868 0.57810 0.57752 17833 22313 32586 48646 70487 39846 80782 83997 70519 61954 373 055 389 631 718 0.550 1.73325 30178 67395 237 c-y 11 0.57694 98103 80486 695 C-i)8 [ 1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL FUNCTIONS Table FUNCTION ez X 127 4.4 e-2 0.550 0.551 0.552 0.553 0.554 1.73325 1.73498 1.73672 1.73846 1.74019 30178 71378 29927 05843 99144 67395 00719 21325 65069 69542 237 302 750 647 780 0.57694 0.57637 0.57579 0.57522 0.57464 98103 31489 70638 15546 66205 80486 48877 90464 29163 89465 695 132 548 839 693 0.555 0.556 0.557 0.558 0.559 1.74194 1.74368 1.74542 1.74717 1.74892 09847 37970 83529 46543 27028 74075 19737 49342 07446 40349 399 955 837 121 310 0.57407 0.57349 0.57292 0.57235 0.57178 22611 84758 52640 26251 05586 96436 75715 53518 56633 12420 024 391 425 257 941 0.560 0.561 0.562 0.563 0.564 1.75067 1.75242 1.75417 1.75593 1.75768 25002 40484 73489 24037 92143 96101 24499 77091 07179 69816 083 041 459 036 648 0.57120 0.57063 0.57006 0.56949 0.56892 90638 81402 77873 80045 87911 48814 94320 78013 29541 79121 886 280 522 648 561 0.565 0.566 0.567 0.568 0.569 1.75944 1.76120 1.76297 1.76473 1.76649 77827 81105 01995 40515 96682 21815 21742 29927 08459 21189 104 902 989 520 621 0.56836 0.56779 0.56722 0.56665 0.56609 01467 20706 45624 76213 12469 57540 96153 26884 82224 95233 464 288 125 657 792 0.570 0.571 0.572 0.573 0.574 1.76826 1.77003 1.77180 1.77357 1.77535 70514 62029 71244 98177 42846 33735 13479 29574 52941 56273 152 471 208 024 392 0.56552 0.56496 0.56439 0.56383 0.56326 54386 01959 55181 14047 78551 99537 29326 19358 04955 22004 097 229 370 664 648 0.575 0.576 0.577 0.578 0.579 1.77713 1.77890 1.78068 1.78246 1.78425 05269 85463 83445 99235 32850 14038 02478 99612 85240 40940 362 341 864 377 016 0.56270 0.56214 0.56158 0.56101 0.56045 48688 24451 05837 92838 85449 06955 96822 29181 42170 74490 693 437 224 538 445 0.580 0.581 0.582 0.583 0.584 1.78603 1.78782 1.78961 1.79140 1.79319 84307 53624 40820 45912 68918 50073 97786 71010 58466 50662 382 336 772 414 599 0.55989 0.55933 0.55877 0.55822 0.55766 83665 87480 96888 11884 32463 65402 54726 82846 90701 19791 033 843 320 245 179 0.585 0.586 0.587 0.588 0.589 1.79499 1.79678 1.79858 1.80038 1.80218 09856 68744 45599 40441 53286 39900 20272 87669 39776 76077 067 757 600 313 198 0.55710 0.55654 0.55599 0.55543 0.55488 58618 90344 27635 70486 18892 12173 10464 57836 98018 75294 905 868 621 264 892 0.590 0.591 0.592 0.593 0.594 1.80398 1.80579 1.80760 1.80940 1.81121 84153 33061 00026 85067 88202 97856 08202 12004 15959 28572 940 413 477 787 596 0.55432 0.55377 0.55321 0.55266 0.55211 72847 32345 97380 67948 44043 34507 21050 80873 60481 06930 035 107 848 771 610 0.595 0.596 0.597 0.598 0.599 1.81303 1.81484 1.81666 1.81847 1.82029 09449 48827 06353 82045 75923 60156 22836 30550 99051 45908 569 588 566 264 101 0.55156 0.55101 0.55046 0.54991 0.54936 25658 12789 05431 03577 07222 67829 91340 26176 21601 27429 766 753 649 542 984 0.600 1.82211 88003 90508 975 c-y [1 0.54881 16360 94026 433 C-i)7 [ 1 128 ELEMENTARY TRANSCENDENTAL EXPONENTIAL Table 4.4 FUNCTIONS FUNCTION e-a ez 5 0.600 0.601 0.602 0.603 0.604 1.82211 ii82394 1.82576 1.82759 1.82942 88003 18305 66846 33645 18719 90508 54062 59597 31970 97859 975 083 740 203 499 0.54881 0.54826 0.54771 0.54716 0.54662 16360 30987 51097 76683 07741 94026 72304 13727 70305 94597 433 710 448 543 605 0.605 0.606 0.607 0.608 0.609 1.83125 1.83308 1.83491 1.83675 1.83859 22088 43770 83782 42143 18872 85773 26048 50853 94189 91893 244 479 497 676 312 0.54607 0.54552 0.54498 0.54443 0.54389 44266 86251 33692 86582 44917 39709 59293 07547 39217 09589 413 368 943 140 946 0.610 0.611 0.612 0.613 0.614 1.84043 1.84227 1.84411 1.84596 1.84780 13987 27507 59448 09832 78674 81637 02933 97134 07433 78869 455 750 270 364 496 0.54335 0.54280 0.54226 0.54172 0.54118 08690 77897 52533 32591 18066 74499 90323 13983 02940 15202 787 981 200 922 890 0.615 0.616 0.617 0.618 0.619 1.84965 1.85150 1.85335 1.85521 1.85707 65995 71812 96145 39011 00429 58327 94538 38085 41401 58772 090 381 258 120 725 0.54064 0.54010 0.53956 0.53902 0.53848 08953 05246 06940 14030 26510 09316 44370 79994 76357 94167 571 616 313 053 789 0.620 0.621 0.622 0.623 0.624 1.85892 1.86078 1.86264 1.86451 1.86637 80418 78996 96182 31995 86452 46342 62108 65928 19523 86472 044 121 925 215 402 0.53794 0.53740 0.53686 0.53633 0.53579 44375 67620 96238 30226 69576 94674 39663 91459 12924 67456 492 618 568 149 037 0.625 0.626 0.627 0.628 0.629 1.86824 1.87011 1.87198 1.87385 1.87573 59574 51378 61683 91108 39071 32222 24085 31242 24743 77511 407 530 321 442 543 0.53526 0.53472 0.53419 0.53365 0.53312 14285 64346 19754 80505 46591 18990 31997 71484 02990 92592 242 571 093 602 086 0.630 0.631 0.632 0.633 0.634 1.87761 1.87948 1.88136 1.88325 1.88513 05792 91289 95581 18687 60625 64343 61910 48763 05331 13924 132 454 361 198 678 0.53259 0.53205 0.53152 0.53099 0.53046 18010 94754 76818 64198 56888 06897 13047 78717 72114 61974 190 683 927 344 883 0.635 0.636 0.637 00.E . 1.88702 1.88891 1.89079 1.89269 1.89458 21414 01074 99623 17079 53463 58737 25849 03226 80722 50084 766 565 199 703 912 0.52993 0.52940 0.52887 0.52834 0.52781 54883 58177 66765 80641 99802 17568 08694 05682 79390 01207 489 574 485 975 673 0.640 0.641 0.642 0.643 0.644 1.89648 1.89837 1.90027 1.90217 1.90408 08793 83087 76365 88646 19949 04951 40855 55225 47391 18580 353 140 865 502 301 0.52729 0.52676 0.52623 0.52571 0.52518 24240 53951 88930 29172 74670 43048 77357 77105 15790 67436 557 426 369 242 140 0.645 0.646 0.647 0.648 0.649 1.90598 1.90789 1.90980 1.91171 1.91362 70292 39696 28178 35758 62456 71922 12453 47112 84748 36119 692 188 287 384 674 0.52466 0.52413 0.52361 0.52309 0.52256 25421 81418 42656 09131 80836 06592 08335 48263 02500 47694 872 432 478 807 830 0.650 1.91554 08290 13896 070 0.52204 57767 61016 048 C-i)7 c1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL 129 FUNCTIONS FUNCTION Table 4.4 e-z X 0.650 0.651 0.652 0,653 0.654 1.91!554 08290 13896 070 1.91'745 73279 32661 108 1.91'337 57443 08913 867 1.92129 60800 61070 883 1.92321 83371 09468 067 0.52204 0.52152 0.52100 0.52048 0.51996 57767 61016 048 39919 20157 530 27286 03334 394 19862 89283 277 17644 57261 823 0.655 0.656 0.657 0.658 0.659 1.92514 25173 76362 630 1.92706 86227 85934 997 1.92899 66552 64290 740 1.92'092 66167 39462 496 1.92128585091 41411 902 0.51944 0.51892 0.51840 0.51788 0.51736 20625 87048 156 28801 58940 364 42166 53755 974 60715 52831 438 84443 38021 612 0.660 0.661 0.662 0.663 0.664 1.9:!479 1.93672 1.93866 1.94060 1.94254 23344 02031 522 80944 55146 776 57912 36517 879 54266 83841 774 70027 36754 070 0.51685 0.51633 0.51581 0.51530 0.51478 13344 91699 238 47414 96754 426 86648 36594 140 31039 95141 674 80584 56836 146 0.665 0.666 0.667 0.668 0.669 1.94449 1.94643 1.94838 1.95033 1.95228 05213 36830 982 59844 27591 272 33939 54498 192 27518 64961 432 40601 08339 065 0.51427 0.51375 0.51324 0.51273 0.51222 35277 06631 974 95112 29998 365 60085 12918 798 30190 41890 516 05423 03924 002 0.670 0.671 0.672 0.673 0.674 1 =15423 73206 35939 496 1:(:,5619 25354 01023 417 1.05814 97063 58805 754 1.')6010 88354 66457 630 1.96206 99246 83108 314 0.51170 0.51119 0.51068 0.51017 0.50966 85777 86542 478 71249 77781 383 61833 66187 865 57524 40820 271 58316 91247 632 0.675 0.676 0.677 0.678 0.679 l/36403 29759 69847 187 ii 96599 79912 89725 700 1.96796 49726 07759 335 1.96993 39218 90929 575 1.97190 48411 08185 868 0.50915 0.50864 0.50813 0.50763 0.50712 64206 07549 157 75186 80313 718 91254 00639 348 12402 60132 723 38627 50908 661 0.680 0.681 0.682 0.683 0.684 1.97387 1.97585 1.97782 1.97980 1.98178 77322 30447 594 25972 30606 040 94380 83526 371 82567 66049 605 90552 56994 589 0.50661 0.50611 0.50560 0.50509 0.50459 69923 65589 610 06285 97305 142 47709 39691 448 94188 86890 827 45719 33551 185 0.685 0.686 0.687 0.688 0.689 1.98377 18355 37159 979 1: 98575 65995 89326 220 1,98774 33493 98?57 531 1,98973 20869 50703 885 1.99172 28142 35403 001 0.50409 0.50358 0.50308 0.50258 0.50207 02295 74825 526 63913 06371 449 30566 24350 644 "2250 25428 387 .3960 06773 037 0.690 0.691 0.692 0.693 0.694 1.99371 1.99571 1.99770 1.99970 2.00170 55332 43082 329 02459 66461 043 69544 00252 033 56605 41163 899 63663 87902 948 0.50157 0.50107 0.50057 0.50007 0.49957 60690 66055 534 47437 01448 895 39194 11627 713 35956 95767 658 37720 53544 971 0.695 0.696 0.697 0.698 0.699 Z!.OO37090739 41175 193 :!.00571 37852 03688 356 :!.00772 05021 80153 865 2.00972 92268 77288 865 2.01173 99613 03818 219 0.49907 0.49857 0.49807 0.49757 0.49708 44479 85i.35 969 56229 91216 541 72965 72961 653 94682 32044 844 21374 70637 732 0.700 2.01375 27074 70476 522 0.49658 53037 91409 515 (-;I6 !c-y 1 L [I 1 ELEMENTARY Table TRANSCENDENTAL EXPONENTIAL 4.4 FVNCTIONS FUNCTION e-z ez X 0.700 0.701 0.702 0.703 0.704 2.01375 2.01576 2.01778 2.01980 2.02182 27074 74673 42430 30365 38498 70476 90010 77179 48759 23544 522 108 065 247 296 0.49658 0.49608 0.49559 0.49509 0.49460 53037 89666 31256 77802 29299 91409 97526 92651 80943 67056 515 471 465 451 976 0.705 0.706 0.707 0.708 0.709 2.02384 2.02587 2.02789 2.02992 2.03195 66849 15438 84286 73414 82840 22347 68004 85374 01341 44819 653 586 210 511 374 0.49410 0.49361 0.49312 0.49262 0.49213 85742 47126 13446 84698 60875 56141 53841 66295 00135 62485 685 826 756 445 987 0.710 0.711 0.712 0.713 0.714 2.03399 2.03602 2.03806 2.04010 2.04214 12586 62672 33118 23945 35173 46750 40109 59906 43184 29026 612 996 288 280 822 0.49164 0.49115 0.49066 0.49017 0.48968 41974 27990 18916 14750 15485 60965 03682 99240 56730 85736 102 649 129 197 169 0.715 0.716 0.717 0.718 0.719 2.04418 2.04623 2.04827 2.05032 2.05237 66822 18913 91467 84503 98043 58556 74939 23384 51146 07529 873 531 083 049 226 0.48919 0.48870 0.48821 0.48772 0.48723 21117 31641 47053 67346 92516 96331 99079 05032 25731 73205 534 460 312 153 263 0.720 0.721 0.722 0.723 0.724 2.05443 2.05648 2.05854 2.06060 2.06266 32106 86714 61886 57644 74008 43887 13628 72211 77154 88034 743 106 257 626 189 0.48675 0.48626 0.48577 0.48529 0.48480 22559 57469 97243 41873 91357 59971 99034 03884 88500 67343 650 560 990 207 253 0.725 0.726 0.727 0.728 0.729 2.06473 2.06679 2.06886 2.07093 2.07300 10999 68637 46943 45938 65642 66486 76210 82971 54598 60992 529 896 273 438 036 0.48432 0.48384 0.48335 0.48287 0.48239 45689 04864 68878 37725 11401 55362 67990 21146 31229 15125 467 997 315 734 923 0.730 0.731 0.732 0.733 0.734 2.07508 2.07715 2.07923 2.08131 2.08339 06076 67261 49218 51967 75529 74122 68033 18844 04749 06025 645 852 323 882 589 0.48190 0.48142 0.48094 0.48046 0.47998 89900 73219 61352 54295 52042 90202 74309 85778 43422 66536 427 180 027 238 031 0.735 0.736 0.737 0.738 0.739 2.08548 2108756 2.08965 2109174 2.09384 19925 85175 71302 78325 06265 05027 86196 36056 43220 98392 819 344 419 868 173 0.47950 0.47902 0.47854 0.47806 0.47759 54589 61931 74064 90982 12680 74894 88751 28841 16377 73052 090 082 182 589 052 0.740 0.741 0.742 0.743 0.744 2.09593 2.09803 2.10013 2.10223 2.10433 55144 24983 15801 27621 60464 94364 26026 90360 86450 15478 563 109 816 725 007 0.47711 0.47663 0.47616 0.47568 0.47520 39155 70400 06412 47186 92716 21034 82972 81989 41687 86144 388 004 423 803 466 0.745 0.746 0.747 0.748 0.749 2.10644 2.10854 2.11065 2.11277 2.11488 14349 89299 85335 02477 40747 80727 87586 43551 58226 43325 065 641 917 625 155 0.47473 0.47425 0.47378 0.47331 0.47283 42999 98029 57801 22312 91555 39912 28019 75969 09739 55779 416 867 767 326 537 0.750 2.11700 00166 12674 C-l)3 669 0.47236 65527 41014 707 [ 1 [ (96 1 ELEMENTARY TRANSCENDENTAL EXPONENTIAL 131 FVNCTIONS FUNCTION Table ez X 4.4 e-z 0.750 0.751 0.752 0.753 0.754 2.11700 2.11911 2.12123 2.12336 2.12548 00166 12674 669 80754 a2217 212 a2534 70011 a30 05526 96236 688 49752 a3191 190 0.47236 0.47189 0.47142 0.47095 0.47048 65527 41014 707 44222 92841 982 27637 39130 a75 15766 08222 791 08604 28930 562 0.755 0.756 0.757 0.758 0.759 2.12761 2.12974 2.13187 2.13400 2.13613 15233 55298 098 0.1990 39105 663 10044 63289 745 39417 58655 946 90130 58141 739 0.47001 0.46954 0.46907 0.46860 0.46813 06147 30537 969 08390 42799 274 15328 95938 749 26958 20650 211 43273 48096 543 0.760 0.761 0.762 0.763 0.764 2.13827 2.14041 2.14255 2.14470 2.14684 62204 96818 602 55662 11894 152 70523 42714 282 06810 30765 301 64544 19676 075 0.46766 0.46719 0.46673 0.46626 0.46579 64270 09909 234 89943 38187 907 20288 65499 852 55301 24879 557 94976 49828 242 0.765 0.766 0.767 0.768 0.769 2.14899 2.15114 2.15329 2.15545 2.15760 43746 55220 173 44438 a5318 010 66642 60038 993 10379 31603 678 75670 54385 916 0.46533 0.46486 0.46440 0.46394 0.46347 39309 74313 393 88296 32768 297 41931 60091 573 00210 91646 708 63129 63261 598 0.770 0.771 0.772 0.773 0.774 2.15976 2.16192 2.16409 2.16625 2.16842 62537 a4915 008 71002 ala77 a66 01087 06121 167 52812 20653 514 26199 90647 604 0.46301 0.46255 0.46208 0.46162 0.46116 30683 11228 073 02866 72301 444 79675 83700 034 61105 83104 714 47152 08658 446 0.775 0.77 0.77 t 0.778 0.779 2.17059 2.17276 2.17493 2.17711 2.17929 21271 a3442 386 38049 68545 234 76555 17634 114 36810 04559 757 18836 05347 a30 0.46070 0.46024 0.45978 0.45932 0.45886 37809 98965 ala 33074 93092 580 32942 30565 la9 37407 51370 344 46465 95954 527 0.780 0.781 0.782 0.783 0.784 2.18147 2.18365 2.18583 2.18802 2.19021 22654 98201 117 48288 63501 691 95758 a3813 099 65087 43882 545 56296 30643 070 0.45840 60113 05223 545 Oi45794 78344 20542 069 0.45749 01154 a3733 175 0.45703 28540 37077 a90 0.45657 60496 23314 727 0.785 0.786 0.787 0.788 0.789 2.19240 2.19460 2.19679 2.19899 2.20119 69407 33215 744 04442 42911 a52 61423 53235 086 40372 59883 740 41311 60752 903 0.45611 0.45566 0.45520 0.45475 0.45429 97017 85639 236 38100 67703 540 83740 13615 885 33931 67940 176 88670 75695 532 0.790 0.791 0.792 0.793 0.794 2.20339 2.20560 2.20780 2.21001 2.21222 64262 55936 659 09247 47730 288 76288 40632 465 65407 41347 466 76626 58787 377 0.45384 0.45339 0.45293 0.45248 0.45203 47952 82355 822 11773 33849 215 80127 76557 724 53011 57316 754 30420 23414 649 0.795 0.796 0.797 0.798 0.799 2.21444 2.21665 2.21887 2.22109 2.22331 09968 04074 299 65453 90542 561 43106 33740 936 42947 51434 850 64999 63608 607 0.45158 0.45112 0.45067 0.45022 0.44977 12349 22592 237 98794 03042 379 89750 13409 518 a5213 02789 227 a5178 20727 758 0.800 2.22554 09284 92467 605 C-l)3 [ 1 0.44932 a9641 17221 591 (96 [1 1 132 ELEMENTARY Table TRANSCENDENTAL EXPONENTIAL 4.4 FUNCTIONS FUNCTION @ X ecz 0.800 0.801 0.802 0.803 0.804 2.22554 2.22776 2.22999 2.23222 2.23446 09284 75825 64644 75762 09202 92467 62440 00181 34513 96726 605 556 717 111 759 0.44932 0.44887 0.44843 0.44798 0.44753 89641 98597 12042 29971 52381 17221 42716 48109 84743 04412 591 986 530 691 369 0.805 0.806 0.807 0.808 0.809 2.23669 2.23893 2.24117 2.24341 2.24566 64988 43140 43681 66635 12022 19986 39932 94378 23379 69230 909 270 249 186 599 0.44708 0.44664 0.44619 0.44574 0.44530 79265 10621 46442 86726 31467 59356 02264 86271 64960 92358 447 340 556 242 738 0.810 0.811 0.812 0.813 0.814 2.24790 2.25015 2.25240 2.25466 2.25691 79866 70189 83014 18363 76258 76471 91886 64507 45618 88752 419 242 569 061 788 0.44485 0.44441 0.44396 0.44352 0.44308 80662 34305 92392 54918 21880 22941 11626 13780 85209 82167 134 826 063 512 806 0.815 0.816 0.817 0.818 0.819 2.25917 2.26143 2.26369 2.26596 2.26823 56723 59779 85450 33758 04725 49701 86510 59486 31195 66470 480 786 532 979 087 0.44263 0.44219 0.44175 0.44131 0.4408'7 93273 69092 49333 33992 23064 61351 79898 95392 65856 49756 106 654 332 218 146 0:822 0.823 0.824 2.27049 2.27277 2.27504 2.27732 2.27960 98375 14729 53812 15645 00251 32405 98368 35993 19188 24138 781 215 046 700 650 0.44043 0.43999 0.43955 0.43911 0.43867 16545 14429 16714 23395 34466 05999 93933 73347 04469 47967 263 588 574 662 847 0.825 0.826 0.827 0.828 0.829 2.28188 2.28416 2.28644 2.28873 2.29102 07653 37874 90936 66863 65678 29303 15424 65522 64904 01164 690 217 506 998 583 0.43823 0.43779 0.43735 0.43692 0.43648 49924 69765 93983 22575 55537 64949 16959 65982 74441 05193 237 611 985 171 342 0.830 0.831 0.832 0.833 0.834 2.29331 2.29561 2.29790 2.30020 2.30251 87402 32060 99674 90267 03862 64182 46132 41479 46985 61709 888 567 593 553 945 0.43604 0.43561 0.43517 0.43474 0.43430 92863 34549 80592 30987 85729 21535 87200 66356 23608 23995 593 502 699 428 109 0.835 0:838 0.839 2.30481 2.30712 2.30942 2.31173 2.31405 40482 00151 82890 88724 17676 87012 26555 86305 74537 01834 474 358 628 437 366 0.43387 0.43344 0.43300 0.43257 0.43214 44814 08238 75996 48084 24498 32990 16504 40877 72886 79740 906 293 616 664 233 0.840 0.841 0.842 0.843 0.844 2.31636 2.31868 2.32100 2.32332 2.32565 69767 45023 43465 65117 10003 81091 27518 58641 94304 56672 734 913 644 351 462 0.43171 0.43127 0.43084 0.43041 0.42998 05234 90286 79652 73326 71304 29079 88978 27942 14906 19239 693 558 052 679 788 0.845 0.846 0.847 0.848 0.849 2.32797 2.33030 2.33263 2.33497 2.33730 78145 69567 84292 22343 83745 70234 61805 60527 97872 07647 734 575 370 812 233 0.42955 0.42912 0.42869 0.42827 0.42784 73582 80155 91020 06172 25606 10739 59632 36577 12659 59395 148 516 204 654 005 0.850 2.33964 68519 25990 C-i)3 937 0.42741 49319 48726 670 8*Ei 8. $76 [ 1 [1 (536 ELEMENTARY TRANSCENDENTAL EXPONENTIAL 133 FUNCTIONS FUNCTION Table ez X 4.4 e-2 0.850 0.851 0.852 0.853 0.854 2.33964 2.34198 2.34433 2.34667 2.34902 68519 76689 08280 63314 41814 25990 91381 44636 28914 89719 937 538 295 459 607 0.42741 0.42698 0.42656 0.42613 0.42570 49319 77306 09563 46085 86869 48726 53025 45091 98148 85850 67n 901 0.855 0.856 0.857 0.858 0.859 2.35137 2.35372 2.35608 2.35843 2.36079 43805 69310 18352 90954 87141 74901 34660 21547 90464 98674 997 911 002 656 336 0.42528 0.42485 0.42443 0.42400 0.42358 31910 81204 34746 92533 54560 82274 61924 99730 71047 51652 123 574 893 281 373 0.860 0.861 0.862 0.863 0.864 2.36316 2.36552 2.36789 2.37026 2.37263 06937 50363 17445 08206 22669 05794 73806 67050 52237 98442 948 196 946 586 400 0.42316 0.42273 0.42231 0.42189 0.42147 20823 91317 66039 44983 28147 17748 45962 13343 97363 75917 817 841 840 945 606 0.865 0.866 0.867 0.868 0.869 2.37500 2.37738 2.37976 2.38214 2.38452 60859 22799 08513 18024 51357 77111 62065 29496 57978 28460 933 359 863 010 126 0.42105 0.42063 0.42021 0.41979 0.41937 15526 07115 02910 02908 07103 27321 30312 64050 08114 42503 165 439 296 234 963 0.870 0.871 0.872 0.873 0.874 2.38691 2.38929 2.39168 2.39408 2.39647 08535 89582 94522 23379 76177 24276 31145 37171 32849 11065 682 671 999 872 184 0.41895 0.41853 0.41811 0.41769 0.41727 15492 28071 44834 65779 90901 47638 04358 93919 97998 98691 983 162 324 822 126 0.875 0.876 0.877 0.878 0.879 2.39887 2.40127 2.40367 2.40608 2.40849 52939 53690 78455 27255 00117 67097 98624 05720 90861 58929 915 518 327 947 666 0.41686 0.41644 0.41602 0.41561 0.41519 20196 53660 91288 33076 79020 78508 20380 07652 24088 53866 403 096 513 408 560 0.880 0.881 0.882 0.883 0.884 2.41089 2.41331 2.41572 2.41814 2.42056 97064 18119 63308 32654 26181 17209 75397 45597 42330 82530 851 361 956 708 413 0.41478 0.41436 0.41395 0.41354 0.41312 29116 83360 41748 04276 70938 81581 92242 71274 04515 78218 367 420 097 140 250 0.885 0.886 0.887 0.888 0.889 2.42298 2.42540 2.42783 2.43026 2.43269 43914 85877 52094 42590 57387 85550 73163 69565 01380 97656 015 018 911 593 799 0.41271 0.41230 0.41188 0.41147 0.41106 41732 16653 95698 78861 66139 79049 94088 10827 17170 01433 666 753 593 568 949 0.890 0.891 0.892 0.893 0.894 2.43512 2.43756 2.44000 2.44244 2.44488 96512 59989 47841 60092 96769 89874 11946 00220 93481 32956 527 472 460 882 134 0.41065 0.41024 0.40983 0.40942 0.40901 57527 53022 52620 56315 64106 52345 59044 11078 98410 11406 488 001 959 082 922 0.895 0.896 0.897 0.898 0.899 2.44733 2.44978 2.45223 2.45468 2.45714 57894 43493 53589 88208 47374 62311 27659 77561 63026 37516 060 394 203 343 904 0.40860 0.40819 0.40779 0.40738 0.40697 75986 91952 12001 36127 64327 40848 77922 14226 41763 52948 458 685 207 826 135 0.900 2.45960 31111 56949 664 C-l)3 [1 d, - ?A7 I”. 726 193 0.40656 96597 40599 112 (-;I5 [ 1 ' ELEMENTARY TRANSCENDENTAL FUNCTIONS Table EXPONENTIAL 4.4 FUNCTION eI X e+ 0.900 0.901 0.902 0.903 0.904 2.45960 2.46206 2.46452 2.46699 2.46946 31111 56949 664 39444 79698 548 72398 66597 083 29997 80940 863 12266 88490 006 0.40656 0.40616 0.40575 0.40535 0.40494 96597 40599 112 32932 97943 710 73330 18615 453 17784 96654 028 66293 26504 879 0.905 0.906 0.907 0.908 0.909 2.47193 2.47440 2.47688 2.47935 2.48183 19230 57471 626 50913 58582 298 07340 64990 529 88536 52339 232 94525 98748 200 0.40454 0.40413 0.40373 0.40333 0.40292 18851 03018 802 75454 21451 540 36098 77463 377 00780 67118 736 69495 86885 773 0.910 0.911 0.912 0.913 0.914 2.48432 25333 84816 587 2.48680 80984 93625 386 2.48929'61504 10739 912 2.49178 66916 24212 291 2.49427 97246 24583 942 0.40252 0.40212 0.40171 0.40131 0.40091 42240 33635 975 19010 04643 753 99800 97586 047 84609 10541 915 73430 41992 136 0.915 0.916 0.917 0.918 0.919 2.49677 2.49927 2.50177 2.50427 2.50678 52519 04888 075 32759 60652 177 37992 89900 513 68243 93156 620 23537 73445 810 0.40051 0.40011 0.39971 0.39931 0.39891 66260 90818 809 63096 56304 950 63933 38134 089 68767 36389 877 77594 51555 677 0.920 0.921 0.922 0.923 0.924 2.50929 2.51180 2.51431 2.51682 2.51934 03899 36297 671 09353 89748 577 39926 44344 189 95642 13141 971 76526 11713 703 0.39851 0.39812 0.39772 0.39732 0.39692 90410 84514 173 07212 36546 962 27995 09334 165 52755 04954 021 81488 25882 492 0.925 0.926 0.927 0.928 0.929 2.52186 2.52439 2.52691 2.52944 2.53197 82603 58147 991 13899 73052 794 70439 79557 936 52249 03317 633 59352 72513 022 0.39653 0.39613 0.39573 0.39534 0.39494 14190 74992 866 50858 55555 360 91487 71236 720 36074 26099 830 84614 24603 311 0.930 0.931 0.932 0.933 0.934 2.53450 2.53704 2.53958 2.54212 2.54466 91776 17854 680 49544 72585 166 32683 72481 544 41218 55857 927 75174 63568 010 0.39455 0.39415 0.39376 0.39337 0.39297 37103 71601 130 93538 72342 199 53915 32469 987 18229 58022 122 86477 55429 996 0.935 0.936 0.937 0.938 0.939 2.54721 2.54976 2.55231 2.55486 2.55742 34577 39007 611 19452 28117 220 29824 79384 537 65720 43847 026 27164 75094 464 0.39258 0.39219 0.39180 0.39140 0.39101 58655 31518 373 34758 93504 997 14784 49000 198 98728 06006 497 86585 72918 221 0.940 0.941 0.942 0.943 0.944 2.55998 2.56254 2.56510 2.56767 2.57024 14183 29271 496 26801 65080 189 65045 43782 593 28940 29203 299 18511 87732 007 0.39062 0.39023 0.38984 0.38945 0.38906 78353 58521 102 74027 71991 894 73604 22897 977 77079 21196 971 84448 77236 34i 0.945 0.946 0.947 0.948 0.949 2.57281 2.57538 2.57796 2.58054 2.58312 33785 88326 089 74788 02513 161 41544 04393 651 34079 70643 376 52420 80516 117 0.38867 0.38829 0.38790 0.38751 0.38712 95709 01753 010 10856 05872 971 29886 01110 896 52794 99369 747 79579 12940 390 0.950 2.58570 96593 15846 199 C-l)3 [1 0.38674 10234 54501 207 [C-l)5 1 ' ELEMENTARY TRANSCENDENTAL EXPONENTIAL x 135 FUNCTIONS FUNCTION Table er 4.4 e-= 0.950 0.951 0.952 0.953 0.954 2.58570 2.58829 2.59088 2.59347 2.59607 96593 66622 62535 84356 32112 15846 61051 03133 31686 38890 199 072 898 135 126 0.38674 0.38635 0.38596 0.38558 0.38519 10234 44757 83143 25389 71491 54501 37117 74242 79713 67755 207 707 140 111 194 0.955 0.956 0.957 0.958 0.959 2.59867 2.60127 2.60387 2.60647 2.60908 05829 05532 31248 83003 60823 19521 70952 93153 88696 62757 695 740 828 799 366 0.38481 0.38442 0.38404 0.38365 0.38327 21445 75247 32893 94380 59703 52978 50378 75335 43613 71361 545 516 273 409 560 0.960 0.961 0.962 0.963 0.964 2.61169 2.61430 2.61692 2.61954 2.62216 64734 94761 50932 33272 41807 23117 80169 46914 38971 74573 718 136 592 373 688 0.38289 0.38251 0.38212 0.38174 0.38136 28859 01844 78654 59286 43734 75112 71780 78665 13447 94189 023 368 061 076 517 0.965 0.966 0.967 0.968 0.969 2.62478 2.62741 2.63004 2.63267 2.63530 76564 37569 24848 38428 78334 74575 62452 64304 08861 27480 291 101 825 583 539 0.38098 0.38060 0.38022 0.37984 0.37946 31997 24069 19947 19628 23107 39337 67716 98534 51378 46217 233 437 325 697 574 0.970 0.971 0.972 0.973 0.974 2.63794 2.64058 2.64322 2.64587 2.64851 44593 37232 56276 01753 73689 54152 25503 80798 61940 13478 532 708 158 558 808 0.37908 0.37870 0.37832 0.37794 0.37756 30381 41445 56296 74931 97345 03398 43649 88076 58165 75777 818 757 798 054 964 0.975 0.976 0.977 0.978 0.979 2.65116 2.65381 2.65647 2.65913 2.66179 72109 97042 48512 26548 31174 82606 19166 75651 07209 71643 682 470 628 434 642 0.37719 0.37681 0.37643 0.37606 0.37568 23535 53497 87227 24721 65976 63156 42920 38065 71965 68367 913 859 949 147 855 0.980 0.981 0.982 0.983 0.984 2.66445 2.66712 2.66979 2.67246 2.67513 62419 20308 04868 16127 54109 29417 43654 80145 07345 96380 138 602 169 099 441 0.37531 0.37493 0.37456 0.37418 0.37381 10988 59753 12267 68527 28529 51399 45561 75729 67156 45466 539 350 751 142 482 0.985 0.986 0.987 0.988 0.989 2.67781 2.68049 2.68317 2.68585 2.68854 18844 10356 28673 73822 45830 21049 57826 85862 86989 45722 708 547 418 272 235 0.37343 0.37306 0.37269 0.37232 0.37194 92269 59743 30948 05880 84535 36660 67113 63571 53155 63358 918 412 361 231 181 0.990 0.991 0.992 0.993 0.994 2.69123 2.69392 2.69662 2.69932 2.70202 44723 70528 23273 02984 09688 49262 87498 53013 41079 49668 289 962 016 142 652 0.37157 0.37120 0.37083 0.37046 0.37009 66910 53000 42802 36313 33529 22045 57455 98195 73247 11961 691 187 674 362 296 0.995 0.996 0.997 0.998 0.999 2.70472 2.70743 2.71013 2.71285 2.71556 43412 04184 92030 06977 49053 79452 33802 18796 43219 18566 181 382 637 755 687 0.36972 0.36935 0.36898 0.36861 0.36824 34445 39058 47366 59363 75046 44058 99632 09141 03418 13662 983 024 744 822 921 1.000 2.71828 18284 59045 C-l)3 235 0.36787 94411 (-!I5 71442 322 [ 1 [ 1 ELEMENTARY TRANSCENDENTAL FUrVCTIONS Table EXPONENTIAL 4.4 FUNCTION e-l e2 2 1.00000 1.10517 1.22140 1.34985 1.49182 00000 00000 09180 75648 27581 60170 88075 76003 46976 41270 1.00000 00000 0.90483 0.81873 0.74081 0.67032 74180 35959 57316 07530 77981 85867 82206 81717 86607 00460 35639 30074 1.64872 1.82211 2.01375 2.22554 2.45960 12707 00128 88003 90509 27074 70477 09284 92468 31111 56950 0.60653 0.54881 0.49658 0.44932 0.40656 06597 12633 42360 16360 94026 43263 53037 91409 51470 89641 17221 59143 96597 40599 11188 2.71828 3.00416 3.32011 3.66929 4.05519 18284 59045 60239 46433 69227 36547 66676 19244 99668 44675 0.36787 0.33287 0.30119 0.27253 0.24659 94411 71442 32160 10836 98079 55329 42119 12202 09664 17930 34012 60312 69639 41606 47694 4.48168 4.95303 5.47394 6.04964 6.68589 90703 38065 24243 95115 73917 27200 74644 12946 44422 79269 0.22313 0.20189 0.18268 0.16529 0.14956 01601 48429 82893 65179 94655 40849 35240 52734 65022 88882 21586 53830 86192 22635 05264 22:10 s-23 2:4 7.38905 8.16616 9.02501 9.97418 11.02317 60989 30650 99125 67650 34994 34121 24548 14721 63806 41602 0.13533 0.12245 0.11080 0.10025 0.09071 52832 36612 69189 64282 52981 91022 31583 62333 88333 88437 22803 73373 79532 89412 50338 2.5 12.18249 13.46373 14.87973 16.44464 18.17414 39607 03473 80350 01690 17248 72834 67710 97050 53694 43061 0.08208 0.07427 0.06720 0.06081 0.05502 49986 23898 79517 35782 14333 88043 55127 39749 76513 00626 25217 96500 32200 56407 22903 20.08553 22.19795 24.53253 27.11263 29.96410 69231 87668 12814 41631 01971 09349 89206 57887 00473 97013 0.04978 0.04504 0.04076 0.03688 0.03337 70683 67863 94298 92023 93557 80607 22039 78366 21517 31674 01240 00545 32699 60326 07948 33.11545 36.59823 40.44730 44.70118 49.40244 19586 92314 44436 77988 43600 67391 44933 00823 91055 30174 0.03019 0.02732 0.02472 0.02237 0.02024 73834 22318 50074 37224 47292 56080 35264 70339 39120 07718 56165 59578 19114 45804 38847 t-23 414 54.59815 60.34028 66.68633 73.69979 81.45086 00331 44239 75973 61969 10409 25142 36995 95797 86649 68117 0.01831 0.01657 0.01499 0.01356 0.01227 56388 88734 18029 26754 01761 24754 55768 20477 70621 85590 12200 93176 73399 03068 44118 i-56 4: 7 4.8 4.9 90.01713 99.48431 109.94717 121.51041 134.28977 13005 21814 56419 33809 24521 23499 75187 34881 96849 35485 0.01110 0.01005 0.00909 0.00822 0.00744 89965 38242 30650 18357 44633 58164 52771 01695 81709 97470 49020 02884 65830 70924 34052 E t: 0: 4 0. 5 i-76 0: 8 0.9 1. 0 1.1 1'3 1:4 1. 5 1.6 11-i 1:9 3:; ;:t 3. 0 z 3:s 3.4 3.5 3:: z 210 00000 00000 148.41315 91025 76603 0.00673 79469 99085 46710 5.0 From C. E. Van Orstrand, Tablesof the exponentialfunction and of the circular sine and cosine to radian arguments, Memoirsof the National Academy of Sciences, 14, Fifth Memoir. U.S. vol. GovernmentPrinting Office, Washington,D.C., 1921(with permission) e-",x12.4. for ELEMENTARY TRANSCENDENTAL EXPONENTIAL 137 FUNCTIONS FUNCTION Table ez 4.4 e-2 148.41315 164.02190 181.27224 200.33680 221.40641 91025 77 72999 02 18751 51 99747 92 62041 87 0.00673 0.00609 0.00551 0.00499 0.00451 79469 99085 46710 67465 65515 63611 65644 20760 77242 15939 06910 21621 65809 42612 66798 244.69193 270.42640 298.86740 330.29955 365.03746 22642 20 74261 53 09670 60 99096 49 78653 29 0.00408 0.00369 0.00334 0.00302 0.00273 67714 38464 06699 78637 16482 93082 59654 57471 27277 75547 45375 81475 94448 18768 36923 403.42879 34927 35 445.85777 00825 17 492.74904 10932 56 544.57191 01259 29 601.84503 78720 82 0.00247 0.00224 0.00202 0.00183 0.00166 87521 76666 35842 28677 19485 80247 94306 36295 73436 63047 77028 90683 15572 73173 93450 665.14163 735.09518 812.40582 897.84729 992.27471 30443 62 92419 73 51675 43 16504 18 56050 26 0.00150 0.00136 0.00123 0.00111 0.00100 34391 92977 57245 03680 37547 89342 09119 02673 48118 37751 47844 80308 77854 29048 51076 1096.63315 84284 59 1211.96707 44925 77 1339.43076 43944 18 1480.29992 75845 45 1635.98442 99959 27 0.00091 0.00082 0.00074 0.00067 0.00061 18819 65554 51621 51049 23265 40427 65858 08376 67937 55387 75193 84424 12527 61129 57256 1808.04241 44560 63 1998.19589 51041 18 2208.34799 18872 09 2440.60197 76244 99 2697.28232 82685 09 0.00055 0.00050 0.00045 0.00040 0.00037 30843 70147 83358 04514 33440 61070 28271 82886 79706 97349 78979 78671 07435 40459 08837 2980.95798 70417 28 3294.46807 52838 41 3640.95030 73323 55 4023.87239 38223 10 4447.06674 76998 56 0.00033 0.00030 0.00027 0.00024 0.00022 54626 27902 51184 35391 38078 86666 46535 69972 14233 85168 27107 95202 48673 24178 84827 4914.76884 02991 34 5431.65959 13629 80 6002.91221 72610 22 6634.24400 62778 85 7331.97353 91559 93 0.00020 0.00018 0.00016 0.00015 0.00013 34683 69010 64417 41057 93667 57912 65858 10987 63341 07330 75095 47660 63889 26482 01145 8103.08392 75753 84 8955.29270 34825 12 9897.12905 87439 16 10938.01920 81651 84 12088.38073 02169 84 0.00012 0.00011 0.00010 0.00009 0.00008 34098 04086 67955 16658 08490 11474 10394 01837 09335 14242 31478 17334 27240 65556 63226 13359.72682 96618 72 14764.78156 55772 73 16317.60719 80154 32 18033.74492 78285 11 19930.37043 82302 89 0.00007 0.00006 0.00006 0.00005 0.00005 48518 29887 70059 77287 36490 85387 12834 95053 22210 54515 99432 17698 01746 82056 17530 22026.46579 48067 17 0.00004 53999 29762 48485 ELEMENTARY TRANSCENDENTAL FUNCTIONS Table EXPONENTIAL 4.4 FUNCTION e* 0)1.00000 00000 00000 000 0)2.71828 18284 59045 235 I 112.00855 60989 30650 227 e 0)7.38905 36923 18766 774 ( lj5.45981 50033 14423 908 1.48413 4.03428 1.09663 2.98095 3)8.10308 :: 12 13 14 15910 25766 034 79349 27351 226,-31584 28458 599 79870 41728 275 39275 75384 008 -- 4)2.20264 5.98741 1.62754 4.42413 1.20260 65794 80671 65241715 19781 846 -79141 90039 208 39200 89205 03342841 64776 778--c ( 6)3.26901 6)8.88611 7)2.41549 6.56599 1.78482 73724 72110 639 05205 07872 637 52.75357529 821 69137 33051 114~ 30036 31872 608 4.85165 1.31881 3.58491 9.74480 2.64891 19540 97902 78057344 83214 69728461 31591 562-p 34462 48902 60022129 84347 229 25 26 27 28 29 7.20048 1.95729 5.32048 1.44625 3.93133 30 e-= 1.00000 3.67879 1.35335 4.97870 1.83156 00000 00000 000 44117 14423 216 28323 66126 919 68367 86394 298 38888 73418 029 Ii f- 3j6.73794 69990 85467 097 - 3 2.47875 21766 66358 423 4 1.23409 96555 45162 080 3.35462 62790 66795 388 9.11881 80408 25118 495 4.53999 1.67017 6.14421 2.26032 8.31528 29762 48485 154 00790 24565 931 23533 28209 759 94069 81054 326 71910 35678 841 !- 713.05902 32050 18257 884 Ii - 7 1.12535 17471 85166 844 9 5.60279 79744 71262 660 8 4.13993 77187 92591 145 1.52299 64375 37267 540 2.06115 7.58256 2.78946 1.02618 3.77513 36224 38557 828 04279 11906 728 80928 68924 808 79631 70189 030 45442 79097 752 99337 38587 252--. 60942 88387 643 24060 17986 167 70642 91475 17442971 44042 074- -11 1.38879 -12 5.10908 -12 1.87952 -13 6.91440 I -13 I 2.54366 43864 96402 059 90280 63324 720 88165 39083 295 01069 40203 009 56473 76922 910 1.06864 2.90488 7.89629 2.14643 5.83461 74581 52446 215, 49665 24742 5232 60182 68069 516 57978 59160 646~ 74252 74548 814 -14 9.35762 -14 3.44247 -14 1.26641 -15 4.65888 I -15 I 1.71390 29688 40174 605 71084 69976 458 65549 09417 572 61451 03397 364 84315 42012 966 (15)1.58601 4.31123 1.17191 3.18559 (16)8.65934 34523 13430 728 15471 15195 227, 42372 80261 1311 31757 11375 622~ 00423 99374 695 6.30511 2.31952 8.53304 3.13913 1.15482 67601 46989 386 28302 43569 388 76257 44065 794 27920 48029 629 24173 01578 599 (17)2.35385 6.39843 1.73927 4.72783 (19)1.28516 26683 70199 854 49353 00549 492 49415 20501 047 94682 29346 561 00114 35930 828 4.24835 1.56288 5.74952 2.11513 7.78113 42552 91588 995 21893 34988 768 22642 93559 807 10375 91080 487 22411 33796 516 45 46 47 48 49 I 19I 9.49611 28861 48509 535 20 7.01673 71057 90067 875 3.49342 59120 02448 396 2.58131 94206 97631 739 2.86251 1.05306 3.87399 1.42516 5.24288 85805 49393 644 17357 55381 238 76286 87187 113 40827 40935 106 56633 63463 937 50 (21)5.18470 55285 87072 464 :: ;34 40 2 43 44 (21)1.90734 65724 95099 691 (-22)1.92874 98479 63917 783 ELEMENTARY TRANSCENDENTAL EXPONENTIAL FUNCTIONS 139 FUNCTION Table ez 4.4 ec2 5.18470 1.40934 3.83100 1.04137 2.83075 55285 90824 80007 59433 33032 87072 26938 16576 02908 74693 464 796 849 780 900 1.92874 7.09547 2.61027 9.60268 3.53262 98479 41622 90696 00545 85722 63917 84704 67704 08676 00807 783 139 805 030 030 7.69478 2.09165 5.68571 1.54553 4.20121 52651 94960 99993 89355 04037 42017 12996 35932 90103 90514 138 154 223 930 255 1.29958 4.78089 1.75879 6.47023 2.38026 14250 28838 22024 49256 64086 07503 85469 24311 45460 94400 074 081 649 326 606 1.14200 3.10429 8.43835 2.29378 6.23514 73898 79357 66687 31594 90808 15684 01919 41454 69609 11616 284 909 489 879 883 8.75651 3.22134 Iv.18506 4.35961 1.60381 07626 02859 48642 00000 08905 96520 92516 33981 63080 48637 338 089 006 974 853 (28)1.69488 4.60718 1.25236 3.40427 9.25378 92444 66343 31708 60499 17255 10333 31291 42213 31740 87787 714 543 781 521 600 2.51543 6.83767 1.85867 5.05239 1.37338 86709 12297 17452 36302 29795 19167 62743 84127 76104 40176 006 867 980 195 188 3.97544 1.46248 5.38018 1.97925 7.28129 97359 62272 61600 98779 01783 08646 51230 21138 46904 21643 808 947 414 554 834 3.73324 1.01480 2.75851 7.49841 2.03828 19967 03881 34545 69969 10665 99001 13888 23170 90120 12668 640 728 206 435 767 2.67863 9.85415 3.62514 1.33361 4.90609 69618 46861 09191 48155 47306 08077 11258 43559 02261 49280 944 029 224 341 566 (34j5.54062 1.50609 4.09399 1.11286 3.02507 23843 73145 69621 37547 73222 93510 85030 27454 91759 01142 053 548 697 412 338 1.80485 6.63967 2.44260 8.98582 3.30570 13878 71995 07377 59440 06267 45415 80734 40527 49380 60734 172 401 679 670 298 8.22301 2.23524 6.07603 1.65163 4.48961 27146 66037 02250 62549 28191 22913 34715 56872 94001 74345 510 047 150 856 246 1.21609 4.47377 1.64581 6.05460 2.22736 92992 93061 14310 18954 35617 52825 81120 82273 01185 95743 564 735 651 885 739 1.22040 3.31740 9.01762 2.45124 6.66317 32943 00983 84050 55429 62164 17840 35742 34298 20085 10895 802 626 931 786 834 8.19401 3.01440 1.10893 4.07955 1.50078 26239 87850 90193 86671 57627 90515 65374 12136 77560 07394 430 553 379 158 888 1.81123 4.92345 1.33833 3.63797 9.88903 90828 82860 47192 09476 03193 89023 12058 04269 08804 46946 282 400 500 579 771 5.52108 2.03109 7.47197 2.74878 1.01122 22770 26627 23373 50079 14926 28532 34810 42990 10214 10448 732 926 161 930 530 (43)2.68811 For 1zj>lOO see Example 11. 71418 16135 448 II f-29)5.90009 05415 97061 391 -30 2.93748 20113 03639 412 -29 1.08063 92777 86978 947 7.98490 21117 10802 808 2.17052 42456 07278 495 (-44)3.72007 59760 20835 963 140 Table ELEMENTARY 4.5 TRANSCENDENTAL RADIX @O- TABLE FUNCTIONS OF THE EXPONENTIAL fl FUNCTION e-2210-n 1.00000 1.00000 1I00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 00001 00002 00003 00004 00005 00006 00007 00008 00009 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00002 00004 00008 00012 00018 00024 00032 00040 50000 00000 50000 00000 50000 00000 50000 00000 50000 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99999 99998 99997 99996 99995 99994 99993 99992 99991 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00002 00004 00008 00012 00018 00024 00032 00040 50000 00000 50000 00000 50000 00000 50000 00000 50000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 00010 00020 00030 00040 00050 00060 00070 00080 00090 00000 00000 00000 00000 00000 00000 00000 00000 00000 00050 00200 00450 00800 01250 01800 02450 03200 04050 00000 00000 00000 00000 00000 00000 00001 00001 00001 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99990 99980 99970 99960 99950 99940 99930 99920 99910 00000 00000 00000 00000 00000 00000 00000 00000 00000 00050 00200 00450 00800 01250 01800 02449 03199 04049 00000 00000 00000 00000 00000 00000 99999 99999 99999 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 00100 00200 00300 00400 00500 00600 00700 00800 00900 00000 00000 00000 00000 00001 00001 00002 00003 00004 05000 00002 20000 00013 45000 00045 80000 00107 25000 00208 80000 00360 45000 00572 20000 00853 05000 -01215 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99900 99800 99700 99600 99500 99400 99300 99200 99100 00000 00000 00000 00000 00001 00001 00002 00003 00004 04999 19999 44999 79999 24999 79999 44999 19999 04999 99998 99987 99955 99893 99792 99640 99428 99147 98785 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 01000 02000 03000 04000 05000 06000 07000 08000 09000 00005 00020 00045 00080 00125 00180 00245 00320 00405 00000 00000 00000 00001 00002 00003 00005 00008 00012 01667 13333 45000 06667 08333 60000 71667 53334 15000 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.94999 99000 98000 97000 96000 95000 94000 93000 92000 91000 00004 00019 00044 00079 00124 00179 00244 00319 00404 99999 99999 99999 99998 99997 99996 99994 99991 99987 98333 86667 55000 93333 91667 40000 28333 46667 85000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 10000 20000 30000 40000 50000 60000 70000 80000 90000 00500 02000 04500 08000 12500 18000 24500 32000 40500 00016 00133 00450 01066 02083 03600 05716 08533 12150 66667 33340 00034 66773 33594 00540 67667 35040 02734 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 90000 80000 70000 60000 50000 40000 30000 20000 10000 00499 01999 04499 07999 12499 17999 24499 31999 40499 99983 99866 99550 98933 97916 96400 94283 91466 87850 33334 66673 00034 33440 66927 00540 34334 68373 02734 Compiled from C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences,vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission), ELEMENTARY RADIX TABLE TRANSCENDENTAL OF THE 141 FTJNCTIONS EXPONENTIAL FUNCTION Table 4.5 1.00001 1.00002 1.00003 1.00004 1.00005 1.00006 1.00007 1.00008 1.00009 50000 00001 50004 00010 50020 00036 50057 00085 50121 16666 33334 50003 66677 83359 00054 16766 33504 50273 70833 00000 37502 33342 37526 00065 70973 00273 37992 0.99999 0.99998 0.99997 0.99996 0.99995 0.99994 0.99993 0.99992 0.99991 00000 00001 00004 00007 00012 00017 00024 00031 00040 49999 99998 49995 99989 49979 99964 49942 99914 49878 83333 66667 50003 33343 16692 00053 83433 66837 50273 1.00010 1.00020 1.00030 1.00040 1.00050 1.00060 1.00070 1.00080 1.00090 00050 00200 00450 00800 01250 01800 02450 03200 04051 001% 01333 04500 10667 20835 36005 57176 85350 21527 67083 40000 33752 73341 93776 40064 67223 40273 34242 34167 26668 02510 86724 04384 80648 40801 10308 14882 0.99990 0.99986 0.99970 0.99960 0.99950 0.99940 0.99930 0.99920 0.99910 00049 30199 00449 00799 01249 01799 02449 03199 04048 99833 98666 95500 89334 79169 64005 42843 14683 78527 33749 99167 73333 06668 33747 97512 39991 27057 39935 33609 73060 33257 46724 29384 20648 95801 30307 99880 1.00100 1.00200 1.00300 1.00400 1.00501 1.0060.1 1.00702 1.00803 1.00904 05001 20013 45045 80106 25208 80360 45572 20855 06217 66708 34000 03377 77341 59401 54064 66848 04273 73867 34166 26675 02601 87235 06338 86485 55523 43117 81406 80558 55810 29341 88080 35662 55845 16000 20736 25705 0.99900 0.99800 0.99700 0.99600 0.99501 0.99401 0.99302 0.99203 0.99104 04998 19986 44955 79893 24791 79640 44429 19148 03787 33374 67333 03372 43991 92682 53935 33235 37060 72883 99166 06675 97601 47235 31335 26474 10490 63033 66216 80554 55302 20662 23064 25642 44988 47970 98697 45648 1.01005 1.02020 1.03045 1.04081 1.05127 1.06183 1.07250 1.08328 1.09417 13 00000 00002 00004 00008 00012 00018 00024 00032 00040 01670 13400 45339 07741 10963 65465 81812 70676 42837 84168 26755 53516 92388 76024 45359 54216 74958 05210 05754 81016 85561 22675 03969 62222 47905 55443 35787 21655 01439 24400 70448 75176 46849 31039 59878 28976 0.99004 0.98019 0.97044 0.96078 0.95122 0.94176 0.93239 0.92311 0.91393 98337 86733 55335 94391 94245 45335 38199 63463 11852 49168 06755 48508 52323 00714 84248 05948 86635 7122‘8 05357 30222 17693 20943 00909 70953 22885 78291 18674 39060 08141 25284 92107 14253 71528 79726 07598 73535 09180 27581 88075 46976 12707 88003 27074 09284 31111 75647 60169 76003 41270 00128 90508 70476 92467 56949 62481 83392 10398 31782 14684 97487 52162 60457 66380 17078 10720 37443 48530 86508 53677 45494 95375 01266 0.90483 0.81873 0.74081 0.67032 0.60653 0.54881 0.49658 0.44932 0.40656 74180 07530 82206 00460 06597 16360 53037 89641 96597 35959 77981 81717 35639 12633 94026 91409 17221 40599 57316 85866 86606 30074 42360 43262 51470 59143 11188 42491 99355 68738 44329 37995 84589 48001 01024 34542 18284 59045 23536 02875 0.36787 94411 71442 32159 55238 9 1 1 1.10517 1.22140 1.34985 1.49182 1.64872 1.82211 2.01375 2.22554 2.45960 1 0 2.71828 54 ; 6 8' ; 37500 33333 37498 99991 70807 99935 37360 33060 37008 142 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES 0.000 sin z 0.00000 00000 00000 00000 000 0.001 0.002 0.003 0.004 0.00099 0.00199 0.00299 0.00399 99998 99986 99955 99893 33333 66666 00002 33341 34166 93333 02499 86666 0.005 0.006 0.007 0.008 0.009 0.00499 0.00599 0.00699 0.00799 0.00899 99791 99640 99428 99146 98785 66692 00064 33473 66939 00492 0.010 0.011 00. “0:: 0: 014 0.00999 0.01099 0.01199 0.01299 0.01399 98333 97781 97120 96338 95426 0.015 0.016 0.017 0.018 0.019 0.01499 0.01599 0.01699 0.01799 0.01899 0.020 0.021 0.022 0.023 0.024 I FUNCTIONS FOR RADIAN ARGUMENTS cos x 667 331 957 342 1.00000 0.99999 0.99999 0.99999 0.99999 00000 95000 80000 55000 20000 00000 00041 00666 03374 10666 00000 66666 66657 99898 66097 000 528 778 750 778 70831 79994 39150 73291 07405 783 446 327 723 100 0.99998 0.99998 0.99997 0.99996 0.99995 75000 20000 55001 80001 95002 26041 53999 00041 70666 73374 64496 93520 50326 30257 26188 529 004 542 El’9 857 34166 68008 02073 36427 71148 66468 75446 59289 42921 51241 254 684 053 659 801 0.99995 0.99993 0.99992 0.99991 0.99990 00004 95006 80008 55011 20016 16665 10039 63995 90034 00656 27778 20617 85281 96278 20901 026 059 066 551 438 94375 93173 91811 90280 88568 06328 42071 78498 15746 53967 09109 41340 72691 27852 31431 944 585 726 832 205 0.99988 0.99987 0.99985 0.99983 0.99981 75021 20027 55034 80043 95054 09359 30643 80008 73952 29976 17975 36508 14243 76107 32558 106 430 829 331 650 0.01999 0.02099 0.02199 0.02299 0.02399 86666 84565 82253 79722 76960 93333 34033 76279 20302 66354 07936 81764 77175 18277 28999 649 335 771 769 311 0.99980 0.99977 0.99975 0.99973 0.99971 00066 95081 80097 55116 20138 66577 03255 60509 59836 23734 77841 88132 19593 06320 58193 270 556 878 750 002 0.025 0.026 0.027 0.028 0.029 0.02499 0.02599 0.02699 0.02799 0.02899 73959 70707 67196 63414 59353 14712 65676 19572 76750 37589 33066 53973 14955 38952 48577 217 517 411 746 881 0.99968 0.99966 0.99963 0.99960 0.99957 75162 20190 55221 80256 95294 75702 40237 42836 09997 69215 58624 62215 92299 38394 53557 967 698 214 779 207 0.030 0.031 0.032 0.033 0.034 0.02999 0.03099 0.03199 0.03299 0.03399 55002 50350 45389 40108 34497 02495 71904 46280 26119 11951 66076 13288 11602 81908 44553 853 752 188 762 435 0.99955 0.99951 0.99948 0.99945 0.99942 00337 95384 80436 55494 20556 48987 78809 89175 11581 78521 51627 04381 38584 32936 14926 216 810 710 824 773 0.035 0.036 0.037 0.038 0.039 0.03499 0.03599 0.03699 0.03799 0.03899 28546 22245 15584 08553 01142 04336 03869 11180 26937 51841 19281 25183 80633 03228 09720 702 461 489 414 085 0.99938 0.99935 0.99931 0.99927 0.99923 75625 20699 55780 80868 95963 23488 80976 86478 76484 88487 57581 76116 24487 91840 98862 460 700 902 819 358 0.040 0.041 0.042 0.043 0.044 0.03998 0.04098 0.04198 0.04298 0.04398 93341 85141 76530 67500 58040 86634 32096 89047 58349 40905 15945 36751 85918 76078 18626 255 449 946 755 492 0.99920 0.99915 0.99911 0.99907 0.99903 01066 96177 81296 56424 21561 60977 33444 46376 41262 60588 94031 49770 58494 28564 80138 457 040 043 524 853 0.045 0.046 0.047 0.048 0.049 0.04498 0.04598 0.04698 0.04798 0.04898 48140 37790 26980 15701 03941 37660 49604 77774 23249 87159 23632 99745 54095 92191 17808 066 054 689 340 403 0.99898 0.99894 0.99889 ii99884 0.99879 76708 21865 57033 82211 97401 47842 47508 05071 67013 80818 40921 41817 12480 76767 48087 992 869 849 299 272 0.050 0.04997 91692 70678 32879 487 0.99875 02603 94966 24656 287 C-79)6 For conversion from degrees to radians see Example 13. For use and extension of the table see Examples 15-17. From C. E. Van Orstrand, Tables of the exponential functionand of thecircular sine and cosine to radian arguments,Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission). Known errors have been corrected. ELEMENTARY CIRCULAR SINES AND COSINES TRANSCENDENTAL FOR RADIAN ARGUMENTS sin 2 2 143 FUNCTIONS Table co9 4.6 5 0.050 0.051 0.052 0.053 0.054 0.04997 0.05097 0.05197 0.05297 0.05397 91692 70678 32879 487 78943 75032 37375 800 65685 01496 29184 649 51906 51396 03981 925 37598 26109 55099 505 0.99875 0.99869 0.99864 0.99859 0.99854 02603 94966 24656 287 97818 58936 84647 237 83046 23208 81242 407 58287 39259 37585 623 23542 59564 41634 531 0.055 0.056 0.057 0.058 0.059 0.05497 0.05597 0.05696 0.05796 0.05896 22750 27067 73387 446 07352 55755 47070 891 91395 13712 61601 567 74868 02534 99503 794 57761 23875 40214 896 0.99848 0.99843 0.99837 0.99831 0.99826 78812 37598 40913 005 24097 27834 37163 704 59397 85743 80900 770 84714 67796 65862 676 00048 31461 23365 235 0.060 0.061 0.062 0.063 0.064 0.05996 0.06096 0.06196 0.06295 0.06395 40064 79444 59919 909 21768 71012 31380 500 02863 00408 23757 982 83337 69523 02430 343 63182 80309 28803 166 0.99820 0.99814 0.99807 0.99801 0.99795 05399 35204 16554 766 00768 38490 34561 437 86156 01782 86552 769 61562 86542 95687 334 26989 55229 92968 628 0.065 0.066 0.067 0.068 0.069 0.06495 0.06595 0.06694 0.06794 0.06894 42388 34782 60114 361 20944 35022 49232 601 98840 83173 44449 361 76067 81445 89264 458 52615 32117 22165 004 0.99788 0.99782 0.99775 0.99768 0.99762 82436 71301 10999 144 27904 99211 77634 635 63395 04415 09538 592 88907 53362 05636 926 04443 13501 40472 866 0.070 0.071 0.072 0.073 0.074 0.06994 0.07094 0.07193 0.07293 0.07393 28473 37532 76397 655 03632 00106 79734 071 78081 22323 54229 480 51811 06738 15974 250 24811 55977 74838 360 0.99755 0.99748 0.99740 0.99733 0.99726 10002 53279 57462 091 05586 42140 62048 084 91195 50526 14757 726 66830 49875 24157 139 32492 12624 39707 777 0.075 0.076 0.077 0.078 0.079 0.07492 0.07592 0.07692 0.07792 0.07891 97072 72742 34208 684 68584 59805 90718 980 39337 20017 33972 485 09320 56301 46257 015 78524 71660 02252 478 0.99718 0.99711 0.99703 0.99695 0.99688 88181 12207 44522 774 33898 23055 48023 568 69644 20596 78496 785 95419 81256 75551 417 11225 82457 82476 279 0.080 0.081 0.082 0.083 0.084 0.07991 0.08091 0.08190 0.08290 0.08390 46939 69172 68730 688 14555 51998 04247 389 81362 23374 58826 394 47349 86621 73635 718 12508 45140 80655 638 0.99680 0.99672 0.99663 0.99655 0.99647 17063 02619 38497 771 12932 21157 70937 933 98834 18485 87272 823 74769 76013 67091 212 40739 76147 53953 598 0.085 0.086 0.087 0.088 0.089 0.08489 0.08589 0.08689 0.08788 0.08888 76828 02416 02338 544 40298 62015 51260 514 02910 27592 29764 492 64653 02885 29594 973 25516 91720 31524 112 0.99638 0.99630 0.99621 0.99613 0.99604 96745 02290 47151 570 42786 38841 93367 506 78864 71197 78234 626 04980 85750 17797 412 21135 69887 49872 388 0.090 0.091 0.092 0.093 0.094 0.08987 0.09087 0.09187 0.09286 0.09386 85491 98011 04969 125 44568 25760 07600 919 02735 79059 84943 819 59984 62093 69966 323 16304 79136 82662 751 0.99595 0.99586 0.99577 0.99567 0.99558 27330 11994 25309 284 23565 01450 99152 586 09841 28634 21703 483 86159 84916 29482 217 52521 62665 36090 844 0.095 0.096 0.097 0.098 0.099 0.09485 0.09585 0.09684 0.09784 0.09883 71686 34557 29625 724 26119 32817 03609 347 79593 78472 83083 006 32099 76177 31775 683 83627 30679 98210 683 0.99549 0.99539 0.99529 0.99520 0.99510 08927 55245 22976 426 55378 57015 30094 649 91875 63330 46473 881 18419 70541 00679 686 35011 75992 51179 796 0.100 0.09983 34166 46828 15230 681 0.99500 41652 78025 76609 556 144 ELEMENTARY TRANSCENDENTAL FUNCTIONS Table 4.6 CIRCULAR SINES AND COSINES FOR RADIAN cos x sin x X ARGUMENTS 0.100 0.101 0.102 0.103 0.104 0.09983 0:10082 0.10182 0;10281 0.10381 34166 46828 15230 681 83707 29567 99512 975 32239 83945 51074 864 79754 15107 52769 040 26240 28302 69768 897 0.99500 0.99490 0.99480 0.99470 0.99459 41652 78025 76609 556 38343 75976 65937 840 25085 70176 08533 469 01879 61949 84132 117 68726 53618 52703 737 0.105 0.106 0.107 0.108 0.109 0.10480 0.10580 0.10679 0.10779 0.10878 71688 28882 49043 655 16088 22302 18823 209 59430 14121 88052 588 01704 10007 45835 941 42900 15731 60869 939 0.99449 0.99438 0.99428 0.99417 0.99406 25627 48497 44220 501 72583 50896 48325 268 09595 66120 03900 596 36665 00466 88538 307 53792 61230 07909 607 0.110 0.111 0.112 0.113 0.114 0.10977 0.11077 0.11176 0.11275 0.11375 83008 37174 80866 495 22018 80326 31964 714 59921 51285 18131 952 96706 56261 20553 909 32364 01575 97013 636 0.99395 0.99384 0.99373 0.99362 0.99350 60979 56696 85035 784 58226 96148 49459 483 45535 89860 26316 578 22907 49101 25308 652 90342 86134 29576 080 0.115 0.116 0.117 0.118 0.119 0.11474 0.11574 0.11673 0.11772 0.11871 66883 93663 81259 372 00256 39072 82361 097 32471 44465 84055 722 63519 16621 44080 790 93389 62434 93496 613 0.99339 0.99327 0.99316 0.99304 0.99292 47843 14215 84471 755 95409 47595 86235 439 33043 01517 70568 768 60744 92218 01110 921 78516 36926 57814 950 0.120 0.121 0.122 0.123 0.124 0.11971 0.12070 0.12169 0.12269 0.12368 22072 88919 35996 735 49559 03206 47206 615 75838 12547 73970 447 00900 24315 33626 003 24735 46003 13267 407 0.99280 0.99268 0.99256 0.99244 0.99232 86358 53866 25224 810 84272 62252 80653 067 72259 82294 82259 329 50321 35193 57029 382 18458 43142 88655 070 0.125 0.126 0.127 0.128 0.129 0.12467 0.12566 0.12665 0.12765 0.12864 47333 85227 68995 744 68685 49729 25157 389 88780 47372 73569 978 07608 86148 72735 909 25160 74174 47043 273 0.99219 76672 29329 05314 910 Oi99207 24964 17930 67355 462 0.99194 63335 34118 54873 474 0.99181 91787 04055 55198 803 0.99169 10320 54896 50278 123 0.130 0.131 0.132 0.133 0.134 0.12963 0.13062 0.13161 0.13260 0.13359 41426 19694 85954 121 56395 31083 43179 968 70058 16843 35844 433 82404 85608 43632 907 93425 46144 07929 171 0.99156 0.99143 0.99130 0.99116 0.99103 18937 14788 03959 451 17638 12868 49177 481 06424 79267 75039 751 85298 45107 13813 659 54260 42499 27814 325 0.135 0.136 0.137 0.138 0.139 0.13459 0.13558 0.13657 0.13756 0.13855 03110 07348 30938 844 11448 78252 74799 575 18431 68023 60677 867 24048 85962 67852 453 28290 41508 32784 107 0.99090 0.99076 0.99063 0.99049 0.99035 13312 04547 96193 339 62454 65348 01628 375 01689 59985 16913 714 31018 24535 91451 667 50441 96067 37644 937 0.140 0.141 0.142 0.143 0.144 0.13954 0.14053 0.14152 0.14251 0.14350 31146 44236 48171 799 32607 03861 61995 092 32662 30237 76542 691 31302 33359 47427 025 28517 23362 82584 791 0.99021 0.99007 0.98993 0.98979 0.98964 59962 12637 17189 895 59580 13293 27270 829 49297 38073 86655 145 29115 28007 21689 546 99035 25111 52197 214 0.145 0.146 0.147 0.148 0.149 0.14449 0.14548 0.14647 0.14746 0.14844 24297 10526 41263 332 18632 05272 32992 773 11512 18167 16543 800 02927 59922 98870 997 92868 41398 34041 627 0.98950 0.98936 0.98921 0.98906 0.98892 59058 72394 77275 984 09187 13854 60997 551 49421 94478 18007 704 79764 60241 99027 617 00216 58111 76256 193 0.150 0.14943 81324 73599 22149 773 p-p1 L ’ J 0.98877 10779 36042 28673 498 c1 c -y ELEMENTARY ~IR~UIAR SINES TRANSCENDENTAL FUNCTIONS AND COSINES FOR RADIAN ARGUMENTS Table 4.6 cos x sin z X 145 0.150 0.151 0.152 0.153 0.154 0.14943 0.15042 Oil5141 0.15240 0.15339 81324 68286 53744 37687 20107 73599 67680 34944 86847 34994 22149 08215 81070 72225 54727 773 725 532 604 267 0.98877 0.98862 0.98847 0.98831 0.98816 10779 36042 11454.42977 02243 28849 83147 44579 54168 42076 28673 27245 20028 17178 75856 498 283 611 614 382 0.155 0.156 0.157 0.158 0.159 0.15438 0.15536 0.15635 0.15734 0.15833 00992 80334 58122 34347 08998 91143 67205 75247 27490 36311 41996 86651 79319 47428 53983 190 555 902 529 354 0.98801 0.98785 0.98770 0.98754 0.98738 15307 66566 07947 39451 61079 74239 94954 59094 22522 42087 85038 50224 78054 60814 60855 006 794 663 736 150 0.160 0.161 0.162 0.163 0.164 0.15931 0.16030 0.16129 0.16227 0.16326 82066 53540 23412 91670 58306 14245 73987 28387 90460 73379 96331 04906 41960 00278 01876 146 020 095 226 705 0.98722 0.98706 0.98690 0.98674 0.98658 72833 74715 66727 48869 21144 75626 81965 20914 53272 40826 94904 18284 09029 51905 22328 095 099 574 638 234 0.165 0.166 0.167 0.168 0.169 0.16425 0.16523 0.16622 0.16721 0.16819 23309 86670 48378 08424 66798 90480 55265 81396 82704 73183 96685 61216 97208 30268 08481 825 228 916 843 981 0.98641 0.98625 0.98608 0.98592 0.98575 83553 36098 78780 11602 34564 46347 33596 67316 13241 38088 70185 03560 72356 51818 25966 554 791 233 712 434 0.170 0.171 0.172 0.173 0.174 0.16918 0.17016 0.17115 0.17213 0.17312 23490 78490 31789 83376 33241 66996 78473 22117 12595 64750 01015 96702 02607 42577 55773 762 805 812 560 865 0.98558 0.98541 0.98524 0.98507 0.98490 47669 50917 44312 27854 01546 09560 96348 68126 95555 50280 70917 38117 37476 20391 62691 193 998 124 598 158 0.175 0.176 0.177 0.178 0.179 0.17410 0.17509 0.17607 0.17706 0.17804 81375 27769 72411 15292 56403 93595 14318 42278 93011 82230 95189 26146 24778 76492 74417 433 505 176 317 975 0.98472 0.98455 0.98437 0.98419 0.98402 65389 19384 63534 97840 22304 04933 33129 09469 09537 09903 47463 47797 09416 33225 57745 670 052 699 443 046 0.180 0.181 0.182 0.183 0.184 0.17902~95734 0.180Dl 33274 0.18099 69014 0.18198 02944 0.18296 35054 25824 39859 40581 44417 67974 17834 10581 59452 72574 57756 180 029 980 233 116 0.98384 0.98366 0.98348 0.98330 0.98311 36927 41713 36661 21775 97056 88121 22728 93246 80179 65017 41459 45058 13586 58485 39552 272 522 083 974 448 0.185 0.186 0.187 0.188 0.;89 0.18394 0.18492 0,18591 0.18689 0.18787 65335 93776 20368 45100 67964 28041 41589 25775 97940 75611 20836 64000 84083 70855 05288 370 231 224 554 013 0.98293 0.98275 0.98256 0.98237 0.98219 62506 18126 63919 99886 26029 30231 59276 36591 47595 78693 46781 82121 41132 94537 69683 122 799 959 971 022 0.190 0.191 0.192 0.193 0.194 0.18885 0.18984 0.19082 0.19180 0.19278 88949 08046 25244 40533 53905 76500 18510 19732 98445 73120 57799 86484 35325 32380 87958 285 571 424 691 485 0.98200 0.98181 0.98162 0.98143 0.98124 42351 48852 45535 32402 09455 17270 51693 71313 66461 28451 31896 65751 56228 69777 35290 788 875 034 178 214 0.195 0.196 0.197 0.198 0.199 0.19376 0.19474 0.19572 0.19670 Oil9768 65349 74855 82414 88016 91650 62421 85204 60517 07604 45907 92769 16058 03723 76404 27565 058 510 204 820 917 0.98104 0.98085 0.98065 0.98046 0.98026 76695 34125 81746 19561 47571 49577 23115 43322 05437 05677 24965 35080 66661 06062 05434 723 479 867 170 796 0.200 0.19866 93307 95061 21545 941 c-y c1 0.98006 65778 41241 63112 420 c-y 1 146 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN ARGUMENTS sin x cos x 0.200 0.201 0.202 0.203 0.204 0.19866 93307 95061 21545 941 0.19964 92978 74900 91597 545 OI20062 90653 05459 37903 151 0.20160 86321 06969 25571 640 0.20258 79972 99863 82615 083 0.98006 65778 41241 63112 420 0.97986 74185 10310 03887 090 0.97966 72793 12041 59192 306 0.97946 61604 46575 47187 084 OI97926 40621 15030 52742 047 0.205 0.206 0.207 0.208 0.209 0.20356 0.20454 0.20552 0.20650 0.20748 71599 04777 97905 397 61189 42549 19110 856 48734 34218 50612 330 34224 01031 51399 175 17648 64439 32944 665 0.97906 0.97885 0.97865 0.97844 0.97823 09845 19505 07327 536 69278 63076 68803 784 18923 49802 01113 156 58781 84716 53874 491 88855 73834 41879 553 0.210 0.211 0.212 0.213 0.214 0.20845 0.20943 0.21041 0.21139 0.21237 98998 46099 57060 871 78263 67877 33732 895 55434 51846 18932 346 30501 20289 12409 982 03453 95699 55467 398 0.97803 0.97782 0.97761 0.97740 0.97718 09147 24148 24491 614 19658 43628 84946 201 20391 41225 09554 014 11348 26863 66806 039 92531 11448 86380 882 0.215 0.216 0.217 0.218 0.219 0.21334 0.21432 0.21530 0.21627 0.21725 74283 00782 28707 677 42978 58454 49764 905 09530 91846 71012 439 73930 24303 77249 851 36166 79385 83368 434 0.97697 0.97676 0.97654 0.97633 0.97611 63942 06862 38054 344 25583 25963 10511 247 77456 82586 90059 555 19564 91546 39246 782 51909 68630 75378 736 0.220 0.221 0.222 0.223 0.224 0.21822 0.21920 0.22018 0.22115 0.22213 96230 80869 31995 179 54112 52747 91115 124 09802 19233 51671 977 63290 04757 25146 920 i4566 33970 41115 484 0.97589 0.97567 0.97545 0.97523 0.97501 74493 30605 48940 602 87317 95212 21920 392 90385 81168 46034 788 83699 08167 40857 388 67259 96877 71849 392 0.225 0.226 0.227 0.228 0.229 0.22310 0.22408 0.22505 0.22602 0.22700 63621 31745 44782 417 10445 23176 94494 428 55028 33582 59230 720 97360 88504 16071 214 37433 13708 47642 363 0.97479 0.97457 0.97434 0.97412 0.97389 41070 68943 28292 737 05133 46983 01125 708 59450 54590 60681 052 04024 16334 34326 607 38856 57756 84008 477 0.230 0.231 0.232 0.233 0.234 0.22797 0.22895 0.22992 0.23089 0.23187 75235 35188 39540 462 10757 79163 77732 354 43990 72082 45933 437 74924 40621 22962 869 03549 11686 80075 884 0.97366 0.97343 0.97320 0.97297 0.97274 63950 0.537483696 773 79306 86678 96733 940 84929 30133 53085 695 80819 65176 26494 602 66980 22218 11536 294 0.235 0.236 0.237 0.238 0.239 0.23284 0.23381 0.23478 0.23575 0.23673 29855 12416 78273 112 53832 70180 65586 809 75472 12580 74343 904 94763 67453 18405 752 11697 62868 90384 520 0.97251 0.97228 0.97204 0.97181 0.97157 43413 32643 00578 389 10121 28807 60642 091 67106 44041 10166 529 14371 12644 95675 843 51917 69892 68349 034 0.240 0.241 0.242 0.243 0.244 0.23770 0.23867 0.23964 0.24061 0.24158 26264 27134 58836 079 38453 88793 65429 334 48256 76627 22091 869 55663 19655 08131 828 60663 47136 67335 933 0.97133 0.97109 0.97086 0.97062 0.97037 79748 52029 60492 618 97865 96272 61916 095 06272 40809 96210 262 04970 24800 96928 391 93961 88375 83670 294 0.245 0.246 0.247 0.248 0.249 0.24255 0.24352 0.24449 0.24546 0.24643 63247 88572 05043 522 63406 73702 85196 546 61130 32513 27365 389 56408 95231 03750 445 49232 92328 36159 337 0.97013 0.96989 0.96965 0.96940 0.96915 73249 72635 38069 313 42836 19650 79682 233 02723 72463 41782 166 52914 75084 47054 425 93411 72494 83195 397 0.250 0.24740 39592 54522 92959 685 c-:)3 2 [1 0.96891 24217 10644 78414 459 c 1 -y1 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL .4ND COSINES FOR RADIAN ARGUMENTS Table 4.6 cos x sin x X 147 FUNCTIONS 0.250 0.251 0.252 0.253 0.254 0.24740 0.24837 0.24934 0.25030 0.25127 39592 54522 92959 685 27478 12778 86007 332 12879 98307 67549 922 95788 42569 27105 742 76193 77272 88317 722 0.96891 0.96866 0.96841 0.96816 0.96791 24217 10644 78414 459 45333 36453 76838 955 56762 97810 13822 250 58508 43570 91154 897 50572 23561 52178 941 0.255 0.256 0.257 0.258 0.259 0.25224 0.25321 0.25418 0.25514 0.25611 54086 34378 05782 506 29456 46095 61854 486 02294 44888 63424 714 72590 63473 38674 587 40335 34820 33804 209 0.96766 0.96741 0.96715 0.96690 0.96664 32956 88575 56805 375 05664 90374 56434 780 68698 81687 68781 180 22061 16211 52599 126 65754 48609 82314 035 0.260 0.261 0.262 0.263 0.264 0.25708 0.25804 0.25901 0.25997 0.26094 05518 92155 09735 339 68131 68959 38788 820 28163 98972 01336 401 85606 16189 82426 844 40448 54868 68386 239 0.96638 0.96613 0.96587 0.96561 0.96535 99781 34513 22555 822 24144 30519 02595 835 38845 94190 90687 131 43888 84058 68308 107 39275 59618 04309 520 0.265 0.266 0.267 0.268 0.269 0.26190 0.26287 0.26383 0.26480 0.26576 92681 49524 43392 399 42295 34933 86023 278 89280 46135 65779 278 33627 18431 39579 372 75325 87386 48230 942 0.96509 0.96483 0.96456 0.96430 0.96403 25008 81330 28964 923 01091 10622 07924 537 67525 09885 16072 584 24313 42476 11288 118 71458 72716 08109 368 0.270 0.271 0.272 0.273 0.274 0.26673 0.26769 0.26865 0.26962 0.27058 14366 88831 12873 229 50740 58861 31394 301 84437 33839 74821 451 15447 50396 83684 915 43761 45431 64354 828 0.96377 0.96350 0.96323 0.96296 0.96269 08963 65890 51301 623 36830 88248 89328 696 55063 07004 47727 972 63662 90334 02389 084 62633 07377 52736 246 0.275 0.276 0.277 0.278 0.279 0.27154 0.27250 0.27347 0.27443 0.27539 69369 56112 85351 302 92262 19879 73627 557 12429 74443 10825 981 29862 57786 29507 043 44551 08166 09350 952 0.96242 0.96215 0.96188 0.96160 0.96133 51976 28237 94814 248 31695 23980 94278 169 01792 66634 59286 807 62271 29189 13299 879 13133 85596 67778 997 0.280 0.281 0.282 0.283 0.284 0.27635 0.27731 0.27827 0.27923 0.28019 56485 64113 73331 967 65656 64435 83865 270 72054 48215 38926 293 75669 54812 68142 411 76492 23866 28856 909 0.96105 54383 10770 94792 459 0.96077 86021 80586 99523 878 0.96050 08052 71880 92684 682 0.96022 20478 62449 62830 504 Oi95994 23302 31050 48581 495 0.285 0.286 0.287 0.288 0.289 0.28115 0.28211 0.28307 0.28403 0.28499 74512 95294 02165 110 69722 09293 88922 591 62110 06345 05725 374 51667 27208 80861 997 38384 12929 50237 384 0.95966 0.95938 0.95909 0.95881 0.95852 16526 57401 10746 590 00154 22179 04351 746 74188 07021 50572 193 38630 94525 08568 713 93485 68245 47227 984 0.290 0.291 0.292 0.293 0.294 0.28595 0.28691 0.28786 0.28882 0.28978 22251 04835 53268 394 03258 44540 28750 981 81396 73943 10698 841 56656 35230 24153 475 29027 70875 80965 551 0.95824 0.95795 0.95767 0.95738 0.95709 38755 12697 16807 013 74442 13353 20481 688 00549 56644 85799 478 17080 29961 36036 308 24037 21649 61457 636 0.295 0.296 0.297 0.298 0.299 0.29073 Oi29169 0.29265 Oi29360 0.29456 98501 23642 75547 489 65067 36583 80597 155 28716 53042 42792 582 89439 16653 78457 616 47225 71345 69198 389 0.95680 21423 21013 90483 768 Oi95651 09241 18315 60759 429 0.95621 87494 04772 90127 632 Oi95592 56184 72560 47507 858 0.95563 15316 14809 23678 590 0.300 0.29552 02066 61339 57510 532 0.95533 64891 25606 01964 231 [C-f)4 1 [ (-;)I 1 148 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN cos x sin x X ARGUMENTS 0.300 0.301 0.302 0.303 0.304 0.29552 0.29647 0.29743 0.29838 0.29933 02066 61339 57510 532 53952 31151 42357 025 02873 25592 74716 586 48819 89771 53102 518 91782 69093 19051 897 0.95533 0.95504 0.95474 0.95444 0.95414 64891 25606 01964 231 04912 99993 28826 414 35384 33968 84359 763 56308 24485 52692 116 67687 69450 92289 242 0.305 0.306 0.307 0.308 0.309 0.30029 0.30124 0.30220 0.30315 0.30410 31752 09261 52585 026 68718 56279 67635 045 02672 56451 07447 613 33604 56380 39950 549 61505 02974 53093 365 0.95384 0.95354 0.95324 0.95294 0.95263 69525 67727 06164 084 61825 19130 11990 559 44589 24430 12121 945 17820 85350 63513 878 81523 04568 47552 001 0.310 0.311 0.312 0.313 0.314 0.30505 0.30601 0.30696 0.30791 0.30886 86364 43443 50156 564 08173 25301 45030 632 26921 96367 57464 615 42601 04767 08284 189 55200 98932 14579 138 0.95233 0.95202 0.95172 0.95141 0.95110 35698 85713 39784 281 80351 33367 79558 038 15483 53066 39561 711 41098 51295 95271 383 57199 35494 94302 111 0.315 0.316 0.317 0.318 0.319 0.30981 i:j1076 0.31171 0.31266 0.31361 64712 27602 84860 120 71125 39828 14184 658 74430 84966 79252 234 74619 12688 33468 402 71680 72974 01977 833 0.95079 0.95048 0.95017 0.94986 0.94954 63789 14053 25664 080 60870 96311 88923 617 48447 92562 63269 094 26523 14047 76481 749 95099 72959 73811 467 0.320 0.321 0.322 0.323 0.324 0.31456 0.31551 0.31646 0.31741 0.31836 65606 16117 76666 176 56385 92727 11130 659 44010 53724 15619 332 28470 50346 51938 844 09756 34148 28330 674 0.94923 0.94892 0.94860 0.94828 0.94796 54180 82440 86757 531 03769 56583 01754 395 43869 10427 28762 501 74482 59963 69764 173 95613 22130 87164 613 0.325 0.326 0.327 0.328 0.329 0.31930 0.32025 0.32120 0.32215 0.32309 87858 57000 94315 718 62767 71094 35507 128 34474 28937 68391 319 02968 83360 35077 048 68241 87512 98012 460 0.94765 0.94733 0.94701 0.94668 0.94636 07264 14815 72098 048 09438 56853 12639 034 02139 68025 61918 976 85370 69063 06147 877 59134 81642 32541 351 0.330 0.331 0.332 0.333 0.334 0.32404 0.32498 0.32593 0.32687 Oij2782 30283 94868 34670 020 89085 59222 32199 224 44637 34694 82047 011 96929 75730 74545 756 45953 37100 93468 777 0.94604 0.94571 0.94539 0.94506 0.94473 23435 28386 97152 941 78275 32866 92611 768 23658 19598 15765 535 59587 14042 35228 939 86065 42606 58837 502 0.335 0.336 0.337 0.338 0.339 0.32876 0.32971 0.33065 0.33160 0.33254 91698 73903 10553 241 34156 41562 79990 386 73316 95834 32882 957 09170 92801 71669 766 41708 88879 64517 288 0.94441 0.94408 0.94375 0.94341 0.94308 03096 32643 01006 864 10683 12448 49997 577 08829 11264 35085 413 97537 59275 93637 243 76811 87612 38092 499 0.340 0.341 0.342 0.343 0.344 0.33348 0.33442 0.33537 0.33631 0.33725 70921 40814 39678 177 96799 05684 79816 635 19332 40903 16300 519 38512 04216 23460 104 54328 53706 12813 399 0.94275 0.94242 0.94208 0.94174 0.94141 46655 28346 22850 264 07071 14493 11062 025 58062 80011 41330 105 99633 59801 94311 834 31786 89707 59229 468 0.345 0.346 0.347 0.348 0.349 0.33819 0.33913 0.34007 0.34101 0.34195 66772 47791 27257 928 75834 45227 35228 880 81505 05108 24823 531 83774 86866 97891 850 82634 50276 64093 188 0.94107 0.94073 0.94039 0.94005 0.93971 54526 06513 00285 905 67854 47944 22986 218 71775 52668 40365 059 66292 60293 39119 944 51409 11367 45650 473 0.350 0.34289 78074 55451 34918 963 0.93937 27128 47378 92003 503 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL AND COSINES FUNCTIONS FOR RADIAN 149 ARGUMENTS Table 4.6 cos x sin z 0.34289 78074 55451 34918 963 0.34383 70085 62847 17681 237 0.93937 0.93902 93454 0.34477 0.34571 0.34665 58658 0.93799 36103 03447 99266 461 0.355 0.356 0.357 0.358 0.359 0.34759 0.34852 0.34946 0.35040 03652 78377 49617 17363 0.93764 0.93729 0.93694 0.93659 0.93624 64888 84296 94332 19349 88839 59978 27409 87337 89202 412 915 418 86299 04124 73578 312 0.360 0.361 0.362 0.363 0.364 0.35227 0.35320 0.35414 0.35508 0.35601 42332 99538 53211 03343 49923 75089 05683 26351 01729 0.365 0.366 0.367 0.368 0.369 0.35694 0.35788 92944 32396 76911 07756 39203 863 0.35975 0.36068 00552 29239 86229 67042 93504 91160 0.36161 0.36254 0.36347 54319 75783 93621 64961 47479 82448 25290 28534 86047 16625 0.350 0.351 0.352 0.353 0.354 33263 09467 102 43783 27841 91058 778 25451 08071 20819 319 35784 73161 82729 27364 28543 09276 17091 852 237 064 58840 a91 0.35133 al604 70292 87868 632 97684 15610 96384 15734 991 866 608 065 96801 63913 294 27128 47378 92003 503 10755 al724 321 0.93868 50389 44865 55613 a41 0.93833 97937 94014 57391 a69 94998 al762 72980 716 0.93589 68236 77934 85835 091 40815 54999 29438 322 0.93519 04038 88060 13742 042 0.93483 57910 30795 02492 a55 0.93448 02433 37816 78462 165 0.93554 37611 63448 79948 354 721 0.93412 0.93376 0.93340 0.93304 a7113 33740 a7371 606 0.93268 84948 14096 19348 a71 97803 21412 31502 729 373 al3 0.36441 07825 38085 52343 006 0.36534 la384 a2970 56131 067 0.93232 0.93196 0.93160 73456 52640 22505 06034 70704 42320 74082 381 737 70188 65151 560 0.93123 a3054 67499 62553 347 0.93087 34291 26582 73524 125 0.375 0.376 0.377 0.378 0.379 0.36627 0.36720 56137 99809 291 733 0.93050 0.93014 0.92977 0.92940 0.92903 76219 12314 29114 948 0.380 0.37092 04694 12982 67184 549 0.37184 89484 33562 49909 aal 0.92866 46355 76510 24949 253 i* 33:: 0: 383 0.37277 0.37370 0.37463 70556 0.92792 0.92754 09378 03592 76777 471 0.385 0.386 0.37555 0.37648 0.37741 91367 57472 0.390 0.391 0.392 0.393 0.394 0.38018 0.38111 0.38203 0.38296 a4151 23161 42823 118 31339 34675 51860 671 74716 33087 43373 349 14272 94059 55222 774 0.38388 49999 0.395 0.396 0.370 0.371 0.372 0.373 0.374 0.384 41380 647 0.35881 68268 55391 65142 021 0.36813 28105 44381 61843 251 0.36906 0.36999 23995 16194 39357 71964 05224 37211 34164 88020 926 758 096 47899 99862 72083 la4 21506 a9741 70366 479 21610 27762 089 945 07984 05404 97425 897 739 922 08841 90501 47704 265 32163 27881 98417 211 46186 92123 64451 a36 50916 51824 06312 328 0.92829 32508 36638 24806 a58 76968 49686 81063 030 0.92717 35283 48161 29943 792 84326 73184 70454 235 0.92679 0.92642 0.92604 0.92566 24101 54613 99852 0.92490 0.92452 0.92414 0.92376 90598 84090 68337 43342 57313 51063 04145 22192 068 776 35142 86457 070 0.92299 0.92261 0.92222 65643 12947 51025 63117 40199 06064 25225 97879 84939 693 040 589 it ;z; 0:399 0.38480 ala88 08245 02477 888 0.38573 09928 14697 01854 707 0.38665 34110 90188 34186 658 0.38757 54427 12300 79611 426 0.38849 70867 59002 a3601 363 0.400 0.38941 a3423 08650 49166 631 0.92106 09940 02885 08279 a53 2 ;:i 01389 47501 46155 64673 67846 04751 19812 59093 46681 397 0.37833 78378 60081 a4790 240 0.37926 33161 23263 89706 110 93636 r(-_sbq L '_I 29011 366 66966 223 04187 63679 438 75863 63138 47019 143 oI9252a 87857 54580 07941 297 16481 39537 314 0.92338 09109 a9547 07898 401 0.92183 79880 46904 06602 584 0.92144 99517 49832 05558 150 cC-f)’ 1 150 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL FUNCTIONS SIR’ES AND COSINES FOR RADIAN cos x sin x X ARGUIMENTS 0.400 0.401 0.402 0.403 0.404 0.38941 0.39033 0.39125 0.39217 0.39309 83423 92084 96842 97687 94611 08650 39988 32150 64660 17434 49166 29019 17700 43663 631 595 358 363 61324 955 0.92028 03157 16118 16919 248 0.91988 85959 56976 45007 979 0.91949 59563 09315 43137 110 0.405 0.406 0.407 0.408 0.409 0.39401 0;j9493 0.39585 0.39677 0.39769 87603 76656 61759 42903 20080 70780 05398 02384 43226 09812 43071 820 0.91910 29995 97324 36782 816 356 508 0.91870 0.91831 0.91791 0.91751 0.410 0.411 0.412 0.413 0.414 0.39860 0.39952 93279 62493 84422 49738 89359 65238 380 251 71230 202 0.40044 27711 88838 35528 558 0.40135 88925 85200 23958 010 0.40227 46126 22702 98524 766 0.40318 99303 85626 63109 550 0.40410 48449 58653 49047 645 0.40501 93554 26869 06660 654 0.415 0.416 0.417 0.418 0.419 0.40593 0.40684 34608 71603 75762 91229 96747 82037 0.420 0.421 0.422 0.423 0.424 0.40776 0.40867 0.40958 0.41049 0.41140 04530 33379 58142 78808 95370 59570 67491 02108 50946 01936 0.425 0.426 0.421 0.428 0.429 0.41232 0.41323 07817 43424 16141 64165 31825 593 0.41414 20333 53326 15081 889 0.41505 20384 00488 14189 067 0.41596 16283 95646 32014 0.430 0.431 0.432 0.433 0.434 0.41687 0.41777 0.41868 0.41959 0.42050 08024 95595 78989 58196 33207 29210 92007 75279 70687 70310 0.435 0.436 0.42141 0,42231 0.42322 04013 70605 32974 ii* t;; 0:439 0.42412 0.92106 0.92067 0.91712 0.91672 0.91632 0.91592 0.91551 09940 11151 02885 95020 08279 86221 853 075 23971 79189 25219 62067 89735 65774 19913 66209 00060 17781 72800 45073 81253 73416 44815 745 295 568 956 737 08228 17549 16605 94682 10547 37232 03796 61266 92832 564 150 202 649 194 17704 51081 08695 90527 85785 99696 0.91511 63204 94631 73753 232 939 655 0.91471 0.91430 0.91390 0.91349 26730 81109 26344 62441 73322 39416 97475 52975 31180 03880 01872 65972 180 251 722 725 18597 47203 84671 15143 81337 279 546 703 980 201 0.91308 0.91268 0.91227 0.91186 0.91145 89403 07233 15937 15518 05981 12308 82776 12591 90901 47728 27243 66357 72866 04379 45647 609 915 996 332 576 75749 435 301 0.91103 0[91062 0.91021 Oi90979 0.90938 87329 59567 22698 76727 21659 54033 21681 63449 93022 24998 67564 86066 20950 54591 90577 373 990 808 701 360 76621 52231 50136 39579 58584 692 243 257 028 774 0.90896 0.90854 0.90813 0.90771 0.90729 57496 84244 01906 10488 09991 74885 59097 94960 00709 95484 12247 41143 95366 47844 84510 591 638 563 729 435 0.42503 66648 52619 21565 91110 67248 45005 83856 04753 26011 11315 81323 79016 684 018 146 456 027 0.90687 Oi90644 0.90602 0.90560 0.90517 00422 81785 54083 17320 71502 99336 33221 19003 79452 38245 62385 67577 73181 97096 59741 731 465 601 848 647 0.440 0.441 0.442 0.443 0.444 0.42593 0.42684 0.42774 0.42865 0.42955 94650 40036 81153 17992 50545 65999 08712 07458 58123 57025 60276 84433 04751 972 381 750 59317 145 0.90475 0.90432 0.90389 0.90346 0.90304 16632 52714 79753 97753 06718 19963 50093 55027 62061 99394 41716 41286 31889 19473 99766 554 061 904 892 305 0.445 0.446 0.447 0.448 0.449 0.43045 0.43136 0.43226 0.43316 0.43406 78803 02755 22395 00908 86947 12746 83666 65073 82453 443 141 917 48696 76203 11100 244 0.90261 0.90217 0.90174 0.90131 0.90088 06653 97562 79449 52319 16175 96132 82279 88745 47341 90780 15457 13291 01061 04523 24210 899 573 718 319 832 0.450 0.43496 55341 11230 21042 084 0.90044 71023 58891 823 37711 76342 50745 219 (-y [1 1 52676 92166 884 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL AND COSINES x 151 FUNCTIONS FOR RADIAN ARGCMENTS Table 4.6 cos x sin x 0.450 0.451 0.452 0.453 0.454 0.43496 0.43586 0.43676 0;43766 0.43856 55341 57635 55571 49140 38331 11230 80759 84561 22842 96246 21042 44573 42243 61170 25020 084 567 681 507 568 0.90044 0.90001 0.89957 Oi89913 0.89870 71023 16866 53709 81556 00412 52676 67546 70803 98765 88646 92166 28580 98337 67474 59552 884 847 319 569 965 0.455 0.456 0.457 0.458 0.459 0.43946 0.44036 0.44125 0.44215 0.44305 23138 03549 79557 51152 18326 05853 53183 40194 69287 43301 23944 04468 59344 17350 33053 492 918 542 215 008 0.89826 0.89782 0.89738 0.89693 0.89649 10281 11168 03076 86010 59975 78561 07522 15441 43127 32287 11933 31966 53089 90836 98759 463 167 030 721 714 0.460 0.461 0.462 0.463 0.464 0.44394 0.44484 0.44573 0.44663 0.44752 81069 39373 93228 42626 87558 65519 39668 69916 60878 17615 76524 23010 42563 89618 92537 151 752 218 275 506 0.89605 0.89560 0.89516 0.89471 0.89426 24975 81014 28097 66229 95414 25525 66339 99127 69179 22683 24253 64298 21110 57699 53342 639 937 867 908 602 0.465 0.466 0.467 0.468 0.469 0.44842 0.44931 0.45020 0745110 0.45199 28014 63986 95465 22442 44907 45634 50888 39782 19166 96343 43101 85958 08029 27868 84976 319 244 479 603 342 0.89382 Oi89337 0.89292 Oi89247 0.89202 15656 26959 29329 22770 07286 06720 69266 59190 26256 21120 58962 52423 93730 80142 01196 873 883 459 134 857 0.470 0.471 0.472 0.473 0.474 0.45288 0.45377 0.45466 0.45555 0.45644 62853 76270 85149 89482 89259 79068 75545 94432 44843 36343 29070 09309 63474 07100 22566 327 736 735 635 671 0.89156 0.89111 0.89066 0.89020 0.88974 82881 49562 07330 56193 96153 95328 01323 92437 22891 47800 93645 96296 04773 26178 33674 402 541 005 292 367 0.475 0.476 0.477 0.478 0.479 0.45733 0.45822 0.45911 0.46000 0.46089 84471 75110 61167 42633 19498 78955 83158 59888 20540 76967 48139 66969 96047 75103 55473 307 994 279 180 739 0.88929 0.88883 0.88837 0.88791 0.88745 27216 49386 62667 67065 62583 23168 05888 53744 25407 80438 20970 56721 38842 48723 05369 288 822 074 197 212 0.480 0.481 0.482 0.483 0.484 0.46177 0.46266 0.46355 0.46443 0.46532 91755 59394 22406 80783 34515 41482 26861 46338 13613 42849 88913 16364 56679 95295 72867 664 968 522 430 132 0.88699 0.88653 0.88606 0.88560 0.88514 49227 27001 95910 55958 07150 79284 83281 54652 56506 52837 19439 47206 44417 20075 90129 995 469 051 401 517 0.485 0.486 0.487 0.488 0.489 0.46620 0.46709 0.46797 0.46886 Oi46974 83594 28011 67757 02823 33201 48672 46175 50915 78918 46678 73849 15033 34040 77761 90760 162 451 104 558 024 0.88467 0.88420 0.88374 0.88327 0.88280 49491 82984 07636 23450 30432 08528 89343 61933 93833 53462 31072 33453 55301 75463 46844 223 094 874 416 214 0.490 0.491 0.492 0.493 0.494 0.47062 0.47150 0.47238 0.47327 Oi47415 58881 79855 96114 07649 14451 71158 69788 60472 61583 91970 03618 21242 11121 91533 19709 136 715 556 149 261 0.88233 0.88186 0.88138 0.88091 0.88044 28586 17916 98427 70125 33014 10121 33995 96151 68537 23984 49570 44058 23994 69230 98588 547 307 541 763 075 0.49'5 0.496 0.497 0.498 0.499 0.47503 0.47591 0.47679 0.47766 0.47854 16512 13823 06374 94157 77164 70950 18319 54345 99774 75827 79950 71693 97532 51191 05452 264 150 118 668 099 0.87996 0.87949 0.87901 0.87853 0.87806 87098 32382 68872 96571 15485 36204 79786 30204 63808 57828 22574 96012 70581 47270 28743 157 154 529 917 023 0.500 0.47942 55386 04203 00027 329 0.87758 25618 r, 90372 -\.-, 71611 628 c(6;W 1 l’-:“l 152 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin x X ARGUMENTS cos 2 0.500 0.501 0.502 0.503 0.504 0.47942 0.48030 0.48117 0.48205 0.48293 55386 04203 00027 329 28813 07080 29394 947 97437 07116 30578 414 61249 27448 70881 314 20240 91696 35573 583 0.87758 0.87710 0.87662 0.87614 0.87565 25618 90372 71611 628 26976 40428 38630 733 19562 87859 50795 903 03383 13407 39357 847 78441 98689 97748 295 0.505 0.506 0.507 0.508 0.509 0.48380 0.48468 0.48555 0.48643 0.48730 74403 23960 15529 617 23727 48823 94818 170 68204 91355 38243 967 07826 77106 78840 928 42584 32116 05316 931 0.87517 0.87469 0.87420 0.87371 0.87323 44744 26201 33418 203 02294 79311 19588 355 51098 42264 46912 391 91160 00180 75052 318 22484 39053 84166 561 0.510 0.511 0.512 0.513 0.514 0.48817 0.48904 0.48992 0.49079 0.49166 72468 82907 49450 013 97471 56492 73435 934 17583 80371 57187 006 32796 82532 85582 104 43101 91455 35667 778 0.87274 0.87225 0.87176 0.87127 0.87078 45076 45751 26310 581 58941 08013 76750 129 64083 14454 85187 176 60507 54560 26898 565 48219 18687 53787 441 0.515 0.516 0.517 0.518 0.519 0.49253 0.49340 0.49427 0.49514 0.49601 48490 36108 63810 364 48953 45953 92799 025 44482 50944 98899 617 35068 81528 98859 309 20703 68647 36861 855 0.87029 0.86979 0.86930 0.86881 0.86831 27222 98065 45347 504 97523 84793 59540 132 59126 71841 83584 429 12036 53049 84660 240 56258 23126 60524 189 0.520 0.521 0.522 0.523 0.524 0.49688 0.49774 0.49861 0.49948 0.50034 01378 43736 71433 446 77084 38729 62299 043 47812 86055 57189 109 13555 18641 78596 658 74302 69914 10484 518 0.86781 0.86732 0.86682 0.86632 0.86582 91796 77649 90038 785 18657 13065 83614 647 36844 26688 33565 898 46363 16698 64378 779 47218 82144 82893 524 0.525 0.526 0.527 0.528 0.529 0.50121 0.50207 0.50294 0.50380 0.50467 30046 73797 84942 748 80778 64718 68796 092 26489 77603 50161 411 67171 47881 24954 981 02815 11483 83349 596 0.86532 0.86482 0.86431 0.86381 0.86331 39416 22941 28399 561 22960 39868 22644 077 97856 34571 19753 996 64109 09560 56071 436 21723 68210 99902 671 0.530 0.531 0.532 0.533 0.534 0.50553 0.50639 0.50725 0.50811 0.50898 33412 04846 96181 366 58953 64911 01306 143 79431 29121 89905 473 94836 35431 92741 999 05160 22300 66364 220 0.86280 0.86230 0.86179 0.86128 0.86077 70705 14761 01380 670 11058 54312 41041 248 42788 92829 81312 894 65901 37140 13920 311 80400 94932 10201 726 0.535 0.536 0.537 0.538 0.539 0.50984 0.51070 0.51156 0.51241 0.51327 10394 28695 79260 534 10529 94093 97962 456 05558 58481 73096 946 95471 62356 25387 754 80260 46726 31605 686 0.86026 0.85975 0.85924 0.85873 0.85822 86292 74755 70140 025 83581 86021 71507 760 72273 39001 18926 068 52372 44824 92837 581 23884 15482 98393 339 0.540 0.541 0.542 0.543 0.544 0.51413 0.51499 0.51585 0.51670 0.51756 59916 53113 10467 728 34431 23551 08484 914 03796 00588 85758 874 68002 27290 01726 969 27041 47234 00855 920 0.85770 0.85719 0.85667 0.85616 0.85564 86813 63824 14253 797 41166 03555 41303 947 86946 49241 51282 623 24160 16304 35326 032 52812 21022 52425 567 0.545 0.546 0.547 0.548 0.549 0.51841 0.51927 0.52012 0.52098 0.52183 80905 04516 98283 861 29584 43752 65410 714 73071 10073 15436 812 11356 49129 88849 675 44432 07094 38858 868 0.85512 72907 80530 77799 957 0.85460 84452 12819 51181 787 0.85408 87450 36734 25018 472 0.85356 81907 71975 12587 703 Oi85304 67829 39096 36027 442 0.550 0.52268 72289 30659 16778 838 C-f)7 0.85252 45220 59505 74280 498 C-J)1 11 [1 ELEMENTARY CIRCULAR SINES AND X TRANSCENDENTAL COSINES FOR 153 FUNCTIONS RADIAN ARGUMENTS sin 2 Table 4.6 cos x 0.550 0.551 0.552 0.553 0.554 0.52268 0.52353 0.52439 0.52524 0.52609 72289 94919 12314 24465 31364 30659 67038 63969 69712 33053 16778 57359 64065 94301 44585 838 653 565 297 976 0.85252 0.85200 0.85147 0.85095 0.85042 45220 14086 74432 26263 69585 59505 55464 50084 67333 32026 74280 10953 82092 23867 20180 498 761 114 il0 431 0.555 0.556 0.557 0.558 0.559 0.52694 0.52779 0.52864 0.52949 0.53033 33002 29370 20460 06264 86773 03301 30292 64391 56488 58002 35674 97627 54824 10933 33815 635 180 757 415 002 0.84990 0.84937 0.84884 0.84831 0.84778 04402 30721 48545 57881 58734 69831 07267 71701 91352 95285 50182 35704 88608 58049 77652 218 287 318 047 517 0.560 0.561 0.562 0.563 0.564 0.53118 0.53203 0.53287 0.53372 0.53457 61979 31872 96446 55691 09598 20883 97610 41195 05179 43639 40385 81418 26300 47726 06347 187 533 543 585 607 0.84725 0.84672 0.84619 0.84565 0.84512 51110 35012 10448 77421 35938 13416 76506 16165 64850 55863 12609 06683 29136 21564 44654 452 799 481 438 991 0.565 0.566 0.567 0.568 0.569 0.53541 0.53626 0.53710 0.53794 0.53878 58160 01367 39212 71686 98780 11183 62956 54637 42441 83121 35362 25057 07291 39926 91211 572 521 168 969 553 0.84458 0.84405 0.84351 0.84297 0.84244 86004 27624 60803 85547 01861 23353 02313 28580 38838 70611 24855 00958 70603 36691 53715 579 945 796 011 445 0.570 0.571 0.572 0.573 0.574 0.53963 0.54047 0.54131 0.54215 0.54299 20487 36797 47702 53195 53266 33969 52812 98021 28505 03714 24099 80524 65614 31859 63213 446 005 465 028 905 0.84190 0.84136 0.84082 0.84027 0.83973 09751 09222 00279 82928 57175 62268 53020 82920 92863 24582 74013 93925 99876 14368 41893 376 658 632 839 605 0.575 0.576 0.577 0.578 0.579 0.54383 0.54467 0;54551 0.54634 0.54718 47906 37109 20865 99165 72002 83642 28825 00342 59818 69423 59158 18694 24296 25797 24232 222 718 136 231 321 0.83919 0.83864 0.83810 0.83755 0.83701 23024 80481 29551 70241 02555 20654 24493 80354 33330 29351 14757 38825 39176 05683 38499 543 019 658 918 807 0.580 0.581 0.582 0.583 0.584 0.54802 0.54886 0.54969 0.55053 0.55136 39367 01252 57649 08548 53942 91873 90432 28912 71672 83624 55618 74682 38538 90300 42652 270 851 382 563 424 0.83646 0.83591 0.83536 0.83481 0.83426 26499 42078 49298 48164 38683 15186 38442 47559 91816 21326 93465 27434 43511 36205 36508 789 927 337 988 907 0.585 0.586 0.587 0.588 0.589 0.55219 0.55303 0.55386 0.55469 0.55552 93823 28181 57009 80299 98041 30227 77494 91989 40829 91685 61353 48692 26889 21434 44380 309 799 504 637 278 0.83371 0.83315 0.83260 0.83205 0.83149 20858 94697 60204 17385 66245 87037 40732 35026 23370 60044 56877 36143 84331 27399 51895 861 543 337 720 332 0.590 0.591 0.592 0.593 0.594 0.55636 0.55719 0.55802 0.55885 0.55968 10229 16852 17904 13375 03258 12783 72905 41388 88127 83575 77572 55827 50056 50327 48880 254 556 192 409 201 0.83094 0.83038 Oi82982 0.82926 Oi82870 06791 39026 62959 78593 85934 00163 99672 15348 04797 26455 49524 61643 23660 09361 75147 800 346 255 243 786 0.595 0.596 0.597 0.598 0.599 0.56050 0.56133 0.56216 0.56299 0.56381 87544 66226 39293 06739 68555 98744 05205 75090 81092 96468 23078 18307 30821 90525 43709 004 516 541 792 545 0.82814 0.82758 0.82702 0.82646 Oi82589 84988 75761 58257 32484 98446 39590 04294 81491 32932 21193 04193 50517 82974 29164 19254 468 407 799 660 799 0.600 0.56464 24733 95035 35720 095 C-f)7 [1 0.82533 56149 09678 29724 095 154 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin x X ARGUMENTS cos x 0.601 0.602 0.603 0.604 0.56464 O;i6546 0.56629 0.56711 0.56793 24733 95035 35720 095 75265 51175 93580 897 20142 39837 08553 336 59356 36531 18642 028 92899 17336 91043 574 0.82533 0.82477 0.82420 0.82363 0:82307 56149 09678 29724 095 05598 62617 27022 123 46800 45065 11146 193 79760 22901 59135 858 04483 62830 68484 934 0.605 0.606 0.607 0.608 0.609 0.56876 20762 58900 04538 687 Oi56958 42938 38434 31827 607 0.57040 59418 33722 21808 719 0.57122 70194 23115 81800 299 0.57204 75257 85537 59705 300 0.82250 0.82193 0.82136 0.82079 0.82022 20976 32380 00471 116 29243 99900 23403 216 29292 34564 55786 102 21127 06368 09403 380 04753 86127 32317 893 0.610 0.611 0.612 0.613 0.614 0.57286 0:$7368 0.57450 O:i7532 0.57614 74601 00481 26119 098 68215 48012 56380 111 56093 08770 12563 221 38225 63966 25415 904 14604 95387 76236 989 0.81964 0.81907 0.81850 0.81792 0.81734 80178 45479 51790 075 47406 56882 17114 225 06443 93612 42372 770 57296 29766 49108 549 99969 40259 08915 198 0.615 0.616 0.617 0.618 0.619 0.57695 0.57777 0.57859 0.57940 0.58022 85222 85396 78697 975 50071 16931 60606 809 09141 73507 45614 047 62426 39217 34861 330 09916 98732 88572 073 0.81677 0.81619 0.81561 0.81503 0.81445 34469 00822 85945 685 60800 88007 79339 051 78970 79180 65565 411 88984 52524 40689 288 90847 87037 62551 318 0.620 0.621 0.622 0.623 0.624 0.58103 0.58184 0.58266 0.58347 0.58428 51605 37305 07584 296 87483 40765 14825 522 17542 95525 36729 641 41775 88579 84595 681 60174 07505 35888 387 0.81387 0.81329 0.81271 0.81213 0.81154 84566 62533 92868 400 70146 59641 39252 335 47593 59801 97147 027 16913 45270 91684 290 78111 99116 19458 331 0.625 0.626 0.627 0.628 0.629 0.58509 0.58590 0.58671 0.58752 0.58833 72729 40462 15480 540 79433 76194 76836 923 80279 04032 83139 861 75257 13891 88356 252 64359 96274 18246 006 0.81096 0.81037 0.80979 0.80920 0.80861 31195 05217 90218 953 76168 48267 68483 556 13038 13768 15067 973 41809 88032 28536 214 62489 58182 86569 178 0.630 0.631 0.632 0.633 0.634 0.58914 0.58995 0.59075 0.59156 0.59237 47579 42269 51311 811 24907 43555 99690 151 96335 92400 89983 484 61856 81661 44033 509 21462 04785 59635 440 0.80802 0.80743 0.80684 0.80625 0.80566 75083 12151 87252 371 79596 38679 90282 722 76035 27315 58094 522 64405 68414 96904 569 44713 53140 97676 566 0.635 0.636 0.637 0.638 0.639 0.59317 0.59398 0.59478 0.59559 0.59639 75143 55812 91193 198 22893 29375 30315 454 64703 20697 86352 425 00565 25599 66873 364 30471 40494 58084 641 0.80507 0.80447 0.80388 0.80328 0.80269 16964 73462 77004 837 81165 22155 17917 411 37320 92798 10598 548 85437 79775 93030 752 25521 78276 91556 338 0.640 0.641 0.642 0.643 0.644 0.59719 0.59799 0.59879 0.59959 0.60039 54413 62392 05188 355 72383 88897 92681 375 84374 18215 24594 757 90376 49145 04673 426 90382 81087 16496 070 0.80209 0.80149 0.80089 0.80030 0.79970 57578 84292 61358 611 81614 94617 26862 715 97636 06847 22056 216 05648 19380 30729 469 05657 31415 26635 842 0.645 0.646 0.647 0.648 0.649 0.60119 0.60199 0.60279 0.60359 0.60439 84385 14041 03535 151 72375 48606 49156 949 54345 85984 56561 576 30288 27978 28662 868 00194 76993 47908 070 0.79909 0.79849 0.79789 0.79729 0.79668 97669 42951 13571 848 81690 54786 65377 243 57726 68519 65855 159 25783 86546 48612 327 85868 12061 36819 444 0.650 0.60518 64057 36039 56037 252 0.600 0.79608 37985 49055 82891 760 ELEMENTARY CIRCULAR SINES AND TRANSCENDENTAL COSINES FOR RADIAN ARGKMENTS sin x X 155 FUNCTIONS Table 4.6 cos x 0.650 0.651 0.652 0.653 0.654 0.60518 0.60598 0.60677 0.60757 0.60836 64057 36039 56037 252 21868 08730 33782 358 73618 99284 80505 ala 19302 12527 93778 646 58909 53891 48897 929 0.79608 0.79547 0.79487 0.79426 0.79365 37985 49055 a2891 760 a2142 02318 08089 927 la343 77432 42041 la3 46596 80778 62180 929 66907 19531 33114 757 0.655 0.656 0.657 0.658 0.659 0.60915 0.60995 0.61074 0.61153 0.61232 92433 29414 78343 652 19865 45745 51174 755 41198 10140 52364 359 56423 30466 62074 073 65533 15201 34867 307 0.79304 0.79243 0.79182 0.79121 0.79060 79281 01659 45900 987 83724 35925 57253 785 80243 31885 28666 909 68843 99886 65458 154 49532 51069 55734 550 0.660 0.661 0.662 0.663 0.664 0.61311 0.61390 0.61469 0.61548 0.61627 68519 73433 78861 515 65375 14865 34819 272 56091 49810 55178 137 40660 a9197 a3019 la6 19075 44570 30974 165 0.78999 0.78937 0.78876 0.78814 0.78753 22314 97365 09278 382 a7197 51494 96354 080 44186 26970 a6436 061 93287 38093 86857 558 34506 99953 81380 523 0.665 0.666 0.667 0.668 0.669 0.61705 0.61784 0.61863 0.61941 0.62020 91327 28086 60071 171 57408 52521 58518 785 17311 31267 20428 576 71027 78333 24475 901 la550 08348 12498 919 0.78691 0.78629 0.78568 0.78506 0.78444 67851 28428 68686 643 93326 40184 00789 551 10938 52672 21368 279 20693 a4132 04022 017 22598 53587 90446 244 0.670 0.671 0.672 0.673 0.674 0.62098 0.62176 0.62255 0.62333 0.62411 59870 36559 68035 744 94980 78835 94799 654 23873 51665 95092 281 46540 72160 48154 700 62974 58052 a8456 349 0.78382 0.78320 0.78257 0.78195 0.78133 16658 80849 28530 294 02880 86510 10376 414 81270 91948 10240 374 51835 19324 22393 698 14579 91581 98907 578 0.675 0.676 0.677 0.678 0.679 0.62489 0.62567 0.62645 0.62723 0.62801 73167 27699 a3921 682 77111 00082 14094 496 74797 94805 48239 849 66220 32101 23383 477 51370 32827 22288 658 0.78070 0.78008 0.77945 0.77882 0.77820 69511 32446 a7358 526 16635 66425 68455 a30 55959 18805 93590 a77 a7488 15655 22308 414 11228 a3820 59699 786 0.680 0.681 0.682 0.683 0.684 0.62879 0.62957 0.63034 0.63112 0.63189 30240 18468 51370 418 02822 11138 la547 018 69108 33578 11028 644 29091 09159 73043 207 a2762 61884 a3499 197 0.77757 0.77694 0.77631 0.77568 0.77505 27187 50927 93718 239 35370 45381 32416 339 35783 96362 41105 566 28434 33829 79438 156 13327 88518 38411 247 0.685 0.686 0.687 0.688 0.689 0.63267 0.63344 0.63422 0.63499 0.63576 30115 16386 33585 498 71140 97929 04308 084 05832 32410 43963 542 34181 46361 45549 306 56180 66947 24110 566 0.77441 0.77378 0.77315 0.77251 0.77188 90470 91938 77293 390 59869 76376 60473 500 21530 74891 94232 293 75460 21318 63436 286 21664 50263 68154 418 0.690 0.691 0.692 0.693 0.694 0.63653 0.63730 0.63807 0.63884 0.63961 71822 21967 94023 743 al098 39859 46216 467 84001 49694 25323 984 80523 81182 06781 a99 70657 64670 73855 200 0.77124 0.77060 0.76997 0.76933 0.76869 60149 97106 60197 354 90922 97998 79579 541 13989 89862 90904 069 29357 10392 19670 418 37030 98049 @SO5 132 0.695 0.696 0.697 0.698 0.699 0.64038 0.64115 0.64192 0.64268 0.64345 54395 31146 94603 464 31729 12236 98782 Ii5 02651 40207 54600 136 67154 47966 i5a92 698 25230 69063 48031 063 0.76805 0.76741 0.76677 0.76612 0.76548 37017 92068 53315 502 29324 32449 39366 321 13956 59961 77279 757 90921 16142 38958 434 60224 43294 73431 759 0.700 0.64421 76872 37691 05367 261 [ c-y1 0.76484 21872 84488 42625 586 [c-y 1 156 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin 5 X ARGUMENTS cos x 0.700 0.701 0.702 0.703 0.704 0.64421 0.64498 0.64574 0.64650 0.64727 76872 37691 05367 261 22071 88685 07414 902 60821 57525 65445 583 93113 80337 88940 870 18940 93892 61979 783 0.76484 21872 84488 42625 586 0.76419 75872 83558 57055 252 Oi76355 22230 85105 i1442 075 0.76290 60953 34492 20253 368 0.76225 92046 77847 53166 023 0.705 0.706 0.707 0.708 0.709 0.64803 0.64879 0.64955 0.65031 0.65107 38295 35607 19561 705 51169 43546 23864 641 57555 56422 40438 747 57446 13597 14335 062 50833 55081 46169 354 0.76161 0.76096 0.76031 0.75966 0.75901 15517 62061 70453 752 31372 34787 58298 030 39617 44439 64022 815 40259 40193 31253 107 33304 71984 34997 406 0.710 0.711 0.712 0.713 0.714 0.65183 0.65259 0.65334 0.65410 0.65486 37710 21536 68121 013 18068 54275 19866 915 91900 95261 24450 173 59199 87111 64083 709 19957 73096 55888 565 0.75836 0.75770 0.75705 0.75640 0.75574 18759 90508 16654 146 96631 47219 18942 159 66925 94330 20755 235 29649 84811 71940 852 84809 72391 28003 128 0.715 0.716 0.717 0.718 0.719 0.65561 0.65637 0.65712 0.65787 0.65863 74166 97140 27566 883 21820 03821 93009 463 62909 38376 27837 851 97427 46694 44880 853 25366 75324 69585 417 0.75509 0.75443 0.75378 0.75312 0.75246 32412 11552 84730 074 72463 57536 12745 203 04970 66335 91983 563 29939 94701 46092 263 47378 00135 76755 558 0.720 0.721 0.722 0.723 0.724 0.65938 0.66013 0.66088 0.66163 0.66238 46719 71473 15361 800 61478 83004 58862 952 69636 58443 15198 027 71185 46973 13079 967 66117 98439 69907 065 0.75180 0.75114 0.75048 0.74982 0.74916 57291 40894 97944 549 59686 75987 70091 576 54570 65174 34189 363 41949 68966 45814 983 21830 48626 09078 707 0.725 0.726 0.727 0.728 0.729 0.66313 0.66388 0.66463 0.66537 0.66612 54426 63349 66778 441 36103 92872 23443 354 11142 38839 73184 280 79534 53748 37633 666 41272 90759 01524 309 0.74849 0.74783 0.74717 0.74650 0.74584 94219 66165 10497 806 59123 84344 52795 369 16549 66673 88624 209 66503 77410 54215 910 08992 81559 02955 103 0.730 0.731 0.732 0.733 0.734 0.66686 0.66761 0.66835 0.66910 0.66984 96350 03697 87373 259 44758 47057 30099 195 86490 75996 51573 181 21539 46342 35102 739 49897 14589 99849 159 0.74517 0.74450 0.74383 0.74317 0.74250 44023 44870 38879 013 71602 33841 50102 364 91736 15714 42167 693 04431 58475 71321 153 09695 30855 77713 862 0.735 0.736 0.737 0.738 0.739 0.67058 0.67132 0.67206 0.67280 0.67354 71556 37903 75177 973 86509 74117 74942 523 94749 81736 71700 537 96269 19936 70863 650 91060 48565 84779 796 0.74183 0.74115 0.74048 0.73981 0.73914 07534 02328 18528 866 97954 43109 01033 791 80963 24156 15559 237 56567 17168 68402 998 24772 94586 14660 158 0.740 0.741 0.742 0.743 0.744 0.67428 0.67502 0.67576 0.67650 0.67723 79116 28145 06748 388 60429 19868 84968 216 34991 85605 96417 996 02796 87900 20669 485 63836 89971 13633 096 0.73846 0.73779 0.73711 0.73644 0.73576 85587 29587 90979 142 39016 96092 48243 787 85068 68756 84181 492 23749 22975 75897 532 55065 34881 12335 582 0.745 0.746 0.747 0.748 0.749 0.67797 0.67870 0.67944 0.68017 0.68090 18104 55714 81235 936 65592 49704 53032 193 06293 37191 55745 803 40199 84105 86745 313 67304 57056 87450 880 0.73508 0.73440 0.73373 0.73305 0.73237 79023 81341 26664 537 95631 39960 28591 681 04894 89077 36602 285 06821 07766 10125 695 01416 75833 81627 975 0.750 0.68163 87600 23334 16673 324 [(-;GJ 1 0.73168 88688 73820 88631 184 [(-;)I 1 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL AND COSINES FOR RADIAN .4RGUMENTS sin x X 157 FUNCTIONS Table 4.6 cos x 0.750 0.751 0.752 0.753 0.754 0.68163 0.68237 0.68310 0.68383 0.68456 87600 23334 16673 324 01079 50908 23885 163 07735 08431 22423 554 07559 65237 62625 080 00545 91345 04892 285 0.73168 0.73100 0.73032 0.72964 0.72895 88688 73820 88631 184 68643 83000 05659 342 41288 85375 76111 160 06630 63683 44059 608 64676 01388 85978 367 0.755 0.756 0.757 0.758 0.759 0.68528 0.68601 0.68674 0.68747 0.68819 86686 57454 92691 917 65974 34953 25484 772 38401 95911 31587 089 03962 13086 40963 419 62647 59922 57950 885 0.72827 0.72758 0.72689 0.72621 0.72552 15431 82687 42395 268 58904 92503 49472 750 95102 16489 70515 436 24030 41026 27404 8k7 45696 53220 31961 494 0.760 0.761 0.762 0.763 0.764 0.68892 0.68964 0.69036 0.69109 0.69181 14451 10551 33914 776 59365 39792 39835 383 97383 23154 38826 030 28497 36835 58582 200 52700 57724 63761 700 0.72483 0.72414 0.72345 0.72276 0.72207 60107 40905 17233 969 67269 92639 68715 814 67190 97707 55489 548 59877 46116 61298 318 45336 28598 15545 123 0.765 0.766 0.767 0.768 0.769 0.69253 0.69325 0.69397 0.69469 0.69541 69985 63401 28295 794 80345 32137 07631 223 83772 42896 10903 039 80259 75335 73038 195 69800 09807 26789 802 0.72138 0.72068 0.71999 0.71930 0.71860 23574 36606 24219 693 94598 62317 00753 084 58415 98627 96800 072 15033 39157 32949 410 64457 78243 29362 010 0.770 0.771 0.772 0.773 0.774 0.69613 0.69685 0.69756 0.69828 0.69900 52386 27356 74701 988 28011 09725 61005 296 96667 39351 43442 524 58347 99368 65024 972 13045 73609 25718 983 0.71791 0.71721 0.71651 0.71581 0.71512 06696 10943 36337 129 41755 33033 64806 626 69642 41008 lb757 355 90364 32078 15581 770 03928 04171 36356 807 0.775 0.776 0.777 0.778 0.779 0.69971 0.70043 0.70114 0.70185 0.70256 60753 46603 54062 747 01464 03580 78713 256 35170 30469 99923 379 61865 13900 60948 949 81541 41203 19385 818 0.71442 0.71372 0.71302 0.71231 0.71161 10340 55931 36051 117 09608 86716 83660 709 01739 96600 90273 093 86740 86370 39059 972 64618 57525 15198 564 0.780 0.781 0.782 0.783 0.784 0.70327 0.70398 0.70469 0.70540 0.70611 94192 00410 18436 790 99809 80256 58108 374 98387 70180 66337 280 89918 60324 70046 581 74395 41535 66131 480 0.71091 0.71020 0.70950 0.70880 0.70809 35380 12277 35721 626 99032 53550 79296 239 55582 84980 15931 435 05038 10910 36614 737 47405 36395 82877 671 0.785 0.786 0.787 0.788 0.789 0.70682 0.70753 0.70823 0.70894 0.70964 51811 05365 92374 614 22158 44073 98290 801 85430 50625 15901 193 41620 18692 30436 730 90720 42656 50970 857 0.70738 0.70668 0.70597 0.70526 0.70455 82691 67199 76290 330 10904 09793 47885 059 32049 71355 67509 330 46135 59771 73107 880 53168 83632 99934 173 0.790 0.791 0.792 0.793 0.794 0.71035 0.71105 0.71175 0.71246 0.71316 32724 17607 80981 403 67624 39345 88841 574 95414 04380 78239 979 16086 09933 58529 620 29633 53937 15005 776 0.70384 0.70313 0.70242 0.70171 0.70099 53156 52236 09691 278 46105 75582 19602 208 32023 64376 31409 812 10917 30026 60306 275 82793 84643 63792 314 0.795 0.796 0.797 0.798 0.799 0.71386 0.71456 0.71526 0.71596 0.71665 36049 35036 79112 713 35326 52590 98579 148 27458 06672 07482 391 12436 98066 96241 109 90256 28277 81536 630 0.70028 0.69957 0.69885 0.69814 0.69742 47660 41039 70466 123 05524 12728 08742 151 56392 13922 35499 779 00271 59535 64661 971 37169 65179 95703 964 0.800 0.71735 60908 99522 76162 718 [c-y1 0.69670 67093 47165 42092 075 [ C-f,9 1 158 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN co5 x sin x X ARGUMENTS 0.800 0.801 0.802 0.803 0.804 0.71735 0.71805 0.71874 0.71944 0.72013 60908 24388 80686 29797 71714 99522 14736 77571 92397 64303 76162 58803 43741 50488 73354 718 753 255 651 263 0.69670 0.69598 0.69527 0.69455 0.69383 67093 90050 06047 15091 17189 47165 22499 08886 24727 89116 42092 59652 74871 13123 26831 075 695 538 218 236 0.805 0.806 0.807 0.808 0.809 0.72083 0.72152 0.72221 0.72290 0.72359 06429 33937 54228 67298 73139 99098 03310 84188 49704 08550 50932 35522 62476 19472 15721 396 503 322 935 677 0.69311 0.69239 0.69166 0.69094 0.69022 12350 00579 81884 56273 23752 21844 43394 74945 38365 56214 23558 94027 40074 02528 89026 425 956 951 784 151 0.810 0.811 0.812 0.813 0.814 0.72428 Oi72497 0.72566 0.72635 0.72703 71743 63105 47217 24072 93664 70142 44620 42849 76416 57636 51092 85175 06266 00277 19583 818 959 069 085 027 0.68949 0.68877 0.68804 0.68732 0.68659 84329 38011 84805 24719 57759 51747 48903 72316 47306 99881 01754 65129 53394 18165 15892 964 158 472 280 545 0.815 0.816 0.817 0.818 0.819 0.72772 Oi72841 0.72909 Oi72977 0.73046 55985 11030 58790 99259 32430 99550 15926 21260 30776 60427 51786 884L4 93542 72343 39565 534 775 651 223 302 0.68586 0.68514 0.68441 0.68368 0.68295 83934 03250 15714 21335 20118 56737 45257 93509 30246 84907 35262 24529 18772 67094 59742 969 414 652 544 692 0.820 0.821 0.822 0.823 0.824 0.73114 0.73182 0.73250 0.73318 0.73386 58297 76852 88089 92001 88581 26895 47595 40670 24998 20187 87938 56503 98872 51414 01366 131 084 320 329 283 0.68222 0.68148 0.68075 0.68002 0.67929 12072 97204 75521 47030 11740 87613 69169 61060 95457 05207 55166 07005 91008 31885 30088 656 802 857 232 213 0.825 0.826 0.827 0.828 0.829 0.73454 0.73522 0.73590 0.73658 0.73725 77822 59718 34261 01446 61264 46578 25249 78008 27404 96715 54873 04953 99391 08557 93150 150 477 793 557 579 0.67855 0.67782 0.67708 0.67635 0.67561 69656 20786 65139 02720 33538 23839 85563 25264 78508 81536 88530 39229 69888 50409 59331 058 106 949 750 781 0.830 0.831 0.832 0.833 0.834 0.73793 0.73860 0.73927 0.73995 0.74062 13711 58777 96458 26746 49635 09962 91899 68020 64557 08481 71872 89026 82039 48913 15603 858 752 434 544 989 0.67487 0.67413 0.67339 0.67265 0.67191 57600 74913 85485 89323 86434 71267 85293 61885 39984 59207 10211 77928 24928 27394 01352 246 481 580 537 983 0.835 0.836 0.837 0.838 0.839 0.74129 0.74196 0.74263 0.74330 0.74397 65117 73186 73836 67059 52849 27503 50074 05390 23383 34732 03320 95758 06248 44844 85324 808 049 576 755 932 0.67117 0.67043 0.66969 0.66895 0.66820 76826 60506 37482 07761 71351 59842 82850 69864 63185 05786 28712 83235 56439 83438 68708 570 098 445 385 357 0.840 0.841 0.842 0.843 0.844 0.74464 0.74531 0.74597 0.74664 0.74730 31199 02103 65554 21545 70070 70859 63927 46848 53275 17609 32125 87199 16798 18184 86260 657 577 923 539 385 0.66746 0.66671 0.66597 0.66522 0.66447 28258 78491 22056 58962 89216 41308 14059 69016 51824 08791 11792 32935 98654 47240 14192 267 396 482 065 152 0.845 0.846 0.847 0.848 0.849 0.74797 0.74863 0.74929 0.74995 0.75062 11121 44693 70779 89371 00464 74999 61339 13273 68190 64233 80133 89598 01550 66317 63922 429 886 724 368 547 0.66373 0.66298 0.66223 0.66148 0.66073 12824 29796 40137 43857 40961 86891 33764 97713 27703 73363 57589 83391 70674 96802 62530 286 100 409 946 783 0.850 0.75128 04051 40292 i-8)9 70271 207 0.65998 31458 [ 71 [c-w 1 84982 7 17039 542 ELEMENTARY CIRCULAR TRANSCENDENTAL SINES AND COSINES FOR RADIAN ARGUMENTS sin z X 159 FUNCTIONS Table 4.6 cos x 0.850 0.851 0.852 0.853 0.854 0.75128 0.75194 0.75259 0.75325 0.75391 04051 00125 88679 69708 43204 40292 36009 91775 48737 48790 70271 23260 88815 26849 57151 207 432 295 594 380 0.65998 0.65923 0.65847 0.65772 0.65697 31458 15356 92661 63381 27524 84982 13509 10556 28392 19944 17039 82909 81024 55410 98010 542 449 321 547 152 0.855 0.856 0.857 0.858 0.859 0.75457 0.75522 0.75588 0.75653 0.75718 09161 67572 18431 61731 97466 34586 49528 37776 44244 14602 25193 67867 79144 75659 62217 237 227 450 143 260 0.65621 0.65546 0.65470 0.65395 0.65319 85097 36108 80564 18474 49843 38799 39199 76042 04884 81933 73388 43373 91635 48193 13861 013 300 218 134 148 0.860 0.861 0.862 0.863 0.864 0.75784 0.75849 0.75914 0.75979 0.76044 25628 46213 59212 64620 62430 95276 33451 77068 74826 76186 97229 58068 06350 53141 24087 459 441 566 684 122 0.65243 0.65167 0.65092 0.65016 0.64940 74681 92995 04791 10079 08865 64051 08756 74216 19251 03332 84627 75966 47091 25131 29254 203 794 357 418 574 0.865 0.866 0.867 0.868 0.869 0.76109 0.76174 0.76239 0.76303 0.76368 52636 35230 10208 77561 37284 31366 91346 07866 33429 21299 24465 04168 22598 13500 49706 750 673 272 144 858 0.64864 0.64787 0.64711 0.64635 0.64559 01156 86962 66288 39144 05536 86580 29767 94312 42282 36391 94718 96858 75010 56369 79782 373 196 176 276 561 0.870 0.871 0.872 0.873 0.874 0.76432 0.76497 0.76561 0.76625 0.76690 89370 33813 70606 99742 21217 25505 00837 02852 87869 12977 07814 32779 02438 91953 38182 480 191 134 834 114 0.64482 0.64406 0.64329 0.64253 0.64176 65472 18960 66007 06621 40810 40001 17117 32390 51117 39234 19477 08727 63447 05733 87329 766 234 280 091 202 0.875 0.876 0.877 0.878 0.879 0.76754 0.76818 0.76882 0.76946 0.77010 35022 41152 39600 30359 13424 36027 15638 11198 82862 91555 03963 42337 60682 84773 22769 458 736 252 027 271 0.64099 0.64022 0.63946 0.63869 0.63792 68581 89942 04901 13466 15643 63325 90610 88955 26862 73477 13035 64049 21244 88380 15258 656 903 528 872 639 0.880 0.881 0.882 0.883 0.884 0.77073 0.77137 0.77201 0.77264 0.77328 88788 56445 16388 68611 13107 98969 67568 60587 42032 76680 29120 68399 79051 37074 19618 965 506 337 497 049 0.63715 0.63638 0.63560 0.63483 0.63406 11441 00868 83931 60638 30997 98580 72592 66570 52208 01835 20801 16079 27264 18529 14874 550 131 710 695 218 0.885 0.886 0.887 0.888 0.889 0.77391 0.77454 0.77518 0.77581 0.77644 49871 78895 00174 13701 19470 30081 68560 59214 69915 69310 68504 53673 36552 33343 78237 290 706 600 321 045 0.63328 0.63251 0.63174 0.63096 0.63018 95014 52699 04059 49102 87834 88415 85546 67460 09021 85724 24894 63485 74481 53235 69127 213 020 571 256 530 0.890 0.891 0.892 0.893 0.894 0.77707 0.77770 0.77832 0.77895 0.77958 17475 07709 90165 64839 31723 26823 12654 97778 53950 53704 86549 17776 38577 85676 28683 033 316 722 211 432 0.62941 0.62863 0.62785 0.62707 0.62629 20265 46402 66252 79824 87125 73696 49694 91105 75942 82849 88020 94643 14919 38222 39581 355 540 057 428 242 0.895 0.896 0.897 0.898 0.899 0.78020 0.78083 0.78145 0.78208 0.78270 90811 42097 85575 21238 49080 70350 77980 51465 66458 99392 32846 21716 39740 14771 20508 443 548 163 667 171 0.62551 0.62473 0.62395 0.62317 0.62239 88163 82946 71482 53778 29842 91096 80578 31818 25961 44779 01810 37587 11458 61790 22654 880 545 656 683 524 0.900 0.78332 69096 27483 38846 138 [c-p1 1 0.62160 99682 70664 45648 472 [ (-f)S 1 160 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES sin X AND COSINES FUNCTIONS FOR RADIAN ARGUMENTS cos x x 0.900 0.901 0.902 0.903 0.904 0.78332 0.78394 0.78456 0.78518 0.78580 69096 27483 38846 138 81278 28730 22159 796 85620 81914 55501 279 82117 66602 18722 439 70762 63143 48518 260 0.62160 99682 70664 45648 472 0.62082 63306 86633 21658 870 0.62004 20722 76323 02558 530 0.61925 71938 23992 22842 983 Oib1847 16961 14519 21204 658 0.905 0.906 0.907 0.908 0.909 0.78642 0.78704 0.78765 0.78827 0.78888 51549 52674 00391 817 24472 17115 10540 713 89524 39174 57664 940 46700 02347 24696 094 95992 90915 60447 888 0.61768 0.61689 0.61611 0.61532 0.61453 55799 33401 62045 040 88460 66755 56924 921 14953 01314 85952 792 35284 24430 19111 466 49462 24068 37523 020 0.910 0.911 0.912 0.913 0.914 0.78950 0.79011 0.79072 0.79134 0.79195 37396 89950 41187 896 70905 85311 32130 474 96513 63647 48850 789 14214 12398 18619 897 24001 19793 41660 812 0.61374 0.61295 0.61216 0.61137 0.61058 57494 88811 54652 118 59390 07856 37447 803 55155 71013 27423 839 44799 68705 61677 674 28329 91968 93848 110 0.915 0.916 0.917 0.918 0,919 0.79256 0.79317 0.79378 0.79438 0.79499 25868 74854 52325 499 19810 67394 80192 738 05820 88020 11086 785 83893 28129 48016 785 54021 79915 72036 860 0.60979 0.60899 0.60820 0.60741 0.60661 05754 32450 15011 758 77080 82406 74518 350 42317 34706 00764 999 01471 82824 21909 476 54552 20845 86522 589 0.920 0.921 0.922 0.923 0.924 0.79560 0.79620 0.79681 0.79741 0.79801 16200 36366 03026 828 70422 91262 60393 471 16683 39183 23692 319 54975 75501 93169 858 85293 96389 50226 129 0.60582 0.60502 0.60422 0.60343 0.60263 01566 43462 84179 741 42522 45973 65991 745 77428 24282 65074 984 06291 74899 16960 980 29120 94936 79945 468 0.925 0.926 0.927 0.928 0.929 0.79862 0.79922 0.79982 0.80042 0.80102 07631 98814 17797 639 21983 80542 20660 537 28343 40138 45653 978 26704 76967 01823 638 17061 91191 80485 294 0.60183 0.60103 0.60023 0.59943 0.59863 45923 82112 55377 043 56708 34746 07885 466 61482 51758 85549 703 60254 32673 40005 791 53031 77612 46494 584 0.930 0.931 0.932 0.933 0.934 0.80161 0.80221 0.80281 0.80340 0.80400 99408 83777 15208 432 73739 56488 41719 806 40048 11892 57726 899 98328 53358 82661 218 48574 85059 17341 371 0.59783 0.59703 0.59622 0.59542 0.59462 39822 87298 23849 491 20635 63051 54424 260 95478 06791 03960 905 64358 21032 41397 846 27284 08887 58618 345 0.935 0.936 0.937 0.938 0.939 0.80459 0.80519 0.80578 0.80637 0.80696 90781 11969 03555 863 24941 39867 83565 545 51049 75339 59525 671 69100 25773 52827 488 79086 99364 63359 313 0.59381 0.59301 0.59220 0.59140 0.59059 84263 74063 90139 324 35305 20863 32740 634 80416 54181 65034 867 19605 79507 66977 785 52881 02922 39319 443 0.940 0.941 0.942 0.943 0.944 0.80755 0.80814 0.80873 0.80932 0.80991 81004 05114 28687 022 74845 52830 83153 915 60605 53130 16899 872 38278 17436 34799 758 07857 57982 15321 017 0.58978 0.58898 0.58817 0.58736 0.58655 80250 31098 22996 099 01721 71298 18462 976 17303 31375 04967 973 27003 19770 59766 388 30829 45514 77276 748 0.945 0.946 0.947 0.948 0.949 0.81049 0.81108 0.81166 0.81225 0.81283 69337 87809 69300 383 22713 20770 98639 669 67977 71528 54920 560 05125 55555 97938 351 34150 89138 54154 591 0.58574 0.58493 0.58412 0.58330 0.58249 28790 18224 88177 827 20893 48104 78446 913 07147 45944 08339 436 87560 23117 31310 012 62139 91583 12874 994 0.950 0.81341 55047 89?73 75068 542 11 c-y 0.58168 30894 63883 49416 618 [ !-;I8 1 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL AND COSINES FOR RADIAN ARGUMENTS sin x X 161 FUNCTIONS Table 4.6 cos x 0:950 0.951 0.952 0.953 0.954 0.81341 0.81399 0.81457 0.81515 0.81573 55047 89373 75068 542 67810 74171 95507 433 72433 62256 91835 411 68910 73166 40081 165 57236 27252 73984 145 0.58168 30894 63883 49416 618 0.58086 93832 53142 86928 810 0.58005 50961 73067 39704 748 0.57924 02290 37944 08966 251 Oi57842 47826 62640 01435 096 0.955 0.956 0.957 0.958 0.959 0.81631 0.81689 0.81746 0.81804 0.81861 37404 45683 42959 322 09409 50441 69980 433 73245 64327 09381 654 28907 10956 04577 644 76388 14762 45701 891 0.57760 0.57679 0.57597 0.57515 0.57433 87578 62601 47846 300 21554 53853 21403 511 49762 52997 56176 536 72210 77213 65441 113 88907 44256 59961 007 0.960 0.961 0.962 0.963 0.964 0.81919 0.81976 0.82033 0.82090 0.82147 15683 00998 27163 322 46785 95734 05121 101 69691 25859 54877 569 84393 19084 28189 263 90886 03938 10495 962 0.57351 0.57270 0.57188 0.57105 0.57023 99860 72456 66212 505 05078 80718 44551 395 04569 88520 07322 513 98342 15912 36911 940 86403 83518 03741 923 0.965 0.966 0.967 0.968 0.969 0.82204 0.82261 0.82318 0.82375 0.82432 89164 09771 78067 694 79221 66757 55069 656 61053 05889 70544 986 34652 58985 15315 328 00014 58683 98799 136 0.56941 0.56859 0.56777 0.56694 0.56612 68763 12530 84208 614 45428 24714 78562 699 16407 42403 28733 004 81708 88498 36093 162 41340 86469 79171 417 0.970 0.971 0.972 0.973 0.974 0.82488 0.82545 0.82601 0.82657 0.82714 57133 38450 05747 662 06003 32571 52898 564 46618 76161 45547 087 78974 05158 34034 750 03063 56326 70155 495 0.56529 0.56447 0.56364 0.56282 0.56199 95311 60354 31303 653 43629 34754 78229 727 86302 34839 35633 190 23338 86340 66624 480 54747 15554 99167 663 0.975 0.976 0.977 0.978 0.979 0.82770 0.82826 0.82882 0.82938 0.82993 18881 67257 63479 226 26422 76369 37592 699 25681 22907 86257 689 16651 46947 29486 397 99327 89390 69534 022 0.56116 0.56034 0.55951 0.55868 0.55785 80535 49341 43450 813 00712 15121 09200 110 15285 40876 22937 736 24263 55149 45183 654 27654 87042 87601 358 0.980 0.981 0.982 0.983 0.984 0.83049 0.83105 0.83160 0.83216 0.83271 73704 91970 46808 453 39776 97248 95697 028 97538 48619 00310 290 46983 90304 50142 703 88107 67360 95650 254 0.55702 0.55619 0.55536 0.55452 0.55369 25467 66217 30087 666 17710 22891 37806 645 04390 87840 78167 757 85517 92397 37748 295 61099 68448 39160 207 0.985 0.986 0.987 0.988 0.989 0.83327 0.83382 0.83437 0.83492 0.83547 20904 25676 03744 902 45368 11970 13205 801 61493 73796 90007 262 69275 59543 82563 379 68708 18432 76889 279 0.55286 0.55202 0.55119 0.55036 0.54952 31144 48435 57861 376 95660 65354 38911 453 54656 52753 13672 322 08140 44732 16453 272 56120 75943 01100 969 0.990 0.991 0.992 0.993 0.994 0.83602 0.83657 0.83712 0.83766 0.83821 59786 00520 51678 926 42503 56699 33299 444 16855 38697 50701 883 82835 99079 90248 385 40439 91248 50455 694 0.54868 0.54785 0.54701 0.54617 0.54534 98605 81587 57534 313 35603 97417 28224 252 67123 59732 24618 647 93173 05380 43512 268 13760 71756 83362 006 0.995 0.996 0.997 0.998 0.999 0.83875 0.83930 0.83984 0.84038 0.84093 89661 69442 96654 953 30495 88741 15567 733 62937 05059 69798 245 86979 75154 52241 668 02618 56621 40408 555 0.54450 0.54366 0.54282 0.54198 0.54114 28894 96802 60547 375 38584 19004 25576 412 42836 77392 79237 026 41661 11542 88693 907 35065 61572 03531 067 1.000 0.84147 09848 07896 50665 250 II 1 C-f)1 0.54030 23058 68139 71740 094 [c-p7 1 162 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN cos x sin x X ARGUMENTS 1.000 1. 001 1.002 1.003 1.004 0.84147 0.84201 0.84254 0.84308 0.84362 09848 07896 50665 250 08662 88256 92390 268 99057 57821 22046 578 81026 77549 97169 747 54565 09246 30271 873 0.54030 0.53946 0.53861 0.53777 0.53693 23058 68139 71740 094 05648 72446 55654 214 82844 16233 47828 237 54653 41780 86864 465 21084 91907 73184 669 1.005 1.006 1.007 1.008 1.009 0.84416 0.84469 0.84523 0.84576 0.84629 19667 15556 42661 273 76327 59970 18177 851 24541 06821 56844 116 64302 21289 28431 774 95605 69397 25943 853 0.53608 0.53524 0.53439 0.53355 0.53270 82147 09970 84748 188 37848 39863 92716 262 88197 26016 77062 668 33202 13394 42130 747 72871 47496 32136 904 1.010 1.011 1.012 1.013 1.014 0.84683 0.84736 0.84789 0.84842 0.84895 18446 18015 19012 310 32818 34859 07211 051 38716 88491 73284 331 36136 48323 36290 466 25071 84612 04660 810 0.53186 0.53101 0.53016 0.52931 0.52846 07213 74355 46620 673 36237 40537 55841 426 59950 93140 16121 808 78362 79791 85137 984 91481 48651 37156 798 1.015 1.016 1.017 1.018 1.019 0.84948 0.85000 0.85053 0.85105 0.85158 05517 68464 29173 940 77468 71835 55845 003 40919 67530 78730 164 95865 29204 92646 111 42300 31363 45804 549 0.52761 0.52677 0.52591 0.52506 0.52421 99315 48406 78219 896 01873 28274 61274 932 99163 37999 01253 921 91194 27850 90098 832 77974 48627 11734 503 1.020 1.021 1.022 1.023 1.024 0.85210 0.85263 0.85315 0.85367 0.85419 80219 49362 92361 655 09617 59411 44882 415 30489 38569 26719 808 42829 64749 24308 778 46633 16717 39374 945 0.52336 0.52251 0.52166 0.52080 0.51995 59512 51649 56988 961 35816 88764 38461 245 06896 12341 05336 792 72758 75271 58150 502 33413 30969 63497 542 1.025 1.026 1.027 1.028 1.029 0.85471 0.85523 0.85575 0.85626 0.85678 41894 74093 41057 997 28609 17351 17949 715 06771 27819 30046 586 76375 87681 60616 931 37417 79977 67982 525 0.51909 0.51824 0.51738 0.51653 0.51567 88868 33369 68691 985 39132 36926 16373 373 84213 96612 59061 276 24121 67920 73657 956 58864 06859 75899 186 1.030 1.031 1.032 1.033 1.034 0.85729 0.85781 0.85832 0.85883 0.85935 89891 88603 37214 627 33792 98311 31744 398 69115 94711 44887 626 95855 64271 51283 734 14006 94317 58248 998 0.51481 0.51396 0.51310 0.51224 0.51138 88449 69955 34753 350 12887 14248 86768 878 32184 97296 50370 116 46351 77168 40101 715 55396 12447 80821 625 1.035 1.036 1.037 1.038 1.039 0.85986 0.86037 0.86088 0.86139 0.86189 23564 73034 57043 938 24523 89466 7.4054 819 16879 33518 21889 224 00625 95953 50385 634 75758 68397 97536 975 0.51052 0.50966 0.50880 0.50794 0.50708 59326 62230 21842 776 58151 86122 51023 535 51880 44242 08807 028 40520 97216 02209 404 24082 06180 18757 138 1.040 1.041 1.042 1.043 1.044 0.86240 0.86291 0.86341 0.86391 0.86442 42272 43338 40328 079 00162 14123 45486 997 49422 74964 20150 131 90049 20934 62441 124 22036 47972 11963 456 0.50622 0.50535 0.50449 0.50363 0.50276 02572 32778 40373 447 76000 39161 57213 919 44374 87986 81451 427 07704 42416 61010 426 65997 66117 93250 711 1.045 1.046 1.047 1.048 1.049 0.86492 0.86542 0.86592 0.86642 0.86692 45379 52878 00206 699 60073 33318 00866 385 66112 87822 80077 424 63493 15788 46561 037 52209 17477 01685 140 0.50190 0.50103 0.50017 0.49930 0.49843 19263 23261 38600 728 67509 78520 34140 520 10745 97070 07134 396 48980 44586 88513 415 82221 87247 26307 756 1.050 0.86742 32255 94016 89438 141 [(-;)I 1 0.49757 10478 91726 99029 085 c-8)7 7 II 1 ELEMENTARY CIRCULAR SINES TRANSCENDENTAL AND COSINES FOR RADIAN ARGUMENTS sin z 5 163 FUNCTIONS Table 4.6 cos x 1.050 1.051 1.052 1.053 1.054 0.86742 0.86792 0.86841 0.86891 0.86940 32255 03628 66321 20330 65651 94016 47403 80499 97035 01611 89438 46316 51123 74685 29477 141 092 146 276 198 0.49757 0.49670 0.49583 0.49496 0.49409 10478 33760 52074 65430 73836 91726 25200 55338 50311 78782 99029 29002 95651 48726 21490 085 975 499 051 510 1.055 1.056 1.057 1.058 1.059 0.86990 0.87039 0.87088 0.87137 0.87186 02276 30203 49427 59941 61741 99694 97621 02601 22711 66899 19162 88046 70443 39954 58660 460 624 529 543 794 0.49322 0.49235 0.49148 0.49061 0.48974 77302 75835 69444 58139 41927 09910 13349 59246 18239 61459 43854 55459 18707 31756 41446 806 008 979 732 534 1.060 1.061 1.062 1.063 1.064 0.87235 0.87284 0.87333 0.87381 0.87430 54823 39181 14811 81707 39866 44986 67663 46494 93918 23243 26228 28925 88556 11299 36468 295 947 345 356 402 0.48887 0.48799 0.48712 0.48625 0.48537 20818 94820 63943 28194 87582 60527 87554 15140 16372 64825 56191 58818 19351 07757 06632 864 317 528 202 362 1.065 1.066 1.067 1.068 1.069 0.87478 0.87527 0.87575 0.87623 0.87671 89281 29948 61863 85020 99415 48654 85211 48845 56366 25458 85179 08932 38105 30362 18969 424 453 753 492 874 0.48450 0.48362 0.48275 0.48187 0.48100 42117 91807 36660 76686 11893 34560 00124 36547 19345 24514 23847 05142 46667 07484 22014 867 311 387 800 811 1.070 1.071 1.072 1.073 1.074 0.87720 0.87768 0.87815 0.87863 0.87911 05042 01898 89976 69274 39784 74681 23471 92149 01900 74797 61030 85627 41877 46904 33716 706 336 919 963 111 0.48012 0.47924 0.47836 0.47749 0.47661 42290 67886 88689 04709 15953 28534 08365 41447 05700 79522 12436 01039 22529 36282 38551 509 904 904 289 762 1.075 1.076 1.077 1.078 1.079 0.87959 0.88006 0.88053 0.88101 0.88148 01504 54428 98551 33868 60376 33788 02703 06248 70011 20461 98997 50816 56244 88879 76291 101 869 731 619 297 0.47573 0.47485 0.47397 0.47309 0.47221 22432 24153 21126 13359 00861 41788 71851 49538 55152 69469 74632 50968 47223 28292 56273 160 911 840 396 392 1.080 1.081 1.082 1.083 1.084 0.88195 0.88242 0.88289 0.88336 0.88383 78068 86941 86990 78210 60596 84947 91699 69831 49337 61096 47373 79609 46247 63390 36998 533 169 031 660 790 0.47132 0.47044 0.46956 0.46868 0.46779 83641 61708 35070 03737 67717 73740 49685 79499 45845 31856 02391 58871 50767 47743 75803 353 547 810 217 727 1.085 1.086 1.087 1.088 1.089 0.88430 Oi88476 0.88523 0.88570 0.88616 34144 98849 54706 01710 39858 36869 09301 11921 79145 46272 09797 08104 88562 84791 53940 534 243 972 522 000 0.46691 0.46602 0.46514 0.46425 0.46337 27019 81651 31624 76945 17623 21135 97750 46239 51605 99315 28984 80991 96791 44159 05181 862 522 014 401 235 1.090 1.091 1.092 1.093 1.094 0.88662 0.88708 0.88755 0.88801 0.88846 69144 89564 01113 03786 97579 49487 25861 13352 50807 77956 23160 35990 98641 26207 88779 860 371 470 951 948 0.46248 0.46159 0.46071 0145982 0.45893 53668 85088 11892 34089 51687 75300 65958 58145 39181 96847 87702 36738 45833 68372 28855 790 852 190 764 783 1.095 1.096 1.097 1.098 1.099 0.88892 0.88938 0.88984 0.89029 0.89075 82488 58507 25633 83860 33184 35422 64713 08227 09252 11966 57470 50354 78315 90807 21527 660 274 047 488 609 0.45804 0.45715 0.45626 0.45537 0.45448 64697 73125 76983 76277 71018 19382 95485 14314 65483 39062 34113 84487 84956 56224 45757 686 142 158 382 688 1.100 0.89120 73600 61435 33995 180 (1 c -;I1 0.45359 61214 25577 38777 137 C-78)6 c1 164 ELEMENTARY Table 4.6 CIRCULAR SINES TRANSCENDENTAL AND COSINES FUNCTIONS FOR RADIAN cos x sin x 2 ARGUMENTS 1.100 1.101 1.102 1.103 1.104 0.89120 0.89166 0.89211 0.89256 0.89301 73600 61435 33995 180 05105 03618 67046 971 27692 85365 80240 901 41359 54417 99171 080 46100 59408 60693 678 0.45359 0.45270 0.45181 0.45092 0.45002 61214 25577 38777 137 46874 16008 69206 400 28007 01790 30573 730 04621 74808 86868 576 76727 27402 83352 928 1.105 1.106 1.107 1.108 1.109 0.89346 0.89391 0.89436 0.89480 0.89525 41911 49863 58063 585 28787 76201 85981 812 06724 89735 85553 594 75718 42671 89157 146 35763 88110 65223 027 0.44913 0.44824 0.44734 0.44645 0.44555 44332 52361 57327 478 07446 42924 48852 689 66077 92780 11424 866 20235 96065 22607 305 69929 47363 94616 628 1.110 1.111 1.112 1.113 1.114 0.89569 0.89614 0.89658 0.89702 0.89747 86856 80047 62924 063 28992 73373 56775 801 62167 23874 91147 427 86375 88234 24683 120 01614 24030 74633 785 0.44466 0.44376 0.44286 0.44197 0.44107 15167 41706 84864 374 55958 74570 06453 951 92312 41874 38633 030 24237 39984 37201 474 51742 65707 44874 890 1.115 1.116 1.117 1.118 1.119 0.89791 0.89835 0.89878 0.89922 0.89966 07877 89740 61099 138 05162 44737 51180 079 93463 49293 03041 321 72776 64577 09884 230 43097 52658 43829 826 0.44017 0.43927 0.43838 0.43748 0.43658 74837 16293 01603 891 93529 89431 54849 166 07829 83253 69812 438 17745 96329 39623 410 23287 27666 95482 777 1.120 1.121 1.122 1.123 1.124 0.90010 0.90053 0.90097 0.90140 0.90183 04421 76504 99711 910 56744 99984 38780 263 00062 87864 32313 880 34371 05813 05144 201 59665 20399 79088 276 0.43568 0.43478 0.43388 0.43298 0.43207 24462 76712 16761 399 21281 43347 41055 736 13752 27890 74199 612 01884 31095 00232 420 85686 54146 91323 845 1.125 1.126 1.127 1.128 1.129 0.90226 0.90269 0.90312 0.90355 0.90398 75940 99095 16291 842 83194 10271 62482 258 81420 23203 90131 256 70615 08069 41527 464 50774 35948 71758 658 0.43117 65167 98666 17655 197 0.43027 40337 66704 57257 452 0.42937 11204 60745 05806 078 0.42846 77777 83700 86372 749 0142756 40066 38914 59134 030 1.130 1.131 1.132 1.133 1.134 0.90441 0.90483 0.90526 0.90568 0.90611 21893 78825 91603 708 83969 09589 10334 160 36996 02030 78425 425 80970 30848 30177 523 15887 71644 26245 348 0.42665 0.42575 0.42485 0.42394 0.42303 98079 30157 31037 122 51825 61627 65422 763 01314 37950 91605 376 46554 64178 14410 540 87555 45785 23669 902 1.135 1.136 1.137 1.138 1.139 0.90653 0.90695 0.90737 0.90779 0.90821 41744 00926 96078 401 58534 96110 80269 960 66256 35516 72815 632 64903 98372 63281 260 54473 64813 78880 126 0.42213 0.42122 0.42031 0.41941 0.41850 24325 88672 03673 585 56874 99161 42580 219 85211 83998 41784 656 09345 50349 25243 478 29285 05800 48758 379 1.140 1.141 1.142 1.143 1.144 0.90863 0.90905 0.90946 0.90988 0.91029 34961 15883 26459 422 06362 33532 34395 940 68673 00620 94400 939 21889 00918 03234 153 66006 19102 04326 885 0.41759 0.41668 0.41577 0.41486 0.41395 45039 58358 09217 519 56618 16446 53794 933 64029 88907 89108 094 67283 85000 90333 707 66389 14400 10281 852 1.145 1.146 1.147 1.148 1.149 0.91071 0.91112 0.91153 0.91194 0.91235 01020 40761 29314 164 26927 52394 39475 912 43723 41410 67087 073 51403 96130 56676 684 49965 05786 06195 821 0.41304 0.41213 0.41122 0.41031 0.40940 61354 87194 88428 529 52190 13888 59906 732 38904 05397 64456 120 21505 73050 55331 381 00004 28587 08169 395 1.150 0.91276 39402 60521 08094 403 0.40848 74408 84157 29815 258 ELEMENTARY CIRCULAR SINES AND TRANSCENDENTAL COSINES FOR RADIAN ARGUMENTS sin x 5 165 FUNCTIONS Table 4.6 cos x 1.150 1.151 1.152 1.153 1.154 0.91276 0.91317 0.91357 Oi91398 0.91439 39402 19712 90890 52933 05835 60521 51391 70367 10330 65075 08094 90306 57146 30107 88579 403 792 165 602 865 0.40848 0.40757 0.40666 0.40574 0.40483 74408 44728 10972 73149 31269 84157 52320 46045 78706 64086 29815 67107 15621 28372 24481 258 284 071 706 224 1.155 1.156 1.157 1.158 1.159 0.91479 0.91519 0.91560 0.91600 0.91640 49594 84204 09663 25966 33109 29314 98669 69679 39800 07401 10465 12711 91743 63815 05261 816 431 383 143 556 0.40391 0.40300 0.40208 0.40117 0.40025 85341 35373 81375 23357 61326 16372 50159 80441 22620 92496 97790 25449 76456 20152 34689 397 945 266 779 958 1.160 1.161 1.162 1.163 1.164 0.91680 0.91720 0.91759 0.91799 0.91839 31087 19898 99536 69999 31282 71766 33100 92520 52063 14682 92661 42911 53200 40902 83374 866 136 023 883 147 0.39933 0.39842 0.39750 0.39658 0.39566 95294 25267 51257 73271 91320 06273 80553 32340 79035 38435 15445 83402 93491 42889 79278 164 355 775 706 377 1.165 1.166 1.167 1.168 1.169 0.91878 0.91918 0.91957 0.91996 0.92035 83380 26291 60010 84533 99857 84250 65556 64310 87139 41592 57652 80075 45798 68222 18336 941 906 178 492 360 0.39475 0.39383 0.39291 0.39199 0.39107 05412 15556 21762 24039 22396 28737 68530 76800 72926 76682 09066 05567 17146 75312 02789 125 898 187 486 366 1.170 1.171 1.172 1.173 1.174 0.92075 0.92114 0.92152 0.92191 0.92230 05977 02889 90590 69076 38343 36135 80158 83968 58796 16793 63957 08886 31967 26061 36915 301 071 851 369 902 0.39015 0.38923 0.38830 0.38738 0.38646 16843 07387 94040 76809 55705 08230 88126 37316 77135 29304 21533 60718 64679 00821 67479 266 072 599 054 575 1.175 1.176 1.177 1.178 1.179 0.92268 0.92307 0.92345 0.92384 0.92422 98386 49203 90789 23140 46253 71033 35513 25145 55777 44173 01956 88974 34733 83468 25312 127 783 097 944 701 0.38554 0.38462 0.38369 0.38277 0.38184 30736 01911 69240 32733 92397 15936 59525 82956 09495 62792 01753 87293 62048 25982 48743 942 547 718 487 902 1.180 1.181 1.182 1.183 1.184 0.92460 0.92498 0.92536 0.92574 0.92612 60124 64748 60123 46244 23108 08020 65932 37446 43024 04056 34610 08156 03329 76141 19188 754 619 642 242 645 0.38092 0.38000 0.37907 0.37814 0.37722 48243 00280 48517 92963 33627 66881 46178 25478 29958 85174 77302 43547 71840 86542 19493 960 271 534 917 444 1.185 1.186 1.187 1.188 1.189 0.92649 0.92687 0.92724 0.92762 0.92799 90710 49047 98116 37912 68432 42853 82657 47634 62876 54404 99516 96383 38942 43819 52606 095 480 352 290 588 0.37629 0.37537 0.37444 0.37351 0.37258 70520 03649 33025 58656 80552 17058 51921 16451 37709 43133 17454 49518 14476 48149 30684 471 342 334 962 752 1.190 1.191 1.192 1.193 1.194 0.92836 0.92874 0.92911 0.92947 0.92984 89672 01628 04297 97675 81758 49166 75038 60825 36260 32004 69260 97404 77546 24192 62877 202 950 899 928 403 0.37165 0.37073 0.36980 0.36887 0.36794 98722 13176 23922 30970 34330 60532 18091 44362 68273 19116 93806 28040 89893 08995 95213 568 589 026 672 382 1.195 1.196 1.197 1.198 1.199 0.93021 0.93058 0.93094 0.93131 0.93167 56542 22025 78201 25068 62622 79650 11719 61664 63866 53638 67095 95146 26876 00337 48349 956 303 083 679 974 0.36701 0.36608 0.36515 0.36422 0.36328 34010 30020 22369 11066 96122 26558 20629 31729 90622 28438 45714 52004 06923 11604 82399 570 819 698 876 631 1.200 0.93203 90859 67226 34967 013 [C-f)1 1 0.36235 77544 76673 57763 837 p-y51 L ’ J 166 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin x X ARGUMENTS cos x 1.200 1.201 1.202 1.203 1.204 0.93203 0.93240 0.93276 0.93312 0.93348 90859 67226 34967 013 09776 41805 91853 542 19369 15485 54567 367 19634 27305 98748 519 10568 17240 76215 175 0.36235 0.36142 0.36049 0.35956 0.35862 77544 76673 57763 837 55343 67184 05108 539 29528 32190 27614 189 00108 04273 71008 651 67092 16376 30309 065 1.205 1.206 1.207 1.208 1.209 0.93383 92167 26196 50966 302 0.93419 64427 96013 35090 992 Oi93455 27346 69465 24584 444 0.93490 80919 90260 35070 567 0.93526 25144 03041 37431 162 0.35769 0.35675 0.35582 0.35488 0.35395 30490 01799 56527 660 90310 94203 63341 607 46564 27606 33727 018 99259 36382 26557 166 48405 55261 83165 039 1.210 1.211 1.212 1.213 1.214 0.93561 0.93596 0.93632 0.93667 0.93702 60015 53385 93341 646 85530 87806 90713 291 01686 53752 79041 926 08478 99608 04663 095 05904 74693 45913 598 0.35301 0.35208 0.35114 0.35021 0.34927 94012 19330 33870 301 36088 64027 04470 775 74644 25144 22698 521 09688 38826 24640 616 41230 41568 61124 730 1.215 1.216 1.217 1.218 1.219 0.93736 0.93771 0.93806 0.93841 0.93875 93960 29266 48199 416 72642 14521 58969 959 41946 82590 62598 617 01870 86543 15169 574 52410 80386 79170 848 0.34833 0.34739 0.34646 0.34552 0.34458 69279 70217 04069 578 93845 61966 52800 358 14937 54360 40329 260 32564 85289 39601 140 46736 92990 69704 455 1.220 1.221 1.222 1.223 1.224 0.93909 0.93944 0.93978 0.94012 0.94046 93563 19067 58093 524 25324 58470 30937 151 47691 55418 86621 257 60660 67676 58302 957 64228 53946 57600 622 0.34364 0.34270 0.34176 0.34082 0.33988 57463 16047 02047 552 64752 93385 66500 405 68615 64277 57501 890 69060 68336 40132 702 66097 45517 56153 996 1.225 1.226 1.227 1.228 1.229 0.94080 0.94114 0.94148 0.94181 0.94215 58391 73872 08723 559 43146 88036 82507 685 18490 57965 30357 157 84419 46123 18091 912 40930 15917 59701 104 0.33894 0.33800 0.33706 0.33612 0.33518 59735 36117 30011 855 49983 80771 74807 668 36852 20455 98234 533 20349 96483 08479 750 00486 50503 20093 523 1.230 1.231 1.232 1.233 1.234 0.94248 0.94282 0.94315 0.94348 0.94381 88019 31697 51002 382 25683 58754 03206 998 53919 63320 76390 684 72724 12574 12870 299 82093 74633 70486 175 0.33423 0.33329 0.33235 0.33140 0.33046 77271 24502 59823 955 50713 60802 72418 427 20823 02059 26391 462 87608 91261 19759 164 51080 71729 85740 328 1.235 1.236 1.237 1.238 1.239 0.94414 0.94447 0.94480 0.94513 0.94545 82025 18562 55790 164 72515 14367 57139 322 53560 32999 77695 223 25157 46354 68328 851 87303 27272 60431 046 0.32952 0.32857 0.32763 0.32668 0.32574 11247 87117 98424 316 68119 81408 78405 786 21705 98914 98386 387 72015 84277 88743 487 19058 82466 43066 054 1.240 1.241 1.242 1.243 1.244 0.94578 0.94610 0.94643 0.94675 0.94707 39994 49538 98628 471 83227 87884 73405 063 17000 17986 53628 942 41308 16467 18984 738 56148 60895 92311 309 0.32479 0.32385 0.32290 0.32195 0.32101 62844 38776 23657 769 03381 98828 67007 475 40681 08569 89227 042 74751 14269 91456 764 05601 62521 65238 364 1.245 1.246 1.247 1.248 1.249 0.94739 0.94771 0.94803 0.94835 0.94866 61518 29788 71844 815 57414 02608 63367 118 43832 59766 12259 472 20779 82619 35461 479 88225 53474 53335 262 0.32006 0.31911 0.31816 0.31721 0.31627 33242 00239 97855 712 57681 74660 77643 341 78930 33339 99262 871 96997 24152 68947 423 11891 95292 09714 116 1.250 0.94898 46193 55586 21434 849 0.31532 23623 95268 66544 754 [c-y1 ELEMENTARY CIRCULAR SINES AND X TRANSCENDENTAL COSINES FOR 167 FUNCTIONS RADIAN ARGUMENTS sin x Table 4.6 cos x 1.250 1.251 1.252 1.253 1.254 0.94898 0.94929 0.94961 0.94992 0.95023 46193 94671 33656 63145 83135 55586 73157 91340 96237 74899 21434 62180 96439 75008 10006 849 713 444 528 196 0.31532 0.31437 0.31342 0.31247 0.31152 23623 32202 37637 39938 39114 95268 72909 77355 58064 64805 66544 11534 49010 20615 10363 754 791 665 601 979 1.255 1.256 1. 257 1.258 1.259 0.95054 0.95085 0.95116 0.95147 0.95178 93623 94605 86078 68040 40486 15326 06469 38232 01466 87975 06166 92038 51091 52726 83188 303 225 729 783 287 0.31057 0.30962 0.30867 0.30772 0.30676 35175 28130 17989 04761 88456 47660 57024 43600 58403 52756 49664 22311 69445 94485 68021 355 242 729 052 196 1.260 1.261 1.262 1.263 1.264 0.95209 0.95239 0.95270 0.95300 0.95330 03415 56824 00708 35065 59892 90515 02793 19468 36151 49407 76385 44617 09200 31003 40886 682 416 227 222 709 0.30581 0.30486 0.30391 0.30295 0.30200 69083 46652 21173 92654 61106 78289 86939 30948 62866 35544 32688 08001 95158 81822 46859 634 291 833 373 693 1.265 1.266 1.267 1.268 1.269 0.95360 0.95390 0.95420 0.95450 0.95480 75186 80944 77163 63840 40972 56753 56660 48552 32808 10759 70045 80258 94039 24694 06289 767 512 032 963 671 0.30105 0.30009 0.29914 0.29819 0.29723 26538 88959 48379 04808 58254 02136 16100 31192 01472 81295 65060 11818 67791 23518 84019 070 814 595 675 121 1.270 1.271 1.272 1.273 1.274 0.95510 0.95539 0.95569 0.95598 0.95627 08555 66588 15067 53989 83351 84692 57849 34427 19578 19409 23509 41432 35209 19640 78657 018 673 944 104 170 0.29628 0.29532 0.29437 0.29341 0.29245 08729 56240 00799 42413 81094 25318 88493 26068 93588 46891 73355 39166 57175 35661 19906 114 425 182 000 579 1.275 1.276 1.277 1.278 1.279 0.95657 0.95686 0.95715 0.95744 0.95772 03150 13383 14048 05142 86661 40985 92326 82408 21166 19488 94719 78101 96095 02109 64678 118 497 419 886 437 0.29150 0.29054 0.28958 0.28863 0.28767 16850 49691 79626 06666 30819 42108 35665 84278 44951 74982 96613 98290 07609 61732 56616 869 890 308 860 726 1.280 1.281 1.282 1.283 1.284 0.95801 0.95830 0.95858 0.95887 0.95915 58602 20964 73742 16935 50539 89224 43180 95120 59764 52796 96370 82604 10371 96853 17957 075 453 286 962 320 0.28671 0.28575 0.28479 0.28383 0.28288 52096 70505 86057 98761 08626 31955 73742 58503 44681 91007 51277 72036 16730 58895 51923 939 934 332 050 831 1.285 1.286 1.287 1.288 1.289 0.95943 0.95971 0.95999 0.96027 0.96055 74551 88969 93790 89011 74629 90853 91535 73400 55966 59710 36739 31748 25260 11427 84322 577 357 814 805 094 0.28192 0.28096 0.28000 0.27904 Oi27808 15663 19881 21288 19896 15714 56494 00438 82417 62291 00198 33192 28157 54428 25809 56310 303 651 993 577 871 1.290 1.291 1.292 1.293 1.294 0.96083 0.96111 0.96138 0.96166 0.96193 50642 17046 73839 21018 58580 06072 17450 17203 29652 80080 65890 33810 49249 84528 50693 556 354 056 675 590 0.27712 0.27615 0.27519 3.27423 0.27327 08750 99015 86519 71271 53280 56557 92064 67693 44692 84588 64138 75651 29289 79480 00512 661 234 769 997 263 1.295 1.296 1.297 1.298 1.299 0.96220 0.96248 0.96275 0.96302 0.96329 86523 04845 13541 12610 02048 94730 00807 26481 00880 54098 24982 78203 02013 36103 95282 339 231 782 915 920 0.27231 0.27135 0.27038 0.26942 0.26846 32557 09111 82951 54087 22529 49177 00534 01003 13198 00008 90379 74605 10035 88600 41057 053 108 206 711 992 1.300 / 0.96355 81854 17192 96470 135 [c-y1 0.26749 88286 24587 40699 798 C-7834 L1 168 ELEMENTARY Table 4.6 CIRCULAR x TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin 2: ARGUMENTS CO8 2 1.300 1.301 1.302 1.303 1.304 0.96355 81854 17192 96470 135 0.96382 52024 22181 85589 331 0.96409 12556 02048 64366 761 0.96435 63446 90740 17032 855 0796462 04694 23167 36927 537 0.26749 0.26653 0.26557 0.26460 0.26364 88286 24587 40699 798 51368 50360 07039 695 11785 41018 09469 650 69546 60519 70890 877 24661 73088 71318 016 1.305 1.306 1.307 1.308 1.309 0.96488 0.96514 0.96540 0.96566 0.96592 36295 35205 53009 126 58247 63694 56266 806 70548 46439 26036 635 73195 22209 56221 061 66185 30740 81411 924 0.26267 0.26171 0.26074 0.25978 0.25881 77140 43213 51456 761 26992 35646 16255 031 74227 15401 38427 774 18854 47755 61955 494 60883 98246 05556 626 1.310 1.311 1.312 1.313 1.314 0.96618 0.96644 0.96669 0.96695 0.96720 49516 12734 02916 926 23185 09856 14689 520 87189 64740 29162 218 41527 20986 02983 276 86195 23159 62656 736 0.25785 0.25688 0.25591 0.25495 0.25398 00325 32669 66133 818 37188 17082 22194 242 71482 17797 37244 030 03217 01385 63156 911 32402 34673 43517 173 1.315 1.316 1.317 1.318 1.319 0.96746 0.96771 0.96796 0.96821 0.96846 21191 16794 3gO85 794 46512 48390 48019 478 62156 65416 05402 607 68121 16306 62628 991 64403 50465 76697 879 0.25301 0.25204 0.25108 0.25011 0.24914 59047 84742 16937 022 83163 18927 20348 457 04758 04816 92269 738 23842 10251 76046 556 40425 03323 23067 996 1.320 1.321 :* 2: 1:324 0.96871 0.96896 0.96920 0.96945 0.96970 51001 18265 26273 590 27911 71045 36648 340 95132 61115 04608 211 52661 41752 23202 252 00495 67204 06414 685 0.24817 0.24720 0.24623 0.24526 0.24429 54516 52372 95957 398 66126 25991 71738 199 75263 93018 44974 865 81939 22539 30889 004 86161 83886 68450 760 1.325 1.326 1.327 1.328 1.329 0.96994 0.97018 0.97042 0.97066 0.97090 38632 92687 13740 188 67070 74387 74662 236 85806 69462 13034 465 94838 36036 71365 051 94163 33208 35004 060 0.24332 0.24235 0.24138 0.24041 0.23944 87941 46638 23445 582 87287 80615 91516 463 84210 55885 01181 759 78719 42753 16828 662 70824 11769 41682 448 1.330 1.331 1.332 1.333 1.334 0.97114 0.97138 0.97162 0.97185 0.97209 83779 21044 56233 768 63683 60583 78261 900 33874 13835 59117 786 94348 43780 95451 405 45104 14372 46235 282 0.23847 0.23750 0.23653 0.23556 0.23458 60534 33723 20751 578 47859 79643 43748 768 32810 20797 47988 097 15395 28690 21258 288 95624 75063 04672 221 1.335 1.336 1.337 1.338 1.339 0.97232 0.97256 0.97279 0.97302 0.97325 86138 90534 56369 230 17450 38163 80187 900 39036 24129 04871 129 50894 16271 73757 0146 53021 83406 09557 931 0.23361 0.23264 0.23167 0.23069 0.22972 73508 31892 95492 805 49055 71391 49935 286 22276 66003 85946 099 93180 88407 85958 358 61778 11512 99624 085 1.340 1.341 1.342 1.343 1.344 0.97348 0.97371 0.97394 0.97416 0.97439 45416 95319 37478 787 28077 22772 08238 616 01000 37498 N994 365 64184 12205 46167 522 17626 20575 48173 349 0.22875 0.22777 0.22680 0.22583 0.22485 28078 08459 46523 264 92090 52617 18849 831 53825 17584 84074 691 13291 77188 87585 859 70500 05482 55305 819 1.345 1.346 1.347 1.348 1.349 0.97461 0.97483 0.97506 0.97528 0.97550 61324 37264 08052 713 95276 37901 46006 501 19479 99092 43832 603 33932 98416 67265 423 38633 14428 88217 916 0.22388 0.22290 0.22193 0.22095 0.21998 25459 76744 96286 212 78180 65480 05279 929 28672 46415 65290 729 76944 94502 50100 463 23007 84913 26774 007 1.350 0.97572 33578 26659 06926 111 (-;I1 [1 0.21900 66870 93041 58142 002 c-p3 II 1 ELEMENTARY CIRCULAR SINES AND x TRANSCENDENTAL COSINES FOR 169 FUNCTIONS RADIAN ARGUMENTS sin x Table 4.6 cos x 1.350 1.351 1.352 1.353 1.354 0.97572 0.97594 0.97615 0.97637 0.97659 33578 18766 94194 59861 15764 26659 15612 62771 50591 62506 06926 73996 12353 39095 87244 111 110 536 407 418 0.21900 0.21803 0.21705 0.21607 0.21510 66870 08543 48036 85358 20520 93041 94501 65124 80961 18281 58142 05261 29854 96725 76154 002 504 627 291 163 1.355 1.356 1.357 1.358 1.359 0.97680 0.97701 0.97723 0.97744 0.97765 61901 98270 24869 41696 48748 82927 97238 91804 53965 72037 27405 89325 83352 21803 40225 609 386 894 706 805 0.21412 0.21314 0.21217 0.21119 0.21021 53530 84399 13137 39753 64257 53567 63517 25046 15278 11553 46271 95410 24434 49048 02083 899 772 790 406 908 1.360 1.361 1.362 1.363 1.364 0.97786 0.97807 0.97828 0.97848 0.97869 46024 33521 11237 79171 37319 35316 34074 59561 04006 60615 18567 02249 23135 20406 61343 849 690 125 864 685 0.20923 0.20826 0.20728 0.20630 0.20532 86658 06968 25195 41349 55440 91419 32637 13175 11211 05130 35767 23964 64404 80880 25435 598 842 112 089 952 1.365 1.366 1.367 1.368 1.369 0.97889 0.97910 0.97930 0.97950 0.97970 85681 24253 53035 72024 81217 23574 88047 50175 07082 56868 61999 07786 73954 45982 39862 774 196 516 521 027 0.20434 0.20336 0.20238 0.20140 0.20042 67477 77471 85432 91369 95291 73521 95182 49113 14517 70801 80524 61151 16990 34489 38946 932 240 457 495 217 1.370 1.371 1.372 1.373 1.374 0.97990 0.98010 0.98030 0.98050 0.98069 80613 70211 50007 20000 80189 98614 32380 59206 81114 01103 22288 30754 93540 49613 68424 769 328 094 233 652 0.19944 0.19846 0.19748 0.19650 0.19552 97209 97133 95072 91036 85036 97572 74640 82010 99890 08682 96568 16515 52911 06852 28380 820 079 545 798 853 1.375 1.376 1.377 1.378 1.379 0.98089 0.98108 0.98128 0.98147 0.98166 30570 71142 01903 22852 33986 23155 52232 94276 56212 45944 69608 42586 66065 27452 42153 920 155 826 479 343 0.19454 0.19356 0.19258 0.19160 0.19062 77079 67178 55340 41577 25898 88987 21600 87511 67905 44156 18444 30840 74132 13553 72884 822 918 912 129 094 1.380 1.381 1.382 1.383 1.384 0.98185 0.98204 0.98223 0.98241 0.98260 35303 26802 08480 80336 42368 72359 45326 75694 75296 56947 72787 48298 82965 95320 26961 813 791 850 221 571 0.18964 0.18865 0.18767 0.18669 0.18571 08312 88831 67462 44217 19105 97834 10696 64691 41955 24813 36320 50314 25395 37980 32156 915 508 757 715 930 1.385 1.386 1,387 1.388 1.389 0.98278 0.98297 0.98315 0.98333 0.98352 94574 36952 69500 92216 05100 34442 22562 37068 94707 13205 61276 42059 92032 31273 95537 561 162 708 673 148 0.18472 0.18374 0.18276 0.18178 0.18079 92135 63119 32665 00183 65884 95776 37540 32988 65185 17379 21451 90577 97169 73489 28124 016 542 360 451 404 1.390 1.391 1.392 1.393 1.394 0.98370 0.98388 0.98405 0.98423 0.98441 08148 01359 84731 58262 21952 11276 08614 25898 84790 07939 54484 29809 13274 84637 29485 004 722 870 207 405 0.17981 0.17882 0.17784 0.17686 0.17587 29776 91871 52177 10704 67464 72999 15656 29142 97424 04651 47659 98336 27690 66173 28756 616 311 484 860 976 1. 39-5 1.396 1.397 1.398 1.399 0.98458 0.98476 0.98493 0.98510 0.98527 75797 19796 53948 78250 92701 18974 42512 04152 30479 49063 56974 17462 20048 50013 86162 360 083 145 670 846 0.17489 0.17390 0.17292 0.17193 0.17095 22464 75715 27228 77011 25074 35146 73409 04115 12112 82423 16514 18192 11759 65937 41718 467 681 690 830 833 1.400 0.98544 97299 947 0.16996 71429 00240 93861 675 [ (-/I1 1 88460 18065 II 1 (-;j3 170 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN cos x sin z 1: ARGUMENTS 1.400 1.401 1.402 1.403 1.404 0.98544 0.98561 0.98578 0.98595 0.98612 97299 88460 18065 947 92043 78208 63203 840 76931 48834 84013 966 51961 31850 04837 776 17131 59751 28769 609 0.16996 0.16898 0.16799 0.16701 0.16602 71429 00240 93861 675 16083 50929 72373 233 59048 20024 23971 842 00332 93227 93533 854 39947 56412 25523 303 1.405 1.406 1.407 1.408 1.409 0.98628 0.98645 0.98661 0.98677 0.98693 72440 66021 54406 982 17886 85129 92502 294 53468 52531 82515 912 79184 04669 09070 631 95031 78970 18307 486 0.16503 0.16405 0.16306 0.16207 0.16109 77901 95615 65404 770 14205 97042 61039 544 48869 47062 64065 184 81902 32209 31258 571 13314 39179 25882 568 1.410 1.411 1.412 1.413 1.414 0.98710 0.98725 0.98741 0.98757 0.98773 01010 13850 34142 909 97117 48711 74427 198 83352 23943 67004 304 59712 80922 65672 895 26197 62012 66048 706 0.16010 0.15911 0.15812 0.15714 0.15615 43115 54831 19016 356 71315 66184 90869 577 97924 60420 32080 359 22952 24876 44997 336 46408 47050 44945 751 1.415 1.416 1.417 1.418 1.419 0.98788 0.98804 0.98819 0.98834 0.98850 82805 10565 21328 142 29533 70919 57953 120 66381 88402 91177 144 93348 09330 40532 586 10430 81005 45199 170 0.15516 0.15417 0.15319 0.15220 0.15121 68303 14596 61477 752 88646 15325 39606 967 07447 37202 41027 471 24716 68347 45317 231 40463 97033 51126 135 1.420 1.421 1.422 1.423 1.424 0.98865 0.98880 0.98895 0.98909 0.98924 17628 51719 79273 627 14939 70753 66940 521 02362 88375 97544 222 79896 55844 40562 021 47539 25405 60478 351 0.15022 0.14923 Oil4824 0.14725 0.14626 54699 11685 77348 698 67432 00880 64281 559 78672 53344 74765 840 88430 57953 95314 499 96716 03732 37224 747 1.425 1.426 1.427 1.428 1.429 0.98939 0.98953 0.98967 0.98982 0.98996 05289 50295 31560 129 53145 84738 52533 174 91106 83949 61159 714 19171 04132 48716 941 37337 02480 74376 619 0.14528 0.14429 0.14330 0.14231 0.14132 03538 79851 37675 648 08908 75628 60810 986 12835 80526 98807 514 15329 84153 72928 666 16400 76259 34563 848 1.430 1.431 1.432 1.433 1.434 0.99010 0.99024 0.99038 0.99052 0.99065 45603 37177 79485 729 43968 67397 01748 121 32431 53301 89307 176 10990 56046 14729 460 79644 37773 88889 346 0.14033 0.13934 0.13835 0.13736 0.13637 16058 46736 66253 390 14312 85619 82699 275 11173 83083 31761 733 06651 29440 95441 799 00755 15144 90849 940 1.435 1.436 1.437 1.438 1.439 0.99079 0.99092 0.99106 0.99119 0.99132 38391 61619 74754 605 87230 91709 01072 941 26160 93157 75959 459 55180 32073 00385 060 74287 75552 81565 735 0.13537 0.13438 0.13339 0.13240 0.13141 93495 30784 71160 849 84881 67086 26554 495 74924 14910 85143 546 63632 65254 13887 244 51017 09245 19491 852 1.440 1.441 1.442 1.443 1.444 0.99145 0.99158 0.99171 0.99184 0.99197 83481 91686 46252 760 82761 49554 53923 766 72125 19229 09874 676 51571 71773 78212 505 21099 79243 94748 990 0.13042 0.12943 0.12844 0.12744 0.12645 37087 38145 49297 752 21853 43347 92153 306 05325 16375 79275 576 87512 48881 85098 002 68425 32647 28105 135 1.445 1.446 1.447 1.448 1.449 0.99209 0.99222 0.99234 0.99247 0.99259 80708 14686 79795 055 30395 52141 50856 088 70160 66639 35228 024 00002 34203 82494 216 19919 31850 76923 086 0.12546 0.12447 0.12348 0.12248 0.12149 48073 59580 71654 525 26467 21717 24785 871 03616 11217 43017 513 79530 20366 29130 391 54219 41572 33939 548 1.450 0.99271 29910 37588 49766 535 c 1 -y1 0.12050 27693 67366 57053 287 (-92 [1 ELEMENTARY CIRCULAR SINES .4ND TRANSCENDENTAL COSINES FOR RADIAN ARGUMENTS sin x X 171 FTJNCTIONS Table 4.6 cos x 1.450 1.451 1.452 1.453 1.454 0.99271 0.99283 0.99295 0.99307 0.99318 29910 37588 49766 535 29974 30417 91459 118 20109 90332 63717 946 00315 98319 11543 325 70591 36356 75120 114 0.12050 0.11950 0.11851 0.11752 0.11653 27693 67366 57053 287 99962 90401 47620 080 71037 03450 05063 327 40925 99404 79804 068 09639 71276 73971 735 1.455 1.456 1.457 1.458 1.459 0.99330 0.99341 0.99353 0.99364 0.99375 30934 87418 01619 777 81345 35468 56903 143 21821 65467 37123 830 52362 63366 80232 355 72967 16112 77380 893 0.11553 0.11454 0.11355 0.11255 0.11156 77188 12194 42103 061 43581 15402 91829 237 08828 74262 84551 407 72940 82249 36104 618 35927 32951 17410 313 1.460 1.461 1.462 1.463 1.464 0.99386 0.99397 0.99408 0.99419 0.99430 83634 11644 84228 683 84362 38896 32148 075 75150 87794 39331 194 55998 49260 21797 223 26904 15209 04300 286 0.11056 0.10957 0.10858 0.10758 0.10659 97798 20069 55117 465 58563 37417 32232 463 18232 78917 88737 835 76816 38604 22199 915 34324 10617 88365 556 1.465 1.466 1.467 : . tb6: 0.99440 0.99451 0.99461 0.99472 0.99482 87866 78550 31137 923 38885 33187 76860 141 79958 74019 56879 043 11085 96938 37979 012 32265 98831 48727 437 0.10559 0.10460 0.10361 0.10261 0.10162 90765 89208 01747 983 46151 68730 36201 884 00491 43646 25487 846 53795 08521 63826 230 06072 58026 06440 584 1.470 1.471 1.472 1.473 1.474 0.99492 0.99502 0.99512 0.99522 0.99531 43,49777580 89785 993 44780 32063 44122 430 36112 62150 87122 898 17493 68709 96604 762 88922 53602 62729 932 0.10062 57333 86931 70090 698 0.09963 07588 90112 33595 391 0.09863 56847 62542 38345 147 0.09764 05119 99295 88804 678 OiO9664 52415 95545 53005 525 1.475 1.476 1.477 1.478 1.479 0.99541 0.99551 0.99560 0.99569 0.99578 50398 19685 97818 664 01919 70812 46063 854 43486 11829 93145 787 75096 48581 75747 356 96749 87906 90969 720 0.09564 0.09465 0.09365 0.09266 0.09166 98745 46561 63028 806 44118 47711 15478 186 88544 94456 71943 189 32034 82355 59452 948 74598 07058 70920 484 1.480 1.481 1.482 1.483 1.484 0.99588 0.99597 0.99606 0.99614 0.99623 08445 37640 05648 408 10182 06611 65569 851 01959 04648 04588 337 83775 42571 53643 374 55630 32200 49677 461 0.09067 0.08967 0.08867 0.08768 0.08668 16244 64309 65577 623 56984 49943 69400 641 96827 59886 75526 752 35783 90154 44661 519 73863 36851 05477 303 1.485 1.486 1.487 1.488 1.489 0.99632 0.99640 0.99649 0.99657 0.99665 17522 86349 44454 246 69452 18829 13277 079 11417 44446 63607 933 43417 79005 43586 693 65452 39305 50450 815 0.08569 0.08469 0.08369 0.08270 0.08170 11075 96168 55002 845 47431 64385 59004 070 82940 37866 52356 240 17612 13060 39407 518 51456 86499 94334 076 1.490 1.491 1.492 1.493 1.494 0.99673 0.99681 0.99689 0.99697 0.99705 77520 43143 38855 320 79621 09312 29093 143 71753 57602 15215 811 53917 08799 73054 448 26110 84688 68141 099 0.08070 0:0797i 0.07871 0.07771 0.07672 84484 54800 61486 832 i6705 i4659 55729 907 48128 62854 62770 926 78764 96243 39483 234 08624 11762 14220 152 1.495 1.496 1.497 :*. t;: 0.99712 ii99720 0.99727 0.99735 0.99742 88334 08049 63530 364 40586 02660 27521 334 62865 93295 41279 821 15173 05727 06360 877 37506 66724 52131 595 0.07572 0.07472 0.07372 0.07273 0.07173 37716 06424 87121 354 66050 77322 30411 478 93638 21620 88691 060 20488 36561 79219 898 46611 19459 92192 943 1.500 0.99749 49866 04054 43094 172 r’-,7”1 L ’ -I 0.07073 72016 67702 91008 819 C-7812 [1 172 ELEMENTARY Table 4.6 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR RADIAN sin x X ARGUMENTS cos x 1.500 1.501 1.502 1.503 1.504 0.99749 0.99756 0799763 0.99770 0.99776 49866 52250 44659 27091 99547 04054 46480 23765 66667 06942 43094 86109 37519 10173 80349 172 251 509 501 750 0.07073 0.06973 0.06874 0.06774 0.06674 72016 96714 20715 44028 66664 67702 78750 50131 79447 64365 91008 12531 67342 39990 89231 819 065 208 761 245 1.505 1.506 1.507 1.508 1.509 0.99783 0.99790 0.99796 0.99802 0.99809 62024 14524 57044 89585 12145 77346 11631 44547 11841 50260 94581 76379 32859 61264 55394 063 092 104 976 397 0.06574 0.06475 0.06375 0.06275 0.06175 88633 09943 30607 50633 70031 02623 92023 30434 15789 46086 48257 24928 01988 37280 63952 343 268 470 758 953 1.510 1.511 1.512 2.513 1.514 0.99815 0.99821 0.99827 0.99833 0.99838 24724 27322 19938 02571 75221 97548 92446 74695 85033 65198 11924 36636 50542 95912 42198 274 332 912 947 118 0.06075 0.05976 0.05876 0.05776 0.05676 88812 06985 24560 41548 57959 19385 33809 87538 78816 05945 90658 01748 57464 94113 24248 160 769 281 053 072 1.515 1.516 1.517 1.518 1.519 0.99844 0.99849 0.99855 0.99860 0.99865 37887 90569 33265 65976 88701 57923 06943 56990 53793 44081 91859 86092 10456 00399 46683 188 495 612 163 784 0.05576 0.05476 0.05377 0.05277 0.05177 73801 89086 03823 18023 31695 67282 61243 86301 40981 23862 36836 97425 48297 08625 74620 851 545 399 609 716 1.520 1.521 1.522 1.523 1.524 0.99871 0.99876 0.99880 0.99885 0.99890 01439 04190 96954 79730 52517 75583 97023 58128 09621 03224 00717 79776 72136 42098 34913 231 634 872 089 328 0.05077 0.04977 0.04877 0.04777 0.04677 44849 57495 69644 81305 92488 33579 68814 28305 10835 15238 19672 94487 27218 23593 67036 613 284 360 598 388 1.525 1.526 1.527 1.528 1.529 0.99895 0.99899 0.99904 0.99908 0.99912 15314 68123 10941 43769 66606 91658 28645 68902 68148 83100 81616 03749 17990 40684 92265 285 180 729 234 762 0.04578 0.04478 0.04378 0.04278 0.04178 03203 13460 23270 32642 41586 40397 85239 48738 29915 27830 18782 17991 81854 05695 63073 371 291 166 871 262 i.530 1.531 1.532 1.533 1.534 0.99916 0.99920 0.99924 0.99928 0.99932 79452 82306 75169 58038 30915 71476 91989 04354 69286 48498 01592 10170 76285 79026 22220 427 755 152 436 463 0.04078 0.03978 0.03878 0.03778 0.03678 50112 58230 65951 73283 80238 41591 70343 13276 69617 38633 05868 64380 47406 42326 15178 899 513 277 008 390 1.535 1.536 1.537 1.538 1.539 0;99935 0.99939 0.99942 0.99946 0.99949 93799 46689 89585 22486 45393 04701 01607 03928 77374 88654 38256 91817 83506 53376 84360 819 592 202 306 752 0.03578 0.03478 0.03378 0.03279 0.03179 86825 93054 98935 04478 09693 19628 11943 14956 28078 50755 10734 52566 43115 63750 74831 312 435 073 505 796 1.540 1.541 1.542 1.543 1.544 0.99952 0:99955 0.99958 0.99961 0.99964 58306 61222 54144 37069 09999 05479 96555 31593 81300 17383 05600 95674 85726 62497 71251 596 180 242 095 832 0.03079 0.02979 0.02879 0.02779 0.02679 14590 19180 23471 27475 31200 82466 22720 71058 27051 90300 15762 05041 40314 98418 35423 248 568 858 526 217 1.545 1.546 1.547 1.548 1.549 0.99966 0.99969 0.99971 0.99974 0.99976 72932 25868 68807 01749 24694 12550 40506 75959 94615 73179 18609 75272 78656 35418 23886 586 821 660 249 150 0.02579 0.02479 0.02379 0.02279 0.02179 34658 37858 40810 43524 46010 60430 37097 19980 08784 03238 86673 66826 69885 69229 17647 867 971 184 328 934 1.550 0.99978 37641 89356 96389 761 [(-;)I 1 0.02079 48278 03092 47364 391 [(-;I9 1 ELEMENTARY CIRCULAR SINES AND TRANSCENDENTAL COSINES FOR RADIAN FUNCTIONS 173 ARGUMENTS Table 4.6 .c 1.550 1.551 1.552 1.553 1.554 0.99978 0.99980 0.99982 0.99984 0.99985 37641 40591 33542 16495 89450 sin x 89356 21853 50374 55624 19308 96389 81488 86102 97539 85428 761 767 606 966 298 0.02079 0.01979 0.01879 0.01779 0.01679 48278 50338 52200 53874 55370 03092 08120 18116 32894 52286 47364 70061 76905 38564 05229 391 827 802 929 507 1.555 1,556 1.557 1.558 1.559 0.99987 0.99989 0.99990 0.99991 0.99993 52406 05363 48321 81281 04241 24131 53795 93 07 279 470 43888 03543 91538 76575 74851 93030 342 676 277 093 623 0.01579 56698 0.01479 57869 0.01379 58891 0.01279 59775 0.01179,60532 76142 04329 36731 73245 13782 06628 52043 30323 09896 38778 284 433 849 874 533 1.560 1.561 1.562 1.563 1.564 0.99994 0.99995 0.99996 0.99996 0.99997 17202 20163 13125 96087 69050 29966 74406 66914 98192 59945 29574 75969 17856 36062 07529 517 172 344 758 731 0.01079 0.00979 0.00879 0.00779 0.00679 61170 61701 62133 62478 62744 58267 06636 58835 14822 74562 44582 34527 95443 93777 75597 392 146 014 062 546 1.565 1.566 1.567 1.568 1.569 0.99998 0.99998 0.99999 0.99999 0.99999 32013 84976 27939 60902 83866 44876 46689 60087 80775 05456 06142 03461 69348 72499 80873 794 318 142 201 162 0.00579 0.00479 0.00379 0.00279 0.00179 62943 63084 63176 63231 63258 38028 05200 76064 50611 28835 66597 72096 77045 46023 23243 372 784 359 436 059 1.570 1.571 1.572 1.573 1.574 0.99999 0.99999 0.99999 0.99999 0.99999 96829 99792 92755 75719 48682 31834 58612 85495 13185 43386 62021 83315 12082 15626 61164 053 895 337 285 539 +0.00079 -0.00020 -0.00120 -0.00220 -0.00320 63267 36732 36729 36714 36677 10733 03695 14450 21533 24944 32548 22583 59042 14087 45343 541 254 804 901 613 1.575 1.576 1.577 1.578 1.579 0.99999 0.99998 0.99998 0.99997 0.99996 11645 64609 07572 40536 63500 78803 23138 81096 58379 61693 15654 45523 16298 92137 35254 423 419 798 261 568 -0.00420 -0.00520 -0.00620 -0.00720 -0.00820 36608 36497 36334 36109 35811 24688 20771 13205 02006 87197 30802 68822 78129 97812 87324 109 280 029 142 647 1.580 1.581 1.582 1.583 1.584 0.99995 0.99994 0.99993 0.99992 0.99991 76464 79429 72395 55361 28327 98740 78223 09847 04315 73330 05255 58361 46545 16554 08844 179 895 499 408 324 -0.00920 35432 68808 26480 539 -0.01020. 34961 46876 15451 796 -0.01120 34388 21448 74764 568 -0.01220 33702 92583 45294 454 -0.01320 32895 60348 88260 743 1.585 1.586 1.587 1.588 1.589 0.99989 0.99988 0.99986 0.99985 0.99983 91295 44263 87233 20204 43177 29595 86814 59691 63927 16226 56407 83504 04289 21344 24106 893 374 313 232 322 -0.01420 -0.01520 -0.01620 -0.01720 -0.01820 31956 30874 29641 28245 26678 24825 86108 44304 99538 51948 85219 38055 68973 20485 55400 553 737 475 440 452 1.590 1.591 1.592 1.593 1.594 0.99981 0.99979 0.99977 0.99975 0.99973 56151 59127 52105 35085 08068 34290 36823 43527 75103 53254 87198 68657 08066 24582 14867 158 422 646 972 933 -0.01920 -0.02020 -0.02120 -0.02220 -0.02320 24929 22987 20843 18488 15910 01692 48945 93900 36773 77799 56809 28070 92788 94801 98151 503 065 583 039 502 1.595 1.596 1.597 1.598 1.599 0.99970 0.99968 0.99965 0.99963 0.99960 71054 24042 67033 00029 23027 00681 41086 99171 00635 72179 50917 77790 11241 35248 99440 259 702 891 219 759 -0.02420 -0.02520 -0.02620 -0.02720 -0.02819 13101 10049 06745 03180 99342 17236 55365 92491 28945 65082 87068 65939 59282 11714 87922 552 492 234 764 093 1.600 0.99957 36030 41505 16434 211 -0.02919 95223 01288 72620 577 C-79)3 For 01.6 cos 2 1 c(-;I1 see Example 16. ;=I.57079 63267 94896 61923 132 [1 u=3.14159 26535 89793 23846 264 174 ELEMENTARY Table TRANSCENDENTAL RADIX 4.7 TABLE OF FUNCTIONS CIRCULAR SINES AND sin .~10-a 0.00000 00001 00000 00000 cos ,110-n 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0;99999 0.99999 0.99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99998 99995 99992 99987 99982 99975 99959 50000 00000 50000 00000 50000 00000 50000 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99950 99800 99550 99200 98750 98200 97550 00000 00000 00000 00000 00000 00000 00000 96800 00000 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99998 99998 99997 95000 80000 55000 20000 75000 20000 55000 00000 00000 00000 00000 00000 00000 00000 99996 80000 00000 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99995 99980 99955 99920 99875 99820 99755 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 99680 00000 00000 99999 99999 99999 99999 99999 99999 99999 99999 99999 99500 98000 95500 92000 87500 82000 75500 00000 00000 00000 00000 00007 00034 00000 00000 00107 00260 68000 00000 00000 0:99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 0.99999 59500 00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 00002 00000 00000 00003 00000 00000 00004 00000 00000 00005 00000 00000 00006 00000 00000 00007 00000 00000 00008 00000 00000 00009 00000 00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 00010 00020 00030 00040 00050 00060 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 0.00000 00069 99999 99999 99999 0.00000 0.00000 00079 00089 99999 99999 99999 99999 00000 00000. 00000 00000 00000 00000 00000 00000 99999 99999 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 00099 00199 00299 00399 00499 00599 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 0.00000 00699 99999 99999 99640 99428 99999 99999 99147 98785 o;ooooo 0.00000 00799 00899 99999 99999 99998 99987 99955 99893 99792 0.00000 00999 99999 99999 98333 0.00000 01999 99999 99999 86667 0.00000 0.00000 0.n0n00 0.00000 0.00000 0.00000 0.00000 02999 03999 114999 05999 06999 07999 08999 99999 99999 99999 99999 99999 99999 99999 99999 99998 99997 99996 99994 99991 99987 COSINES 55000 93333 91667 40000 28333 46667 85000 99595 95950 95000 00000 00000 00000 00000 00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 09999 99999 19999 99999 29999 99999 39999 99999 49999 99999 59999 99999 99963 0.00000 69999 79999 99999 89999 99999 99999 94283 00000 00540 01000 01707 02734 0.00000 0.00001 0.00002 0.00003 0.00004 0.00005 99999 99999 83333 33333 0.99999 99999 50000 00000 041bl 99999 99998 bbbbb bbbbl 0.99999 99996 00000 00000 bbbb7 99999 99999 99999 99999 99995 99989 99979 50000 33333 16666 00002 33342 66693 99987 00000 50000 00026 66667 04167 O.OOOOb 0.00007 99999 99999 99964 99942 00000 83333 bbbbb 00065 33473 bb940 0.99999 0.99999 99968 00492 99959 0.00009 0.00019 0.00029 0.00039 0.00049 0.00059 99833 98666 95500 89333 79166 33333 bbbbb 00002 33341 bbb92 34167 93333 02500 86667 70833 0.99999 0.99999 64000 00064 80000 0.00069 0.00079 99999 99999 99999 99999 99999 99999 99999 39167 0.99999 0.99999 0.99999 0.99999 0.99999 99999 bb939 00492 73333 07499 0.99999 96800 99998 14666 78500 0.99999 0.00099 99998 33333 34166 bbbb5 0.99999 0.00000 0.00000 0.00008 0.00089 99999 99914 99878 42833 99866 99550 98933 97916 96400 91466 87850 50000 33473 33333 bbbb7 00000 33333 66667 00000 99995 99968 50000 33333 66667 0.99999 0.99999 0.99999 0.99999 0.99999 99995 99992 99982 99975 50000 00000 50000 00000 00000 00003 00010 37500 50000 00054 00100 00170 00273 04167 bbbb7 37500 99950 00000 0041b 66667 99800 00000 06666 66666 99550 99200 98750 98200 97550 33749 99990 66610 95950 00000 00001 00002 00005 OOOlD 00017 00027 95000 00041 00000 06666 60416 39999 00416 00000 66450 99352 65033 Obbbb 63026 33749 92619 bb6bb 52118 For )I >lO, sin ,110-n = .rlO-n; cos .rlO-n = 1 -i .c210-2n; to 25~. From C. E. Van Orstrand, Tables of ttie exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission). ELEMENTARY CIRCULAR TRANSCENDENTAL SINES AND COSINES X FOR LARGE 175 FUNCTIONS RADIAN ARGUMENTS Table 4.8 00000 23058 68365 24966 36208 cos x 00000 68139 47142 00445 63611 00000 71740 38699 45727 91463 000 094 757 157 917 0.00000 +0.84147 +0.90929 +0.14112 -0.75680 00000 09848 74268 00080 24953 sin 5 00000 07896 25681 59867 07928 -0.95892 -0.27941 +0.65698 +0.98935 +0.41211 42746 54981 65987 82466 84852 63138 98925 18789 23381 41756 46889 87281 09039 77780 56975 315 156 700 812 627 +0.28366 +0.96017 +0.75390 -0.14550 -0.91113 21854 02866 22543 00338 02618 63226 50366 43304 08613 84676 26446 02054 63814 52586 98836 664 565 120 884 829 -0.54402 -0I99999 -0.53657 +0:42016 +0.99060 11108 02065 29180 70368 73556 89369 50703 00434 26640 94870 81340 45705 97166 92186 30787 475 156 537 896 535 -0.83907 +0.00442 +0.84385 +0.90744 +0.13673 15290 56979 39587 67814 72182 76452 88050 32492 50196 07833 45225 78574 10465 21385 59424 886 836 396 269 893 +0.65028 -0.28790 -0.96139 -0.75098 +0.14987 78401 33166 74918 72467 72096 57116 65065 79556 71676 62952 86582 29478 85726 10375 32975 974 446 164 016 424 -0.75968 -0.95765 -0.27516 +0.66031 +0.98870 79128 94803 33380 67082 46181 58821 23384 51596 44080 86669 27384 64189 92222 14481 25289 815 964 034 610 835 +0.91294 52507 27627 65437 610 +0.83665 56385 36056 03186 648 -0.00885 13092 90403 87592 169 -0.84622 04041 75170 63524 133 -0.90557 83620 06623 84513 579 +0.40808 -0.54772 -0I99996 -0.53283 +0.42417 20618 92602 08263 30203 90073 13391 24268 94637 33397 36996 98606 42138 12645 55521 97593 227 427 417 576 705 -0.13235 17500 97773 +0.76255 84504 79602 +0.95637 59284 04503 +0.27090 57883 07869 -0.66363 38842 12967 02890 73751 01343 01998 50215 201 582 234 634 117 +0.99120 +0.64691 -0:29213 -0.96260 -0.74805 28118 93223 88087 58663 75296 63473 28640 33836 13566 89000 59808 34272 19337 60197 35176 329 138 140 545 519 -0.98803 -0.40403 +0.55142 +0.99991 +0.52908 16240 76453 66812 18601 26861 92861 23065 41690 07267 20023 78998 00604 55066 14572 82083 775 877 156 808 249 +0.15425 +0.91474 +0.83422 -0.01327 -0.84857 14498 23578 33605 67472 02747 87584 04531 06510 23059 84605 05071 27896 27221 47891 18659 866 244 553 522 997 -0.42818 -0.99177 -0.64353 +0.29636 +0.96379 26694 88534 81333 85787 53862 96151 43115 56999 09385 84087 00440 73683 46068 31739 75326 675 529 567 230 066 -0.90369 -0.12796 +0.76541 +0.95507 +0.26664 22050 36896 40519 36440 29323 91506 27404 45343 47294 59937 75984 68102 35649 85758 25152 730 833 108 654 683 ti +0.74511 31604 79348 78698 771 -0.15862 26688 04708 98710 332 -0.91652 15479 15633 78589 899 -0.83177 47426 28598 28820 958 +0.01770 19251 05413 57780 795 -0.66693 -0.98733 -0.39998 +0.55511 +0.99984 80616 92775 53149 33015 33086 52261 23826 88351 20625 47691 84438 45822 29395 67704 22006 409 883 471 483 901 442 47 48 49 +0.85090 +0.90178 +0.12357 -0.76825 -0.95375 +0.52532 -0.43217 -0.99233 -0.64014 +0.30059 19888 79448 54691 43394 25437 17729 84778 50928 69199 43637 69604 29495 71827 73131 08368 746 278 975 294 703 :: 12 13 14 20 t: 23 24 310 ;23 34 ;56 ;78 39 40 t: 35245 83476 31227 46613 26527 34118 48809 45224 23666 59471 00000 50665 69539 22210 25137 000 250 602 074 264 1.00000 +0.54030 -0.41614 -0.98999 -0.65364 42486 18503 00406 79904 81836 238 329 153 497 042 50 +0.96496 60284 92113 27406 896 -0.26237 48537 03928 78591 439 From C. E. Van Orstrand, Tables of the exponential function and of the circular sine and cosine to radian arguments, Memoirs of the National Academy of Sciences, vol. 14, Fifth Memoir. U.S. Government Printing Office, Washington, D.C., 1921 (with permission) for x_<lOO. 176 ELEMENTARY Table 4.8 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS FOR LARGE RADIAN sin x ARGUMENTS co5 x -0.26237 +0.67022 +0.98662 +0,39592 -0.55878 48537 91758 75920 51501 90488 03928 43374 40485 81834 51616 78591 73449 29658 18150 24581 439 435 757 339 787 +0.96496 +0.74215 -0.16299 -0.91828 -0.82930 60284 41968 07807 27862 98328 92113 13782 95705 12.118 63150 27406 53946 48100 89119 14772 896 738 333 973 785 -0.99975 -0.52155 +0.43616 +0.99287 +0.63673 51733 10020 47552 26480 80071 58619 86911 47824 84537 39137 83659 88018 95908 11816 88077 863 741 053 509 123 +0.02212 +0.85322 +0.89986 +0.11918 -0.77108 67562 01077 68269 01354 02229 61955 22584 69193 48819 75845 73456 11396 78650 28543 22938 356 968 300 584 744 -0.30481 -0.96611 -0.73918 +0.16735 +0.92002 06211 77700 06966 57003 60381 02216 08392 49222 02806 96790 70562 94701 86727 92152 68335 565 829 602 784 154 -0.95241 -0.25810 +0.67350 +0.98589 +0.39185 29804 lb359 71623 65815 72304 15156 38267 23586 82549 29550 29269 44570 25288 69743 00516 382 121 783 864 171 +0.82682 -0.02655 -0.85551 -0.89792 -0.11478 86794 11540 99789 76806 48137 90103 23966 75322 89291 83187 46771 79446 25899 26040 22054 021 384 683 073 507 -0.56245 -0.99964 -0.51776 +O, 44014 +0.99339 38512 74559 97997 30224 03797 38172 66349 89505 96040 22271 03106 96483 06565 70593 63756 212 045 339 105 155 +0.77389 +0.95105 +0.25382 -0.67677 -0.98514 06815 46532 33627 19568 62604 57889 54374 62036 87307 68247 09778 63665 27306 62215 37085 733 657 903 498 189 +0.63331 -0.30902 -0.96725 -0.73619 +0.17171 92030 27281 05882 27182 73418 86299 66070 73882 27315 30777 83233 70291 48729 96016 55609 201 749 171 815 845 -0.38778 +0.56610 +0.99952 +0.51397 -0.44411 lb354 76368 01585 84559 26687 09430 98180 80731 87535 07508 43773 32361 24386 21169 36850 094 028 610 609 760 +0.92175 +0.82433 -0.03097 -0.85780 -0.89597 12697 13311 50317 30932 09467 24749 07557 31216 44987 90963 31639 75991 45752 85540 14833 230 501 196 835 703 -0.99388 -0.62988 +0.31322 +0.96836 +0.73319 86539 79942 87824 44611 03200 23375 74453 33085 00185 73292 18973 87856 15263 40435 lb636 081 521 353 015 321 -0.11038 +0.77668 +0.94967 +0.24954 -0.68002 72438 59820 76978 01179 34955 39047 21631 82543 73338 87338 55811 15768 20471 12437 79542 787 342 326 735 720 -0.17607 -0.92345 -0.82181 +0.03539 +0.86006 56199 84470 78366 83027 94058 48587 04059 30822 33660 12453 07696 80260 54487 68362 22683 212 163 211 543 685 -0.98437 -0.38369 +0.56975 +0.99937 +0.51017 66433 84449 03342 32836 70449 94041 49741 65311 95124 41668 89491 84477 92000 65698 89902 821 893 851 442 379 +0.89399 +0.10598 -0.77946 -0.94828 -0.24525 66636 75117 60696 21412 19854 00557 51156 15804 69947 67654 89051 85002 68855 23213 32522 827 021 400 104 044 -0.44807 -ii99436 -0.62644 +0.31742 +0.96945 36161 74609 44479 87015 93666 29170 28201 10339 19701 69987 15236 52610 06880 64974 60380 548 672 027 551 439 +0.68326 +0.98358 +0.37960 -0.57338 -0.99920 17147 77454 77390 18719 68341 36120 34344 27521 90422 86353 98369 85760 69648 88494 69443 958 773 192 922 272 +0.73017 -Oil8043 -0.92514 -Or81928 +0.03982 35609 04492 75365 82452 08803 94819 91083 96413 91459 93138 66479 95011 89170 25267 89816 352 850 475 566 180 -0.50636 56411 09758 79365 656 +0.86231 88722 87683 93410 194 ELEMENTARY CIRCULAR X SINES TRANSCENDENTAL AND COSINES sin 5 FOR LARGE CO8x RADIAN X 177 FUNCTIONS ARGUMENTS Table 4.8 sin z co9 2 102 103 104 -0.50636 +0.45202 +0.99482 -0.62298 -0.32162 564 579 679 863 240 +0.86231 +0.89200 +0.10158 -0.78223 -0.94686 887 487 570 089 801 150 151 152 153 154 -0.71487 +0.20214 +0.93332 +0.80640 -0.06192 643 988 052 058 034 +0.69925 +0.97935 +0.35904 -0.59136 -0.99808 081 460 429 968 109 105 106 107 108 109 -0.97053 -0.72714 +0.18478 +0.92681 +0.81674 528 250 174 851 261 -0.24095 +0.68648 +0.98277 +0.37550 -0.57700 905 655 958 960 218 155 156 157 158 159 -0.87331 -0.88178 -0.07954 +0.79582 +0.93951 198 462 854 410 973 -0.48716 +0.47165 +0.99683 +0.60552 -0.34249 135 229 099 787 478 110 111 112 113 114 -0.04424 -0.86455 -0.88999 -0.09718 +0.78498 268 145 560 191 039 -0.99902 -0.50254 +0.45596 +0.99526 +0.61952 081 432 910 664 061 160 161 162 163 164 +0.21942 -0.70240 -0.97845 -0.35491 +0.59493 526 779 035 018 278 -0.97562 -0.71177 +0.20648 +0.93490 +0.80377 931 476 223 040 546 115 116 117 118 119 +0.94543 +0.23666 -0.68969 -0.98195 -0.37140 533 139 794 217 410 -0.32580 -0.97159 -0.72409 +0.18912 +0.92847 981 219 720 942 132 165 166 167 168 169 +0.99779 +0.48329 -0.47555 -0.99717 -0.60199 728 156 019 329 987 -0.06633 -0.87545 -0.87968 -0.07513 +0.79849 694 946 859 609 619 120 121 122 123 124 +0.58061 +0.99881 +0.49871 -0.45990 -0.99568 118 522 315 349 699 +0.81418 -0.04866 -0.86676 -0.88796 -0.09277 097 361 709 891 620 170 171 172 173 174 +0.34664 +0.97659 +0.70865 -0.21081 -0.93646 946 087 914 053 197 +0.93799 +0.21510 -0.70555 -0.97752 -0.35076 475 527 101 694 911 125 126 127 128 129 -0.61604 +0.32999 +0.97263 +0.72103 -0.19347 046 083 007 771 339 +0.78771 +0.94398 +0.23235 -0.69289 -0.98110 451 414 910 582 552 175 176 177 178 179 -0.80113 +0.07075 +0.87758 +0.87757 +0.07072 460 224 979 534 217 +0.59848 +0.99749 +0.47941 -0.47943 -0.99749 422 392 231 877 605 130 131 132 133 134 -0.93010 -0.81160 +0.05308 +0.86896 +0.88592 595 339 359 576 482 -0.36729 +0.58420 +0.99859 +0.49487 -0.46382 133 882 007 222 887 180 181 182 183 184 -0.80115 -0.93645 -0.21078 +0.70868 +0.97658 264 140 107 041 438 -0.59846 +0.35079 +0.97753 +0.70552 -0.21513 007 734 329 964 471 135 136 137 138 139 +0.08836 -0.79043 -0.94251 -0.22805 +0.69608 869 321 445 226 013 -0.99608 -0.61254 +0.33416 +0.97364 +0.71796 784 824 538 889 410 185 186 187 188 189 +0.34662 -0.60202 -0.99717 -0.47552 +0.48331 118 394 102 367 795 -0.93800 -0.79847 +0.07516 +0.87970 +0.87544 520 804 615 293 489 140 141 142 143 144 +0.98023 +0.36317 -0.58779 -0.9.9834 -0.49102 966 137 501 536 159 -0.19781 -0.93172 -0.80900 +0.05750 +0.87114 357 236 991 253 740 190 191 192 193 194 +0.99779 +0.59490 -0.35493 -0.97845 -0.70238 928 855 836 657 633 +0.06630 -0.80379 -0.93488 -0.20645 +0.71179 686 339 971 273 593 145 146 147 148 149 +0.46774 +0.99646 +0.60904 -0.33833 -0.97464 516 917 402 339 865 +0.88386 +0.08395 -0.79313 -0.94102 -0.22374 337 944 642 631 095 195 196 197 198 199 +0.21945 +0.93953 +0.79580 -0.07957 -0.88179 467 006 584 859 884 +0.97562 +0.34246 -0.60555 -0.99682 -0.47162 270 646 186 859 571 150 -0.71487 643 +0.69925 081 200 -0.87329 730 11ooY +0.48718 768 178 ELEMENTARY Table 4.8 X CIRCULAR SINES TRANSCENDENTAL AND COSINES co9 x sin x FUNCTIONS FOR LARGE 5 RADIAN sin x ARGUMENTS cos x 200 201 202 203 204 -0.87329 -0.06189 +0.80641 +0.93330 +0.20212 730 025 841 970 036 +0.48718 +0.99808 +0.59134 -0.35907 -0.97936 768 296 538 242 069 250 251 252 253 254 -0.97052 -0.32159 +0.62301 +0.99482 +0.45199 802 386 221 373 890 +0.24098 +0.94687 +0.78221 -0.10161 -0.89201 831 771 211 569 850 205 206 207 208 209 -0.71489 -0.97464 -0.33830 +0.60906 +0.99646 751 190 503 793 664 -0.69922 +0.22377 +0.94103 +0.79311 -0.08398 926 033 651 806 947 255 256 257 258 259 -0.50639 -0.99920 -0.57335 +0.37963 +0.98359 163 803 717 563 318 -0.86230 -0.03979 +0.81930 +0.92513 +0.18040 361 076 553 609 080 210 211 212 213 214 +0.46771 -0.49104 -0.99834 -0.58777 +0.36319 852 785 709 062 945 -0.88387 -0.87113 -0.05747 +0.80902 +0.93171 747 260 243 763 141 260 261 262 263 264 +0.68323 -0.24528 -0.94829 -0.77944 +0.10601 970 121 171 719 749 -0.73019 +X96945 -0.31740 +0.62646 +0.99436 416 197 012 794 427 215 216 217 218 219 +0.98024 +0.69605 -0.22808 -0.94252 -0.79041 562 849 161 453 474 +0.19778 -0.71798 -0.97364 -0.33413 +0.61257 403 508 202 697 207 265 266 267 268 269 +0.89401 +0.86005 +0.03536 -0.82183 -0.92344 017 403 818 501 688 +0.44804 -0.51020 -0.99937 -0.56972 +0.38372 667 297 435 556 628 220 221 222 223 224 +0.08839 +0.88593 +0.86895 +0.05305 -0.81162 871 880 084 349 100 +0.99608 517 +Q.46380 216 -0.49489 841 -0.99859 167 -0.58418 435 270 271 272 273 274 -0.17604 +0.73321 +0.96835 +0.31320 -0.62991 595 082 694 015 141 +0.98438 195 +0168000 139 -0.24956 931 -0194968 714 -0.77666 699 225 226 227 228 229 -0.93009 -0.19344 +0.72105 +0.97262 +0.32996 488 382 860 306 237 +0.36731 +0.98111 +0.69287 -0.23238 -0.94399 937 135 409 a42 409 275 276 277 278 279 -0.99388 -0.44408 +0.51400 +0.99952 +0.56608 533 566 431 109 279 +0.11041 +0.89598 +0.85778 +0.03094 -0.82434 720 433 760 490 840 230 231 232 233 234 -0.61606 -0.99568 -0.45987 +0.49873 +0.99881 420 419 672 928 669 -0.78769 +0.09280 +0.88798 +0.86675 +0.04863 594 622 277 206 350 280 281 282 283 284 -0.38780 -0.98515 -0.67674 +0.25385 +0.95106 942 144 976 252 397 -0.92173 -Oil7168 +0.73621 +0:96724 +0.30899 958 765 312 294 406 235 236 237 238 239 +0.58058 -0.37143 -0.98195 -0.68967 +0.23669 664 209 787 611 068 -0.81419 -0.92846 -0.18909 +0.72411 +0.97158 847 012 982 799 506 285 286 287 288 289 +0.77387 -0.11481 -0.89794 -0.85550 -0.02652 159 476 095 437 102 -0.63334 -0.99338 -0.44011 +0.51779 +0.99964 253 692 595 559 826 240 241 242 243 244 +0.94544 +0.78496 -0.09721 -0.89000 -0.86453 515 171 191 935 630 +0.32578 -0.61954 -0.99526 -0.45594 +0.50257 131 428 371 228 038 290 291 292 293 294 +0.82684 +0.92001 +0.16732 -0.73920 -0.96610 563 423 598 100 999 +0.56242 -0.39188 -0.98590 -0.67348 +0.25813 893 496 163 488 076 245 246 247 248 249 -0.04421 +0.81676 +0.92680 +0.18475 -0.72716 256 000 719 212 319 +0.99902 +0.57697 -0.37553 -0.98278 -0.68646 215 756 754 515 463 295 296 297 298 299 -0.30478 +0.63676 +0.99286 +0.43613 -0.52157 191 125 906 763 672 +0.95242 +0.77106 -0.11921 -0.89987 -0.85320 217 103 006 997 439 250 -0.97052 802 +0.24098 831 300 -0.99975 584 -0.02209 662 ELEMENTARY CIRCULAR SINES x sin x TRANSCENDENTAL AND COSINES FUNCTIONS FOR LARGE cos x RADIAN x 179 ARGUMENTS sin x Table 4.8 cos x 300 301 302 303 304 -0.99975 -0.55876 +O. 39595 +O. 98663 +O. 67020 584 405 283 250 680 -0.02209 +O. 82932 +O. 91827 +O. 16296 -0.74217 662 668 085 104 440 350 351 352 353 354 +O. 14114 985 +o. 90930 +O. 84145 997 470 305 306 307 308 309 -0.26240 -0.95376 -0.76823 +O. 12360 +O. 90180 394 171 536 304 137 -0.96495 -0.30056 +O. 64016 +O. 99233 +O. 43215 812 379 750 174 076 355 356 357 358 359 -0.00003 -0.84148 -0.90928 -0.14109 +O. 75682 014 727 488 017 220 -1.00000 000 -0.54027 +O. 41617 +O. 98999 +O. 65362 694 425 675 081 310 311 312 313 314 +O. 85088 +O. 01767 -0.83179 -0.91650 -0.15859 769 179 148 949 291 -0.52534 -0.99984 -0.55508 +O. 40001 +O. 98734 764 384 823 294 406 360 361 362 363 364 +O. 95891 +O. 27938 -0.65700 -0.98935 -0.41209 572 655 932 386 102 -0.28369 -0.96017 -0.75388 +O. 14552 +O. 91114 109 871 245 986 268 315 316 317 318 319 +O. 74513 +O. 96378 +O. 29633 -0.64356 -0.99177 326 735 979 121 500 +O. 66691 -0.26667 -0.95508 -0.76539 +O. 12799 560 199 258 465 359 365 366 367 368 369 +O. 54404 +o. 99999 +O. 53654 -0.42019 -0.99061 640 007 748 439 148 +O. 83905 -0.00445 -0.84387 -0.90743 -0.13670 513 584 013 412 736 320 ;s: 323 324 -0.42815 +O. 52910 +O. 99991 +o. 55140 -0.40406 543 827 226 153 522 +o. 90370 +O. 84855 +O. 01324 -0.83423 -0.91473 511 433 661 998 018 370 371 372 373 374 -0.65026 +O. 28793 +O. 96140 +O. 75096 -0.14990 494 218 579 734 701 +O. 75970 +O. 95765 +O. 27513 -0.66033 -0.98870 752 080 436 935 325 326 327 328 329 -0.98803 -0.66361 +O. 27093 +O. 95638 +O. 76253 627 133 481 473 895 -0.15422 +O. 74807 +O. 96259 +O. 29210 -0.64694 167 753 770 998 231 375 376 377 378 379 -0.91295 755 -0.83663 913 +o. 00888 145 +O. 84623 647 +O. 90556 557 -0.40805 +O. 54775 +O. 99996 +O. 53280 -0.42420 454 448 056 751 631 330 331 332 333 334 -0.13238 -0.90559 -0.84620 -0.00882 +O. 83667 163 115 434 117 215 -0.99119 -0.42415 +O. 53285 +O. 99996 +o. 54770 882 171 853 109 404 380 381 382 383 384 +O. 13232 -0.76257 -0.95636 -0.27087 +O. 66365 187 795 712 677 643 -0.99120 -0.64689 +O. 29216 +O. 96261 +O. 74803 680 634 764 403 752 335 336 337 338 339 +O. 91293 +O. 14984 -0.75100 -0.96138 -0.28787 295 741 715 920 445 -0.40810 -0.98870 -0.66029 +O. 27519 +O. 95766 958 914 407 232 816 385 386 387 388 389 +O. 98802 +o. 40401 -0.55145 -0.99991 -0.52905 697 007 183 146 711 -0.15428 -0.91475 -0.83420 +O. 01330 +O. 84858 123 454 674 689 622 340 341 342 343 344 +O. 65031 +O. 99060 +O. 42013 -0.53659 -0.99999 074 323 968 836 034 +O. 75966 -0.13676 -0.90745 -0.84383 -0.00439 831 708 945 778 555 390 391 392 393 394 +O. 42820 +O. 99178 +O. 64351 -0.29639 -0.96380 991 271 506 737 342 +O. 90367 +O. 12793 -0.76543 -0.95506 -0.26661 930 379 345 471 388 345 346 347 348 349 -0.54399 +O. 41214 +O. 98936 +O. 65696 -0.27944 582 595 263 387 444 +O. 83908 +O. 91111 +o. 14547 -0.75392 -0.96016 793 021 206 186 395 396 397 398 399 -0.74509 +O. 15865 +O. 91653 +O. 83175 -0.01773 306 243 361 801 206 +O. 66696 +O. 98733 +O. 39995 -0.55513 -0.99984 052 450 769 837 277 350 -0.95893 283 -0.28363 328 400 -0.85091 936 -0.52529 634 784 -0.95893 -0.75678 283 279 -0.28363 +O. 65366 +O. 98998 +O. 41611 -0.54032 328 643 824 943 767 010 180 ELEMENTARY Table 4.8 5 CIRCULAR sin 2 SINES TRANSCENDENTAL AND COSINES co9 FUNCTIONS FOR LARGE X x RADIAN sin x ARGUMENTS co9 x 400 401 402 403 404 936 532 321 396 359 -0.52529 +0.43220 +0.99233 +0.64012 -0.30062 634 513 919 118 129 450 451 452 453 454 -0.68328 -0.98358 -0.37957 +0.57340 +0.99920 373 231 985 657 563 -0.73015 +0.18046 +0.92515 +0.81927 -0.03985 296 010 898 096 100 405 406 407 408 409 +0.26234 -0.67025 -0.98662 -0.39589 +0.55881 577 155 268 747 405 -0.96497 -0.74213 +0.16302 +0.91829 +0.82929 394 399 052 472 299 455 456 457 458 459 +0.50633 -0.45205 -0.99482 -0.62296 +0.32165 965 268 985 505 095 -0.86233 -0.89199 -0.10155 +0.78224 +0.94685 414 124 572 967 832 410 411 412 413 414 +0.99975 +0.52152 -0.43619 -0.99287 -0.63671 451 528 188 624 476 -0.02215 -0.85323 -0.89985 -0Ii1915 +0.77109 689 583 368 021 942 460 461 462 463 464 +0.97054 +0.72712 -0.18481 -0.92682 -0.81672 255 181 137 982 521 +0.24092 -0.68650 -0.98277 -0.37548 +0.57702 979 847 401 166 680 415 416 417 418 419 +0.30483 +0.96612 +0.73916 -0.16738 -0.92003 933 555 039 542 785 +0.95240 +0.25807 -0.67352 -0.98589 -0.39182 379 251 944 154 950 465 466 467 468 469 +0.04427 +0.86456 +0.88998 +0.09715 -0.78499 279 660 186 190 906 +0.99901 +0.50251 -0.45599 -0.99526 -0.61949 948 826 593 957 695 420 421 422 423 424 -0.82681 +0.02658 +0.85553 +0.89791 +0.11475 172 129 559 441 487 +0.56247 +0.99964 +0.51774 -0.44017 -0.99339 878 666 401 009 384 470 471 472 473 474 -0.94542 -0.23663 +0.68971 +0.98194 +0.37137 551 211 977 647 611 +0.32583 +0.97159 +0.72407 -0.18915 -0.92848 830 932 641 902 252 425 426 427 428 429 -0.77390 -0.95104 -0.25379 +0.67679 +0.98514 977 534 421 415 108 -0.63329 +0.30905 +0.96725 +0.73617 -0.17174 587 140 824 232 704 475 476 477 478 479 -0.58063 -0.99881 -0.49868 +0.45993 +0.99568 573 376 703 026 978 -0.81416 +0.04869 +0.86678 +0.88795 +0.09274 347 372 212 504 619 430 431 432 433 434 . -0.85091 -0.90177 -0.12354 +0.76827 +0.95374 +0.38775 -0.56613 -0.99951 -0.51395 +0.44413 385 249 922 260 968 -0.92176 -0.82431 +0.03100 +0.85781 +0.89595 296 427 516 859 756 480 481 482 483 484 +0.61601 -0.33001 -0.97263 -0.72101 +0.19350 671 928 707 682 297 -0.78773 -0.94397 -0.23232 +0:69291 +0.98109 308 419 978 756 969 435 436 437 438 439 +0.99389 +0.62986 -0.31325 -0.96837 -0.73316 198 458 741 198 982 +0.11035 -0.77670 -0.94966 -0.24951 +0.68004 728 497 826 093 560 485 486 487 488 489 +0.93dll +0.81158 -0.05311 -0.86898 -0.88591 702 578 369 067 083 +0.36726 -0.58423 -0.99858 -0.49484 +0.46385 329 328 847 603 557 440 441 442 443 444 +0.17610 +0.92347 +0.82180 -0.03542 -0.86008 529 001 066 843 478 +0.98437 +0.38367 -0.56977 -0.99937 -0.51015 134 061 511 222 112 490 491 492 493 494 -0.08833 +0.79045 +0.94250 +0.22802 -0.69610 866 167 438 291 177 +0.99609 +0.61252 -0.33419 -0.97365 -0.71794 050 441 379 577 312 445 446 447 448 449 -0.89398 -0.10595 +0.77948 +0.94827 +0.24522 316 754 495 257 276 +0.44810 +0.99437 +0.62642 -0.31745 -0.96946 056 066 095 729 676 495 496 497 498 499 -0.98023 -0.36314 +0.58781 +0.99834 +0.49099 370 328 939 363 533 +0.19784 +0.93173 +0.80899 -0.05753 -0.87116 312 331 219 262 220 450 -0.68328 373 -0.73015 296 500 -0.46777 181 -0.88384 927 ELEMENTARY CIRCULAR x SINES TRANSCENDENTAL AND COSINES FOR LARGE co9 x sin z RADIAN x 181 FUNCTIONS ARGUMENTS Table 4.8 sin x co9 x 500 501 502 503 504 -0.46777 -0.99647 -0.60902 +0.33836 +0.97465 181 170 011 176 539 -0.88384 -0.08392 +0.79315 +0.94101 +0.22371 927 940 478 611 157 550 551 552 553 554 -0.21948 -0.93954 -0.79578 +0.07960 +0.88181 408 038 759 864 305 -0.97561 -0.34243 +0.60557 +0.99682 +0.47159 608 814 585 620 913 505 506 507 508 509 +0.71485 -0.20217 -0.93333 -0.80638 +0.06195 535 940 135 275 042 -0.69927 -0.97934 -0.35901 +0.59139 +0.99807 236 850 615 399 923 555 556 557 ::98 +0.87328 +0.06186 -0.80643 -0.93329 -0.20209 261 016 623 888 084 -0.48721 -0.99808 -0.59132 +0.35910 +0.97936 400 483 107 055 678 510 511 512 513 514 +0.87332 +0.88177 +0.07951 -0.79584 -0.93950 667 040 849 235 941 +0.48713 -0.47167 -0.99683 -0.60550 +0.34252 502 887 339 389 310 560 561 562 563 564 +0.71491 +0.97463 +0.33827 -0.60909 -0.99646 859 516 666 184 411 +0.69920 -0.22379 -0.94104 -0.79309 +0.08401 771 971 671 970 951 515 516 517 518 519 -0.21939 +0.70242 +0.97844 +0.35488 -0.59495 585 924 413 199 701 +0.97563 +0.71175 -0.20651 -0.93491 -0.80375 593 358 172 110 753 565 566 567 568 569 -0.46769 +0.49107 +0.99834 +0.58774 -0.36322 187 411 883 623 754 +0.88389 +0.87111 +0.05744 -0.80904 -0.93170 157 780 234 534 046 520 521 522 523 524 -0.99779 -0.48326 +0.47557 +0.99717 +0.60197 528 517 670 555 580 +0.06636 +0.87547 +0.87967 +0.07510 -0.79851 701 403 426 603 433 570 571 572 573 574 -0.98025 -0.69603 +0.22811 +0.94253 +0.79039 158 684 096 460 628 -0.19775 +0.71800 +0.97363 +0.33410 -0.61259 448 607 514 856 589 525 526 527 528 529 -0.34667 -0.97659 -0.70863 +0.21084 +0.93647 773 735 787 000 255 -0.93798 -0.21507 +0.70557 +0.97752 +0.35074 430 583 237 059 088 575 576 577 578 579 -0.08842 -0.88595 -0.86893 -0.05302 +0.81163 874 278 592 338 861 -0.99608 -0.46377 +0.49492 +0.99859 +0.58415 251 546 461 327 989 530 531 532 533 534 +0.80111 -0.07078 -0.87760 -0.87756 -0.07069 655 230 424 088 210 -0.59850 -0.99749 -0.47938 +0.47946 +0.99749 837 179 586 522 818 580 581 582 583 584 +0.93008 +0.19341 -0.72107 -0.97261 -0.32993 380 424 948 606 391 -0.36734 -0.98111 -0.69285 +0.23241 +0.94400 740 719 235 774 403 535 536 537 538 539 +0.80117 +0.93644 +0.21075 -0.70870 -0.97657 068 083 160 168 790 +0.59843 -0.35082 -0.97753 -0.70550 +0.21516 592 557 965 828 415 585 586 587 588 589 +0.61608 +0.99568 +0.45984 -0.49876 -0.99881 795 139 996 541 816 +0.78767 -0.09283 -0.88799 -0.86673 -0.04860 737 623 663 702 339 540 541 542 543 544 -0.34659 +0.60204 +0.99716 +0.47549 -0.48334 290 801 876 715 434 +0.93801 +0.79845 -0.07519 -0.87971 -0.87543 565 989 621 726 032 590 591 592 593 594 -0.58056 +0.37146 +0.98196 +0.68965 -0.23671 210 008 357 428 997 +0.81421 +0.92844 +0.18907 -0.72413 -0.97157 597 893 022 878 792 545 546 547 548 549 -0.99780 -0.59488 +0.35496 +0.97846 +0.70236 128 432 654 280 487 -0.06627 +0.80381 +0.93487 +0.20642 -0.71181 678 133 901 324 710 595 596 597 598 599 -0.94545 -0.78494 +0.09724 +0.89002 +0.86452 497 304 191 309 115 -0.32575 +0.61956 +0.99526 +0.45591 -0.50259 281 794 078 545 644 550 -0.21948 408 -0.97561 608 600 +0.04418 245 -0.99902 348 182 ELEMENTARY Table x 4.8 CIRCULAR sin x SINES TRANSCENDENTAL AND COSINES cos x FUNCTIONS FOR LARGE X RADIAN ARGUMENTS cos x sin x 600 601 602 603 604 +0.04418 -0.81677 -0.92679 -0.18472 +0.72718 245 739 586 249 389 -0.99902 -0.57695 +0.37556 +0.98279 +0.68644 348 294 547 072 271 650 651 652 653 654 +0.30475 -0.63678 -0.99286 -0.43611 +0.52160 320 449 546 050 244 -0.95243 -0;77104 +0.11923 +0:89989 +0.85318 136 183 999 312 866 605 606 607 608 609 +0.97052 iO:32156 -0.62303 -0i99482 -0.45197 075 532 579 067 201 -0.24101 -0.94688 -0.78219 +0.10164 +0.89203 756 740 333 568 212 655 656 657 658 659 +0.99975 +0.55873 -0.39598 -0.98663 -0.67018 651 905 051 742 443 +0.02206 -0.82934 -0.91825 -0.16293 +0.74219 648 352 891 130 460 610 611 612 613 614 +0.50641 +0.99920 +0.57333 -0.37966 -0.98359 763 923 248 351 862 +0.86228 +0.03976 -0.81932 -0.92512 -0.18037 834 064 281 465 115 660 661 662 663 664 +0.26243 +0.95377 +0.76821 -0.12363 -0.90181 303 077 607 295 440 +0.96495 +0.30053 -0.64019 -0.99232 -0.43212 021 504 066 802 358 615 616 617 618 619 -0.68321 +0.24531 +0.94830 +0.77942 -0.10604 769 043 128 830 746 +0.73021 +0.96944 +0.31737 -0.62649 -0.99436 475 458 153 144 107 665 666 667 668 669 -0.85087 -0.01764 +0.83180 +0.91649 +0.15856 185 165 821 743 314 +0.52537 +0:99984 +0.55506 -0.40004 -0.98734 329 437 315 057 884 620 621 622 623 624 -0.89402 -0.86003 -0.03533 +0.82185 +0.92343 368 865 805 218 531 -0.44801 +0.51022 +0.99937 +0.56970 -0.38375 972 890 542 079 412 670 671 672 673 674 -0.74515 -0.96377 -0.29631 +0.64358 +0.99177 337 931 100 428 114 -0.66689 +0.26670 +0.95509 +0.76537 -0.12802 314 104 151 525 348 625 626 627 628 629 +0.17601 -0.73323 -0.96834 -0.31317 +0.62993 627 132 941 153 482 -0.98438 -0.67997 +0.24959 +0.94969 +0.77664 726 929 850 658 801 675 676 677 678 679 +0.42812 -0.52913 -0.99991 -0.55137 +0.40409 819 384 266 639 279 -0.90371 -0.84853 -0.01321 +0.83425 +0.91471 802 838 646 660 800 630 631 632 633 634 +0.99388 +0.44405 -0.51403 -0.99952 -0.56605 200 865 017 202 794 -0.11044 -0.89599 -0.85777 -0.03091 +0.82436 716 772 210 477 546 680 681 682 683 684 +0.98804 +0.66358 -0.27096 -0.95639 -0.76251 092 878 382 354 945 +0.15419 -0.74809 -0.96258 -0.29208 +0.64696 188 754 953 115 529 635 636 637 638 639 +0.38783 +0.98515 +0.67672 -0.25388 -0.95107 721 661 757 168 328 +0.92172 +0.17165 -0.73623 -0.96723 -0.30896 789 795 352 528 539 685 686 687 688 689 +0.13241 +0.90560 +0.84618 +0.00879 -0.83668 151 393 828 102 866 +0.99119 +0.42412 -0.53288 -0.99996 -0.54767 483 441 404 136 882 640 641 642 643 644 -0.77385 +0.11484 +0.89795 +0.85548 +0.02649 250 470 421 876 089 +0.63336 +0.99338 +0:44008 -0.51782 -0.99964 586 346 889 138 905 690 691 692 693 694 -0.91292 -0.14981 +0.75102 +0.96138 +0.28784 065 760 706 090 558 +0.40813 +0.98871 +0.66027 -0.27522 -0.95767 710 365 143 130 684 645 646 647 648 649 -0.82686 -0.92000 -0.16729 +0.73922 +0.96610 259 241 626 130 221 -0.56240 +0.39191 +0.98590 +0.67346 -0.25815 400 270 667 260 988 695 696 697 698 699 -0.65033 -0I99059 -0.42011 +0.53662 +0.99999 364 911 233 379 047 -0.75964 +0.13679 +0.90747 +0.84382 +0.00436 871 694 211 161 541 650 +0.30475 320 -0.95243 136 700 +0.54397 052 -0.83910 433 ELEMENTARY CIRCULAR x SINES TRANSCENDENTAL AND COSINES sin x FOR LARGE cos x FUNCTIONS RADIAN X 183 ARGUMENTS Table sin z cos x 4.8 700 701 702 703 704 +0.54397 -0.41217 -0.98936 -0.65694 +0.27947 052 342 702 115 339 -0.83910 -0.91110 -0.14544 +0.75394 +0.96015 433 541 039 186 344 750 751 752 753 754 +0.74507 -0.15868 -0.91654 -0.83174 +0.01776 295 219 566 127 220 -0.66698 -0.98732 -0.39993 +0.55516 +0.99984 298 971 006 345 224 705 706 707 708 709 +0.95894 +0.75676 -0.14117 -0.90932 -0.84143 137 309 969 251 841 +0.28360 -0.65368 -0.98998 -0.41609 +0.54035 437 925 399 202 304 755 756 757 758 759 +0.85093 +0.90176 +0.12351 -0.76829 -0.95373 519 229 330 325 453 +0.52527 -0.43223 -0.99234 -0.64009 +0.30065 069 231 292 802 004 710 711 712 713 714 +O.OOOOb 029 +0.84150 356 +0.90927 234 +0.14106 032 -0.75684 190 +1.00000 +0.54025 -0.41620 -0.99000 -0.65359 000 157 166 100 799 760 761 762 763 764 -0.26231 +0:67027 +0.98661 +0.39586 -0.55883 668 392 776 979 905 +0.96498 +0.74211 -0.16305 -0.91830 -0.82927 184 379 026 665 614 715 716 717 718 719 -0.95890 -0.27935 +0.65703 +0.98934 +0.41206 717 761 205 947 355 +0.28372 +0.96018 +0.75386 -0.14555 -0.91115 000 713 264 968 511 765 766 767 768 769 -0.99975 -0.52149 +0.43621 +0.99287 +0.63669 384 956 901 983 152 +0.02218 +0.85325 +0.89984 +0.11912 -0.77111 703 155 053 028 861 720 721 722 723 724 -0.54407 -0.99998 -0.53652 +0.42022 +0.99061 170 994 204 174 560 -0.83903 +0.00448 +0.84388 +0.90742 +0.13667 873 599 631 145 750 770 771 772 773 774 -0.30486 -0.96613 -0.73914 +0.16741 +0.92004 804 333 009 514 966 -0.95239 -0.25804 +0.67355 +0.98588 +0.39180 460 339 173 649 176 725 726 727 728 729 +0.65024 -0.28796 -0.96141 -0.75094 +0.14993 204 105 408 744 682 -0.75972 -0.95764 -0.27510 +0.66036 +0.98869 712 212 538 198 558 775 776 777 778 779 +0.82679 -0.02661 -0.85555 -0.89790 -Oil1472 477 142 119 114 492 -0.56250 -0.99964 -0.51771 +0.44019 +0.99339 370 585 822 716 730 730 731 732 733 734 +0.91296 +0.83662 -0.00891 -0.84625 -0.90555 985 262 160 253 279 +0.40802 -0.54777 -0.99996 -0.53278 +0.42423 702 970 029 200 360 780 781 782 783 784 +0.77392 +0.95103 +0.25376 -0.67681 -0.98513 886 602 505 634 591 +0.63327 -0.30908 -0.96726 -0.73615 +0.17177 255 007 589 192 673 735 736 737 738 739 -0.13229 +0.76259 +0.95635 +0.27084 -0.66367 199 745 831 775 898 +0.99121 +0:64687 -0.29219 -0.96262 -0.74801 079 335 647 220 752 785 786 787 788 789 -0.38772 +0.56615 +0.99951 +0.51392 -0.44416 606 733 829 674 668 +0.92177 +0.82429 -0.03103 -0.85783 -0.89594 465 720 529 408 417 740 741 742 743 744 -0.98802 -0.40398 +0.55147 +0.99991 +0.52903 232 250 697 106 153 +0.15431 +0.91476 +0.83419 -0.01333 -0.84860 102 672 011 703 217 790 791 792 793 794 -0.99389 -0.62984 +0.31328 +0.96837 +0.73314 531 117 604 950 932 -0.11032 +0.77672 +0.94965 +0.24948 -0.68006 732 396 881 174 770 745 746 747 748 749 -0.42823 -0.99178 -0.64349 +0.29642 +0.96381 715 657 199 616 146 -0.90366 -0.12790 +0.76545 +0.95505 +0.26658 639 390 285 577 483 795 796 797 798 799 -0.17613 -0.92348 -0.82178 +0.03545 +0.86010 497 158 349 855 016 -0.98436 -0.38364 +0.56979 +0.99937 +0.51012 603 277 988 115 519 750 +0.74507 295 -0.66698 298 800 +0.89396 965 -0.44812 751 184 ELEMENTARY Table X 4.8 CIRCULAR sin 5 SINES TRANSCENDENTAL AND COSINES FOR LARGE X cos 5 FUNCTIONS RADIAN sin 2 ARGUMENTS co5 x 800 801 802 803 804 +0.89396 +0.10592 -0.77950 -0.94826 -0.24519 965 756 384 300 354 -0.44812 -0.99437 -0.62639 +0.31748 +0.96947 751 385 745 587 415 850 851 852 853 854 +0.98022 +0.36311 -0.58784 -0.99834 -0.49096 773 519 378 189 907 -0.19787 -0.93174 -0.80897 +0.05756 +0.87117 267 426 447 271 700 805 806 807 808 809 +0.68330 .tO.98357 +0.37955 -0.57343 -0.99920 573 687 196 126 443 +0.73013 -0.18048 -0.92517 -0.81925 +0.03988 237 975 042 368 112 855 856 857 858 859 +0.46779 +0.99647 +0.60899 -0.33839 -0.97466 845 423 620 013 214 +0.88383 +0.08389 -0.79317 -0.94100 -0.22368 517 936 314 591 219 810 811 812 813 814 -0.50631 +0.45207 +0.99483 +0.62294 -0.32167 365 956 291 147 949 +0.86234 +0.89197 +0.10152 -0.78226 -0.94684 940 762 573 845 862 860 861 862 863 864 -0.71483 +0.20220 +0.93334 +0.80636 -0.06198 427 893 217 493 051 +0.69929 +0.97934 +0.35898 -0.59141 -0.99807 390 241 802 830 736 815 816 817 818 819 -0.97054 -0.72710 +0.18484 +0.92684 +0.81670 981 111 099 114 782 -0.24090 +0.68653 +0.98276 +0.37545 -0.57705 054 039 844 372 142 865 866 867 868 869 -0.87334 -0.88175 -0.07948 +0.79586 +0.93949 135 618 845 060 908 -0.48710 $0.47170 +0.99683 +0.60547 -0.34255 870 545 579 989 142 820 821 822 823 824 -0.04430 -0.86458 -0.88996 -0.09712 +0.78501 291 174 811 190 774 -0.99901 -0.50249 +0.45602 +0:99527 +0.61947 814 220 276 249 329 870 871 872 873 874 +0.21936 -0.70245 -0.97843 -0.35485 +0.59498 644 070 790 381 124 -0.97564 -0.71173 +0.20654 +0.93492 +0.80373 254 241 122 180 959 825 826 827 828 829 +0.94541 +0.23660 -0.68974 -0.98194 -0.37134 569 282 159 076 812 -0.32586 -0.97160 -0.72405 +0.18918 +0.92849 680 646 561 862 371 875 876 877 878 879 +0.99779 +0.48323 -0.47560 -0.99717 -0.60195 328 878 322 782 173 -0.06639 -0.87548 -0.87965 -0.07507 +0.79853 709 859 992 597 248 830 831 832 833 834 +0.58066 +0.99881 +0.49866 -0.45995 -0.99569 027 229 090 702 258 +0.81414 -0.04872 -0.86679 -0.88794 -0.09271 596 383 716 118 618 880 881 882 883 884 +0.34670 +0.97660 +0.70861 -0.21086 -0.93648 601 383 660 947 312 +0.93797 +0.21504 -0.70559 -0.97751 -0.35071 385 639 373 423 265 835 836 837 838 839 -0.61599 +0.33004 +0.97264 +0.72099 -0.19353 297 774 407 594 254 +0.78775 +0.94396 +0.23230 -0.69293 -0.98109 165 424 046 929 386 885 886 887 888 889 -0.80109 +0.07081 +0.87761 +0.87754 +0.07066 851 237 869 643 203 +0.59853 +0.99748 +0.47935 -0.47949 -0.99750 252 965 940 167 031 840 841 842 843 844 -0.93012 -0.81156 +0.05314 +0.86899 +0.88589 809 816 379 559 685 -0.36723 +0.58425 +0.99858 +0.49481 -0.46388 525 775 687 983 228 890 891 892 893 894 -0.80118 -0.93643 -0.21072 +0.70872 +0.97657 871 025 213 294 141 -0.59841 +0.35085 +0.97754 +0.70548 -0.21519 177 380 600 692 358 845 846 847 848 849 +0.08830 -0.79047 -0.94249 -0.22799 +0.69612 863 014 431 356 342 -0.99609 -0.61250 +0.33422 +0.97366 +0.71792 316 058 221 264 213 895 896 897 898 899 +0.34656 -0.60207 -0.99716 -0.47547 +0.48337 463 208 649 063 073 -0.93802 -0.79844 +0.07522 +0.87973 +0.87541 610 174 627 159 575 850 +0.98022 773 -0.19787 267 900 +0.99780 327 +0.06624 670 ELEMENTARY CIRCULAR SINES sin 5 TRANSCENDENTAL AND COSINES 185 FUNCTIONS FOR LARGE RADIAN 5 cos 2 ARGUMENTS sin x Table 4.8 cos x 900 901 902 903 904 +0.99780 +0.59486 -0.35499 -0.97846 -0.70234 327 009 472 902 341 +0.06624 -0.80382 -0.93486 -0.20639 +0.71183 670 926 831 374 827 950 951 952 953 954 +0.94546 +0.78492 -0.09727 -0.89003 -0.86450 479 436 191 684 600 +0.32572 -0.61959 -0.99525 -0.45588 +0.50262 431 160 784 862 250 905 906 907 908 909 +0.21951 io.93955 +0.79576 -0.07963 -0.88182 349 070 933 869 727 +0.97560 +0.34240 -0.60559 -0.99682 -0.47157 947 981 984 380 255 955 956 957 958 959 -0.04415 +0.83679 +0.92678 +0.18469 -0.72720 233 478 454 287 458 +0.99902 +0.57692 -0.37559 -0.98279 -0.68642 481 832 341 629 079 910 911 912 913 914 -0.87326 -0.06183 +0.80645 +0.93328 +0.20206 792 008 406 805 131 +0.48724 +0.99808 +0.59129 -0.35912 -0.97937 032 669 676 869 287 960 961 962 963 964 -0.97051 -0.32153 +a.62305 +0.99481 +0.45194 349 677 937 760 512 +0.24104 +0.94689 +0.78217 -0.10167 -0.89204 682 709 455 567 574 915 916 917 918 919 -0.71493 -0.97462 -0.33824 +0.60911 +0.99646 966 841 829 575 158 -0.69918 +0.22382 +0.94105 +0.79308 -0.08404 616 909 690 134 955 965 966 967 968 969 -0.50644 -0.99921 -0.57330 +0.37969 +0.98360 362 043 778 140 406 -0.86227 -0.03973 +0.81934 +0.92511 +0.18034 308 052 009 320 150 920 921 922 923 924 +0.46766 -0.49110 -0.99835 -0.58772 +0.36325 523 037 056 184 562 -0.88390 -0.87110 -0.05741 +0.80906 +0.93168 567 299 224 306 952 970 971 972 973 974 +0.68319 -0.24533 -0.94831 -0.77940 +0.10607 568 966 084 942 744 -0.73023 -0.96943 -0.31734 +0.62651 +0.99435 535 718 294 493 787 925 926 927 928 929 +0.98025 +0.69601 -0.22814 -0.94254 -0.79037 754 520 031 467 781 +0.19772 -0.71802 -0.97362 -0.33408 +0.61261 493 705 827 015 972 975 976 977 978 979 +0.89403 +0.86002 +0.03530 -0.82186 -0.92342 718 327 793 936 374 +0.44799 -0.51025 -0.99937 -0.56967 +0.38378 277 482 648 601 195 930 931 932 933 934 +0.08845 +0.88596 +0.86892 +0.05299 -0.81165 877 676 100 328 622 +0.99607 +0.46374 -0.49495 -0.99859 -0.58413 984 875 080 487 542 980 981 982 983 984 -0.17598 +0.73325 +0.96834 +0.31314 -0.62995 660 181 189 290 823 +0.98439 +0.67995 -0.24962 -0.94970 -0.77662 256 719 769 602 902 935 936 937 938 939 -0.93007 -0.19338 +0.72110 +0.97260 +0.32990 273 467 037 905 546 +0.36737 +0.98112 +0.69283 -0.23244 -0.94401 544 302 061 706 398 985 986 987 988 989 -0.99387 -0.44403 +0.51405 +0.99952 +0.56603 867 164 603 296 309 +0.11047 +0.89601 +0.85775 +0.03088 -0.82438 712 111 661 464 252 940 941 942 943 944 -0.61611 -0.99567 -0.45982 +0.49879 +0.99881 169 859 319 154 962 -0.78765 +0.09286 +0.88801 +0.86672 +0.04857 880 625 049 199 328 990 991 992 993 994 -0.38786 -0.98516 -0.67670 +0.25391 +0.95108 499 179 538 083 260 -0.92171 -0.17162 +0.73625 +0.96722 +0.30893 620 825 392 763 672 945 946 947 948 949 +0.58053 -0.37148 -0.98196 -0.68963 +0.23674 755 806 927 246 926 -0.81423 -0.92843 -0.18904 +0.72415 +0.97157 347 773 062 957 078 995 996 997 998 999 +0.77383 -0.11487 -0.89796 -0.85547 -0.02646 341 465 748 315 075 -0.63338 -0.99338 -0.44006 +0.51784 +0.99964 919 000 182 716 985 +0.94546 479 +0.32572 431 For x>lOOOsee Example 16. 1000 +0.82687 954 +0.56237 908 186 ELEMENTARY Table TRANSCENDENTAL FUNCTIONS 4.9 CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS FOR RADIAN ARGUMENTS 67 35 39 79 x-l-cot 0.00000 0.00333 0.00666 0.01000 0.01333 x 000 335 684 060 476 0.00166 0.00333 0.00500 0.00666 668 349 053 791 20.00833 16.67667 14.29738 12.51334 11.12612 58 09 76 32 53 0.01666 0.02000 0.02334 0.02667 0.03001 944 480 096 805 621 0.00833 0.01000 O.Oll.67 0.01334 0.01501 576 420 334 330 419 09 07 35 99 07 10.01668 9.10926 8.35336 7.71401 7.16624 61 83 70 72 39 0.03335 0.03669 0.04003 0.04338 0.04672 558 628 845 223 776 0.01668 0.01835 0.02003 0.02170 0.02338 614 925 365 946 680 1.01135 1.01293 1.01462 1.01642 1.01832 64 80 61 16 55 6.69173 6.27674 5.91078 5.58566 5.29495 24 65 21 93 84 0.05007 0.05342 0.05677 0.06013 0.06348 516 458 615 000 628 0.02506 0.02674 0.02842 0.03011 0.03180 578 653 915 379 054 49 81 35 77 78 1.02033 1.02246 1.02469 1.02704 1.02950 88 26 78 58 78 5.03348 95 4.79708 57 4.58232 93 4.38639 73 4.20693.71 0.06684 0.07020 0.07357 0.07693 0.08030 512 667 105 841 889 0.03348 0.03518 0.03687 0.03857 0.04027 955 092 477 124 044 3.91631 3.75909 3.61326 3.47760 3.35106 74 41 32 37 28 1.03208 1.03477 1.03759 1.04052 1.04357 50 89 10 27 57 4.04197 3.88983 3.74908 3.61852 3.49708 25 14 94 56 77 0.08368 0.08705 0.09044 0.09382 0.09721 264 978 046 483 302 0.04197 0.04367 0.04538 0.04709 0.04881 250 754 569 707 181 625 751 941 487 688 3.23272 3.12180 3.01759 2.91949 2.82696 81 50 80 61 00 1.04675 1.05005 1.05347 1.05703 1.06072 16 22 94 51 13 3.38386 3.27805 3.17897 3.08600 2.99861 34 83 74 99 68 0.10060 0.10400 0.10740 0.11080 0.11421 519 147 202 697 648 0.05053 0.05225 0.05397 0.05570 0.05744 003 186 744 689 034 876 0:38 0.39 0.36502 0.37640 0.38786 0.39941 0.41105 849 285 316 272 492 2.73951 2.65672 2.57822 2.50367 2.43276 22 80 89 59 50 1.06454 1.06849 1.07258 1.07681 1.08118 02 38 47 50 74 2.91632 2.83869 2.76536 2.69599 2.63027 08 75 87 57 48 0.11763 0.12104 0.12447 0.12790 0.13133 070 976 383 306 759 0.05917 0.06091 0.06266 0.06441 0.06617 792 976 601 678 222 0.40 0.41 0.42 0.43 0.44 0.42279 0.43463 0.44657 0.45862 0.47078 322 120 255 102 053 2.36522 2.30080 2.23927 2.18044 2.12413 24 12 78 95 20 1.08570 1.09036 1.09518 1.10015 1.10527 44 89 36 15 57 2.56793 2.50872 2.45242 2.39882 2.34775 25 20 03 48 15 0.13477 0.13822 0.14167 0.14513 0.14859 758 318 456 185 524 0.06793 0.06969 0.07146 0.07324 0.07502 246 763 789 336 418 0.45 0.46 0.47 0.48 0.49 0.48305 0.49544 0.50796 0.52061 0.53338 507 877 590 084 815 2.07015 2.01837 1.96863 1.92082 1.87480 74 22 61 05 73 1.11055 1.11600 1.12161 1.12740 1.13335 94 60 91 22 91 2.29903 2.25251 2.20805 2.16553 2.12483 27 55 98 72 00 0.15206 0.15554 0.15902 0.16251 0.16600 486 089 348 280 901 0.07681 0.07860 0.08040 0.08220 0.08401 051 247 022 390 366 0.50 0.54630 249 c-y 0.01 0.02 0.03 0.04 tanx 0.00000 0000 0.01000 0333 0.02000 2667 0.03000 9003 0.04002 1347 0.05 0.06 0.07 0.08 0.09 0.05004 0.06007 0.07011 0.08017 0.09024 0.10 0.11 0.12 0.13 0.14 x 0.00 cot x set x csc x 99.990066666 49.99333 32 33.32333 27 24.98666 52 1.00000 1.00005 1.00020 1.00045 1.00080 00 00 00 02 05 100.0:166 50.00333 33.33833 25.00666 1708 2104 4558 1105 3790 19.98333 16.64666 14.26237 12.47332 11.08109 06 19 33 19 49 1.00125 1.00180 1.00245 1.00320 1.00406 13 27 50 86 37 0.10033 0.11044 0.12057 0.13073 0.14092 467 582 934 732 189 9.96664 9.05421 8.29329 7.64892 7.09612 44 28 49 55 94 1.00502 1.00608 1.00724 1.00850 1.00988 0.15 0.16 0.17 0.18 0.19 0.15113 0.16137 0.17165 0.18196 0.19231 522 946 682 953 984 6.61659 6.19657 5.82557 5.49542 5.19967 15 54 68 56 16 0.20 0.21 0.22 0.23 0.24 0.20271 0.21314 0.22361 0.23414 0.24471 004 244 942 336 670 4.93315 4.69169 4.47188 4.27088 4.08635 0.25 0.26 0.27 0.28 0.29 0.25534 0.26602 0.27675 0.28755 0.29841 192 154 814 433 279 0.30 0.31 0.32 0.33 0.34 0.30933 0.32032 0.33138 0.34252 0.35373 0.35 [1 1.83048 77 1.13949 39 C-j)2 II 1 2.08582 96 0.16951 228 [ c-p9 1 csc x-x-l 0.00000 000 0.08582 964 C-l)8 [. 1 Compilation of tansand cots from National Bureau of Standards, Table ofcircularand hyperbolictangentsand cotangents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission). ELEMENTARY CIRCULAR x TANGENTS, TRANSCENDENTAL COTANGENTS, FOR 0.50 0.51 0.52 0.53 0.54 tanx 0.54630 0.55935 0.57256 0.58591 0.59942 0.55 0.56 0.57 0.58 0.59 187 FUNCTIONS SECANTS RADIAN AND COSECANTS Table 4.9 ARGUMENTS 249 872 183 701 962 cot x 1.83048 1.78776 1.74653 1.70672 1.66825 772 154 626 634 255 set x 1.13949 1.14581 1.15231 1.15900 1.16589 39 07 38 77 70 csc x 2.08582 2.04843 2.01255 1.97810 1.94501 96 63 78 89 07 0.61310 0.62694 0.64096 0.65516 0.66955 521 954 855 845 565 1.63104 1.59502 1.56013 1.52632 1.49352 142 471 894 503 784 1.17298 1.18028 1.18778 1.19551 1.20345 68 21 81 06 53 1.91319 1.88257 1.85311 1.82473 1.79739 00 90 45 78 41 0.60 0.61 0.62 0.63 0.64 0.68413 0.69891 0.71390 0.72911 0.74454 681 886 901 473 382 1.46169 1.43078 1.40073 1.37152 1.34310 595 125 873 626 429 1.21162 1.22003 1.22868 1.23758 1.24673 83 59 47 16 39 1.77103 1.74560 1.72106 1.69737 1.67449 22 45 62 57 37 0.65 0.66 0.67 0.68 0.69 0.76020 0.77610 0.79225 0.80866 0.82533 440 491 417 138 611 1.31543 1.28848 1.26222 1.23661 1.21162 569 559 118 155 759 1.25614 1.26583 1.27580 1.28605 1.29660 92 52 04 34 31 1.65238 1.63101 1.61034 1.59034 1.57100 34 05 23 84 01 0.70 0.71 0.72 0.73 0.74 0.84228 0.85952 0.87706 0.89491 0.91308 838 867 790 753 953 1.18724 1.16342 1.14016 1.11742 1.09518 183 833 258 140 285 1.30745 1.31863 1.33013 1.34196 1.35415 93 17 09 77 38 1.55227 1.53413 1.51656 1.49954 1.48304 03 35 54 35 60 0.75 0.76 0.77 0.78 0.79 0.93159 0.95045 0.96966 0.98926 1.00924 646 146 833 154 629 1.07342 1.05213 1.03128 1.01085 0.99083 615 158 046 503 842 1.36670 1.37962 1.39293 1.40664 1.42076 11 24 10 08 67 1.46705 1.45154 1.43650 1.42190 1.40775 27 43 25 99 03 0.80 0.81 0.82 0.83 0.84 1.02963 1.05045 1.07171 1.09343 1.11563 857 514 372 292 235 0.97121 0.95196 0.93308 0.91455 0.89635 460 830 500 085 264 1.43532 1.45032 1.46580 1.48175 1.49821 42 96 02 42 08 1.39400 1.38066 1.36771 1.35513 1.34292 78 78 62 96 52 0.85 0.86 0.87 0.88 0.89 1.13833 1116155 1.18532 1.20966 1.23459 271 586 486 412 946 0.87847 0.86091 0.84365 0.82667 0.80997 778 426 058 575 930 1.51519 1.53271 1.55080 1.56948 1.58878 02 39 46 63 44 1.33106 1.31953 1.30833 1.29745 1.28688 09 53 72 63 25 0.90 0.91 0.92 0.93 0.94 1.26015 1.28636 1.31326 1.34087 1.36923 822 938 370 383 448 0.79355 0.77738 0.76146 0.74578 0.73033 115 169 169 232 510 1.60872 1.62933 1.65065 1.67270 1.69552 58 92 49 52 44 1.27660 1.26661 1.25691 1.24747 1.23830 62 84 05 40 10 0.95 0.96 0.97 0.98 0.99 1.39838 1.42835 1.45920 1.49095 1.52367 259 749 113 827 674 0.71511 0.70010 0.68530 0.67070 0.65630 188 485 649 959 719 1.71914 1.74361 1.76897 1.79525 1.82252 92 84 37 95 32 1.22938 1.22071 1.21228 1.20409 1.19613 40 57 91 77 51 1.00 1.55740 772 0.64209 262 [ Y21 1.85081 57 [(-$1 1 1.18839 51 11 c-412 5 ELEMENTARY Table CIRCULAR 4.9 TRANSCENDENTAL TANGENTS, FOR FUNCTIONS COTANGENTS, SECANTS RADIAN ARGUMENTS AND COSECANTS 262 942 141 260 722 set x 1.85081 1.88019 1.91070 1.94243 1.97542 57 15 a9 08 47 1.18839 1.18087 1.17356 1.16645 1.15954 51 20 01 42 90 0.57361 0.56040 0.54733 0.53441 0.52162 970 467 693 147 342 2.00976 _ 2.04552 2.08279 2.12166 2.16223 32 49 43 31 06 1.15283 1.14632 1.13999 1.13384 1.12787 98 17 02 11 01 97 82 53 01 51 0.50896 0.49644 0.48403 0.47175 0.45958 811 096 759 371 520 2.20460 2.24890 2.29524 2.34378 2.39466 44 16 97 77 75 1.12207 1.11644 1.11098 1.10569 1.10055 33 69 71 05 37 2.23449 2.29579 2.35998 2.42726 2.49789 69 85 11 64 94 0.44752 0.43557 0.42373 0.41198 0.40033 802 829 221 610 638 2.44805 2.50413 2.56310 2.62518 2.69063 57 48 57 99 21 1.09557 1.09074 1.08607 1.08154 1.07715 35 67 04 17 79 1.20 1.21 1.22 1.23 1.24 2.57215 2.65032 2.73275 2.81981 2.91192 16 46 42 57 99 0.38877 0.37731 0.36593 0.35463 0.34341 957 227 119 310 486 2.75970 2.83270 2.90997 2.99188 3.07885 36 55 35 25 30 1.07291 1.06881 1.06485 1.06102 1.05732 64 46 01 06 39 1.25 1.26 1.27 1.28 1.29 3.00956 3.11326 3.22363 3.34135 3.46720 97 91 32 00 57 0.33227 0.32120 0.31020 0.29928 0.28841 342 577 899 023 670 3.17135 3.26993 3.37517 3.48778 3.60853 77 04 57 15 36 1.05375 1.05032 1.04700 1.04382 1.04076 79 05 98 41 14 1.30 1.31 1.32 1.33 1.34 3.60210 3.74708 3.90334 4.07230 4.25561 24 10 78 98 79 0.27761 0.26687 0.25619 0.24556 0.23498 565 440 034 088 350 3.73833 3.87822 4.02940 4.19329 4.37153 41 33 74 31 10 1.03782 1.03499 1.03229 1.02970 1.02723 00 a5 53 88 77 1.35 1.36 1.37 1.38 1.39 4.45522 4.67344 4.91305 5.17743 5.47068 18 12 a1 74 86 0.22445 0.21397 0.20353 0.19314 0.18279 572 509 922 574 234 4.56607 4.77923 5.01379 5.27312 5.56133 06 14 49 60 39 1.02488 1.02263 1.02050 1.01848 1.01656 07 65 39 18 93 1.40 i.41 1.42 1.43 1.44 5.79788 6.16535 6.58111 7.05546 7.60182 37 61 95 38 61 0.17247 0.16219 0.15194 0.14173 0.13154 673 663 983 413 734 5.88349 6.24592 6.65666 7.12597 7.66731 01 80 08 85 76 1.01476 1.01306 1.01147 1.00999 1.00861 51 85 85 43 52 1.45 1.46 1.47 1.48 1.49 8.23809 8.98860 9.88737 10.98337 12.34985 28 76 49 93 64 0.12138 0.11125 0.10113 0.09104 0.08097 732 194 908 6660 2601 8.29856 9.04406 9.93781 11.02880 12.39027 45 25 58 87 66 1.00734 1.00616 1.00510 1.00413 1.00327 05 95 15 62 29 1.50 1.51 1.52 1.53 1.54 14.10141 16.42809 19.66952 24.49841 32.46113 99 17 78 04 89 0.07091 0.06087 0.05084 0.04081 0.03080 4844 1343 0061 8975 6066 14.13683 16.45849 19.69493 24.51881 32.47653 29 92 14 14 83 1.00251 1.00185 1.00129 1.00083 1.00047 13 09 15 27 44 1.55 1.56 1.57 1.58 1.59 48.07848 92.62049 +1255.76559 - 108.64920 - 52.06696 25 63 15 36 96 0.02079 0.01079 + 0.00079 - 0.00920 - 0.01920 9325 6746 6327 3933 6034 48.08888 92.62589 +1255.76598 - 108.65380 - 52.07657 10 45 97 55 18 1.00021 1.00005 1.00000 1.00004 1.00018 63 83 03 24 44 1.60 - tan 2 cot .I 1.00 1.01 1.02 1103 1.04 1.55740 1.59220 1.62813 1.66524 1.70361 77 60 04 40 46 0.64209 0.62805 0.61420 0.60051 0.58698 1.05 1.06 1.07 1.08 1.09 1.74331 1.78442 1.82702 1.87121 1.91709 53 48 82 73 18 1.10 1.11 1.12 1.13 1.14 1.96475 2.01433 2.06595 2.11975 2.17587 1.15 1.16 1.17 1.18 1.19 34.23253 27 For 01.6, use 4.3.44. - 0.02921 1978 (-55’2 [ 3 - csc .x 34.24713 56 1.00042 66 t--p3 [ 1 ELEMENTARY CIRCULAR SINES AND TRANSCENDENTAL COSINES TO 189 FUNCTIONS TENTHS OF A DEGREE Table 4.10 0.00000 0.00174 0.00349 0.00523 0.00698 sin 0 00000 53283 06514 59638 12602 00000 65898 15224 31420 97962 1.00000 0.99999 0.99999 0.99998 0.99997 cos e 00000 84769 39076 62922 56307 00000 13288 57790 47427 05395 90.0° 89.9 89.8 89. 7 89.6 0.00872 0.01047 0.01221 0.01396 0.01570 65354 17841 70008 21803 73173 98374 16246 35247 39145 11821 0.99996 0.99994 0.99992 0.99990 0.99987 19230 51693 53696 25240 66324 64171 65512 60452 09304 81661 89.5 89.4 89.3 89.2 89.1 0.01745 0.01919 0.02094 0.02268 0.02443 24064 74423 24198 73335 21781 37284 99690 83357 72781 52653 0.99984 0.99981 0.99978 0.99974 0.99970 76951 57121 06834 26093 14897 56391 21644 74845 22698 81183 E 88:8 88.7 88.6 0.02617 0.02792 0.02966 0.03141 0.03315 69483 16387 62440 07590 51783 07873 23569 85111 78128 88526 0.99965 0.99961 0.99955 0.99950 C.99945 73249 01150 98601 65603 02159 75557 40354 19384 65732 41757 88.5 88.4 88.3 88.2 88.1 0.03489 0.03664 0.03838 0.04013 0.04187 94967 37087 78090 17925 56537 02501 06556 87520 32560 29200 0.99939 0.99932 0.99926 0.99919 0.99912 08270 83937 29164 43951 28300 19096 78656 10621 14446 98858 88.0 87.9 87.8 87. 7 87.6 0.04361 0.04536 0.04710 0.04884 0.05059 93873 29881 64507 97697 29400 65336 29254 09643 95613 76713 0.99904 0.99897 0.99888 0.99880 0.99871 82215 05697 98749 61373 93571 81858 90715 61970 41434 84186 87.5 87.4 87.3 87.2 87.1 0.05233 0.05407 0.05582 0.05756 0.05930 59562 88129 15049 40269 63735 42944 84775 93164 59567 75962 0.99862 0.99853 0.99844 0.99834 0.99823 95347 66703 07641 18166 98279 54574 26212 81981 14028 23765 87.0 86.9 86.8 86. 7 86.6 0.06104 0.06279 0.06453 0.06627 0.06801 85395 05195 23082 39004 52906 34857 29313 52958 00000 65248 0.99813 0.99802 0.99791 0.99780 0.99768 47984 67284 56182 14682 42788 21867 28272 72179 92050 35605 86.5 86.4 86.3 86.2 86.1 0.06975 0.07149 0.07323 0.07497 0.07671 64737 74443 81971 87268 90281 44125 32686 27632 26328 26819 0.99756 0.99744 0.99731 0.99718 0.99705 40502 07829 44772 51335 27522 59824 30944 24458 25116 26920 86.0 85.9 85.8 85.7 85.6 0.07845 0.08019 0.08193 0.08367 0.08541 90957 89243 85086 78433 69231 27845 28859 30041 32315 37367 0.99691 0.99677 0.99663 0.99649 0.99634 73337 88784 73868 28592 52961 33128 56247 18037 49504 90906 85.5 85.4 85.3 85.2 85.1 0.99619 46980 91746 sin e 85.0 e 0.08715 57427 47658 co.3 6 For conversion from radiansto degrees Example see ‘See page II. 14. 9o”-e 190 ELEMENTARY Table 4.10 CIRCULAR TRANSCENDENTAL SINES .4ND COSINES FUNCTIONS TO TENT.HS sin 0 e 0 OF A DEGREE cos e 90°-e z-23 5: 4 5.5 5. 6 55'87 5:9 6.0 2: 6:3 6.4 6. 5 66.76 6:8 6.9 ::: ;*: 714 77.56 717 ::i 8. 0 88:; 2: $56 8:l 29" 0.08115 0.08889 0.09063 0.09237 0.09410 57427 42968 25801 05874 83133 47658 66442 97780 46562 18514 0.99619 0.99604 0.99588 0.99572 0.99556 46980 91746 10654 10770 43986 15970 46981 84582 19646 03080 85.0' 84.9 84.8 84.7 84.6 0.09584 0.09758 0.09931 0.10105 0.10279 57525 28997 97497 62971 25367 20224 59149 43639 82946 87247 0.99539 0.99522 0.99505 0.99488 0.99470 b1983 73999 81831 55699 61226 07088 28788 28171 17174 84.5 84.4 84.3 84.2 84.1 67653 36233 06023 91045 0.11146 89322 06325 0.99452 18953 68273 0.99433 79441 33205 0.99415 09639 72315 0.99396 09554 55180 0.99376 79191 60596 84.0 83.9 83.8 83.7 83. b 0.11320 0.11493 0.11667 0.11840 0.12013 32137 71504 07370 39683 67907 92867 99333 06501 68388 34647 0.99357 18556 76587 0.99337 27656 00396 0.99317 06495 38486 0.99296 55081 06537 0.99275 73419 29446 83.5 83.4 83.3 83.2 83.1 0.12186 0.12360 0.12533 0.12706 0.12879 93434 14767 32335 46086 55965 05147 40493 64304 01350 77563 0.99254 0.99233 0.99211 0.99189 0.99167 61516 19378 47013 44425 11623 41322 85489 14478 90030 83090 83.0 82.9 82.8 82.7 82.6 0.13052 0.13225 0.13398 0.13571 0.13744 61922 63902 61854 55724 45460 20052 57122 18292 34304 37147 0.99144 0.99121 0.99098 0.99074 48613 55402 31997 78404 73810 51542 14836 71444 Oi99050 94632 38309 82.5 82.4 82.3 82.2 82.1 0.13917 0.14090 0.14262 0.14435 0.14608 31009 12319 89337 62010 30285 bOO65 37583 05512 00973 62412 0.99026 0.99002 0.98977 0.98952 0.98927 80687 36577 62309 57890 23329 41570 lb558 07789 0.14780 0.14953 0.15126 0.15298 0.15471 94111 53434 08202 58362 03862 29611 43710 47219 84038 99468 0.98901 0.98875 0.98849 0.98822 0.98795 58633 61917 63810 47006 38868 08684 83814 46553 0.15643 0.15815 ::1” 44650 40231 0.98768 0.98741 0.98713 0.98685 0.10452 84632 0.10626 0.10799 0.10973 40713 93557 43110 80672 0.15988 11876 0.16160 38211 0.16332 59622 T:Z x 9:9 10. 0 90°-e 54484 91835 03361 41622 0.16504 0.16676 0.16848 0.17020 0.17192 60678 16102 65003 66033 79410 0.17364 81776 66930 cos e * *see page Il. 76058 87467 93795 94991 91002 5 [t--8)7 1 0.98657 67179 68969 62988 98657 69389 83405 38067 62650 57164 21616 95138 50911 72988 06807 06969 82.0 81.9 81.8 81.7 81.6 81.5 81.4 81.3 81.2 81.1 81.0 80.9 80.8 80.7 80.6 0.98628 56015 37231 0.98599 60370 70505 0.98570 34690 88854 0.98540 78984 83490 0.98510 93261 54774 80.5 80.4 80.3 80.2 80.1 C.98480 77530 12208 sin 8 t-l)4 80.0 a c1 191 ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE 4.10 cos e sin 0 e Table 900-e lO.OO 10.1 10.2 10. 3 10. 4 0.17364 81776 66930 0.17536 67260 91987 0.17708 47403 19583 0.17880 22151 16350 i1;18051 91452 50560 0.98480 0.98450 0.98419 0.98388 0.98357 77530 12208 31799 74437 56079 69242 50379 33542 14708 13386 80.0° 79.9 79.8 79.7 79.6 10.5 10.6 10.7 10. 8 10.9 0.18223 0.18395 0.18566 0.18738 0.18909 55254 92147 13506 12720 66153 85577 13145 85725 54429 89891 0.98325 0.98293 0.98261 0.98228 0.98195 49075 63955 53491 49554 27965 43615 72507 28689 87126 96444 79.5 79.4 79.3 79.2 79Il 11.0 11.1 11.2 11.3 11.4 0.19080 0.19252 0.19423 0.19594 0.19765 89953 76545 19665 25907 43512 19972 61442 42518 73403 79126 0.98162 0.98129 0.98095 0.98061 0.98027 71834 47664 26639 92245 51553 49192 46585 46613 11746 21722 79.0 78.9 78.8 78.7 78.6 11. 5 11. 6 11. 7 11.8 11.9 0.19936 0.20107 0.20278 0.20449 0.20620 79344 17197 79211 45965 72953 56512 60518 41790 41853 96630 0.97992 0.97957 0.97922 0.97886 0.97850 47046 20830 52495 99344 28106 21766 73887 61685 89851 01778 78.5 78.4 78.3 78.2 78.1 12. 0 12.1 12.2 12.3 12.4 0.20791 0.20961 0.21132 0.21303 0.21473 16908 17759 85629 03822 47964 55389 03862 74977 53271 67063 0.97814 0.97778 0.97741 0.97704 0.97667 76007 33806 32367 58606 58942 86096 55744 35264 22783 34168 78.0 77.9 77.8 77.7 77.6 12.5 12.6 12.7 12.8 12.9 0.21643 0.21814 0.21984 0.22154 0.22325 96139 38103 32413 96543 62043 52838 84976 19467 01160 10951 0.97629 0.97591 0.97553 0.97514 0.97476 60071 19933 67619 38747 45439 45857 93543 05563 11941 91222 77.5 77.4 77.3 77.2 77.1 13. 0 13.1 13.2 13.3 13.4 0.22495 0.22665 0.22835 0.23004 0.23174 10543 43865 13074 36855 08701 10656 97371 88104 79034 94157 0.97437 0.97397 0.97357 0.97317 0.97277 00647 85235 59672 79052 89028 73160 88727 77088 58782 09397 77.0 76.9 76.8 76.7 76.6 13. 5 13. 6 13.7 13.8 13.9 0.23344 0.23514 0.23683 0.23853 0.24022 53638 55905 21131 02590 81460 65619 34515 78581 80424 71264 0.97236 0.97196 0.97154 0.97113 0.97071 99203 97677 10005 78546 91199 97646 42799 09636 64815 78191 76.5 76.4 76.3 76.2 76.1 14. 0 14.1 14.2 14. 3 14.4 0.24192 0.24361 0.24530 0.24699 0.24868 18955 99668 50117 86023 73858 78803 90127 22743 98871 64855 0.97029 0.96987 0.96944 0.96901 0.96858 57262 75996 20152 84747 53498 95139 57314 06870 31611 28631 76.0 75.9 75.8 75.7 75.6 14. 5 14. 6 14.7 14.8 14.9 0.25038 0.25206 0.25375 0.25544 0.25713 00040 54441 93582 43114 79445 84806 57579 35791 27931 54696 0.96814 0.96770 0.96726 0.96682 0.96637 76403 78108 91704 81971 77527 75877 33886 04459 60793 21329 75.5 75.4 7513 75.2 75.1 15.0 90°-e 0.25881 90451 02521 cose * (--7)l 5 *see c1 page Il. 0.96592 58262 89068 sin e c-y4 II 1 15.0 e ELEMENTARY Table 4.10 CIRCULAR TRANSCENDENTAL SINES AND COSINES e 0.25881 90451 02521 0.2605ti 45086 42648 0.26218 91786 40865 Oi26387 30499 65373 0.26555 61174 86809 15.5 15. 6 15.7 15.8 15.9 0.26723 0:2689i 0.27060 0:27228 0.27395 16.0 16.1 16.2 16.3 16.4 0.21563 0.27731 0.27899 0.28066 0.28234 16.5 16.6 16.1 16.8 16.9 TO TENTHS sin 0 15.0° 15.1 15.2 15. 3 15.4 FUNCTIONS OF A DEGREE cose 90°-e 0.96592 0.96547 0.96501 0.96455 0.96409 58262 89068 26308 79225 64944 72311 14184 5719i 54042 34110 74.9 74.8 74.7 14.6 83760 78257 98206 15266 04459 75864 02470 40574 92186 9'2432 0.96363 0.96316 0.96269 0.96221 0.96174 04532 08623 25667 97658 17464 26479 79935 29285 13095 49211 74.5 74.4 74.3 74.2 74.1 73558 16999 46533 02378 11060 39229 67089 20788 14568 42816 0.96126 0.96011 0.96029 0.95980 0.95931 16959 38319 91541 57594 36856 16943 52919 75187 39745 40058 74.0 73.9 73.8 73.7 73.6 0.28401 0.28568 0.28736 0.28903 0.29070 53447 03923 83674 04974 05198 49712 17969 44472 21935 98252 0.95881 0.95832 0.95782 0.95731 0.95681 97348 68193 25744 65133 24948 45315 94975 32067 35840 57607 73.5 73.4 13.3 73.2 73.1 17.0 17.1 11.2 17.3 17.4 0.29237 0.29404 0.29570 0.29737 0.29904 17041 03252 32304 80500 44047 48740 77786 07922 56087 0.95630 0.95579 0.95527 0.95476 0.95424 47559 63035 30147 98330 83621 22344 07995 02797 03285 16277 73.0 72.9 72.8 72.7 72.6 17.5 17.6 17.7 17.8 17.9 0.30070 0.30236 0.30403 0.30569 0.30735 57995 04273 98907 50445 30609 25490 53049 63106 66177 99807 0.95371 0.95319 0.95266 0.95212 0.95159 69507 48227 06677 92947 14812 53586 93927 42139 44038 79438 72.5 72.4 12.3 72.2 72.1 18.0 18.1 18.2 18.3 18.4 0.30901 0.31067 0.31233 0.31399 0.31564 69943 74947 64296 30732 49185 12233 24559 67405 90369 47102 0.95105 0.95051 0.94997 0.94942 0.94887 65162 95154 57316 27784 20515 24653 54776 41904 60116 44491 72.0 71.9 71.8 71.7 71.6 18. 5 18.6 18.1 18.8 18.9 0.31730 0.31895 0.32061 0.32226 0.32391 46564 05092 93092 98070 29905 85676 56952 30511 74181 98149 0.94832 0.94776 0.94721 0.94664 0.94608 36552 06199 84100 09586 02777 46029 92601 15696 53588 27545 71.5 71.4 71.3 71.2 71.1 19.0 19.1 19.2 19.3 19.4 0.32556 0.32721 0.32886 0.33051 0.33216 81544 57157 78989 79104 66467 38583 43927 13223 11318 83703 0.94551 0.94494 0.94437 0.94380 0.94322 85755 99317 89121 57531 63702 37481 09515 83229 26579 47601 71.0 70.9 70.8 70.7 70.6 19. 5 19.6 19.7 19.8 19.9 0.33380 0.33545 0.33709 0.33813 0.34037 68592 33771 15697 50255 52584 23082 79202 45291 95502 13050 0.94264 0.94205 0.94147 0.94088 0.94028 14910 92178 74521 87297 05448 12038 07689 54225 81270 10419 70.5 70.4 70.3 70.2 70.1 0.93969 26207 85908 70.0 e 20.0 90"-e 22737 0.34202 01433 25669 cos e * [51 (-7)l sin 0 c-:,4 c1 15. o” 193 ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR SINES AND COSINES TO TENTHS OF A DEGREE 4.10 co5e sin t9 0 Table 90°-e 20.0° 20.1 20.2 20.3 20.4 0.34202 0.34365 0.34529 0.34693 0.34857 01433 25669 96945 85616 81989 98535 56515 73256 20473 21815 0.93969 0.93909 0.93849 0.93788 0.93728 26207 85908 42520 94709 30227 59556 89346 11898 19894 91892 70.0° 69.9 69.8 69.7 69.6 20.5 20.6 20.7 20.8 20.9 0.35020 0.35184 0.35347 0.35510 0.35673 73812 59467 16484 04702 48437 79257 69624 08137 79993 19625 0.93667 0.93605 0.93544 0.93482 0.93420 21892 48398 95357 38973 40308 29867 56763 96014 44743 21030 69.5 69.4 69.3 69.2 69.1 21.0 21.1 21.2 21.3 21.4 0.35836 0.35999 0.36162 0.36325 0.36487 79495 45300 68081 20051 45700 82092 12304 72978 67843 37620 0.93358 0.93295 0.93232 0.93169 0.93105 04264 97202 35348 25489 38012 15512 12275 85549 58158 62528 69.0 68.9 68.8 68.7 68.6 21.5 21.6 21.7 21.8 21.9 0.36650 0.36812 0.36974 0.37136 0.37298 12267 24297 45526 84678 67572 73829 78355 50235 77825 75809 0.93041 0.92977 0.92913 0.92848 0.92783 75679 82025 64858 88251 25715 34056 58268 80914 62538 98920 68.5 68.4 68.3 68.2 68.1 22.0 22.1 22.2 22.3 22.4 0.37460 0.37622 0.37784 0.37945 0.38107 65934 15912 42631 39366 07868 18467 61595 29005 03763 50274 0.92718 0.92652 0.92587 0.92520 0.92454 38545 66787 86308 71837 05848 09995 97183 85782 60336 12313 68.0 67.9 67.8 67.7 67.6 22.5 22.6 22.7 22.8 22.9 0.38268 0.38429 0.38590 0.38751 0.38912 34323 65090 53226 59804 60423 24319 55864 52103 39501 40206 0.92387 0.92321 0.92253 0.92186 0.92118 95325 11287 02171 12981 80894 56246 31515 88501 54055 65721 67.5 67.4 67. 3 67.2 67.1 23.0 23.1 23.2 23.3 23.4 0.39073 0.39233 0.39394 0.39554 0.39714 11284 89274 71166 03561 19095 90951 55025 62965 78906 34781 0.92050 0.91982 0.91913 0.91844 0.91775 48534 52440 14973 21738 53392 55234 63813 43087 46256 83981 67.0 66.9 66.8 66.7 66.6 23.5 23.6 23.7 23.8 23.9 0.39874 90689 25246 0.40034 90325 56895 0.40194 77766 55960 0.40354 52963 52390 Oi40514 15867 79863 0.91706 0.91636 0.91566 0.91495 0.91425 00743 85124 27295 62240 25933 39561 96678 49825 39552 34264 66.5 66.4 66.3 66.2 66.1 24.0 24.1 24.2 24.3 24.4 0.40673 66430 75800 0.40833 04603 81385 Oi40992 30338 41573 0.41151 43586 05109 0.41310 44298 24542 0.91354 0.91283 0.91212 0.91140 0.91068 54576 42601 41772 33043 01161 72273 32766 35445 36608 06177 66.0 65.9 65.8 65.7 65.6 24.5 24.6 24.7 24.8 24.9 0.41469 0.41628 0.41786 0.41945 0;42103 0.90996 0.90923 0.90850 0.90777 0.90704 12708 76543 61090 47069 81775 26722 74785 32909 40142 91465 65.5 65.4 0.90630 77870 36650 65.0 e 25.0 90°-e 0.42261 82617 40699 cos0 * ‘See page n. 32426 56239 07922 60401 70738 01077 20824 46177 58133 67491 5 [(-7121 sin 0 cC-i)4 1 E.-23 65.1 ELEMENTARY TRANSCENDENTAL FUNCTIONS Table 4.10 CIRCULAR e SINES AND COSINES TO TENTHS OF A DEGREE cos e sin 0 9o”-e 25.0' 25.1 25.2 25.3 25.4 0.42261 82617 40699 0.42419 94227 45390 Oi42577 92915 65073 0.42735 78633 87192 0.42893 51334 03146 0.90630 0.90556 0.90482 i;90408 0.90333 77870 36650 87990 11140 70524 66020 25496 60778 52928 63301 65.0' 64.9 64.8 64.7 64.6 25.5 25.6 25.7 25.8 25.9 0.43051 0.43208 0.43365 0.43523 0.43680 10968 08295 57488 01982 90845 87544 10993 72328 17883 67702 0.90258 0.90183 0.90107 0.90031 0.89955 52843 49861 25264 05114 70213 22092 87714 02194 77789 55180 64.5 64.4 64.3 64.2 64.1 26.0 26.1 26.2 26.3 26.4 0.43837 0.43993 0.44150 0.44307 0.44463 11467 89077 91698 55915 58527 91745 11908 24180 51791 84927 0.89879 0.89802 0.89725 0.89648 0.89571 40462 99167 75757 60616 83696 74328 64303 83441 17602 39413 64.0 63.9 63.8 63.7 63.6 26.5 26.6 26.7 26.8 26.9 0.44619 0.44775 0.44931 0.45087 0.45243 78131 09809 90878 38770 89986 15897 75406 89431 47093 11783 0.89493 0.89415 0.89337 0.89258 0.89179 43616 02025 42368 39368 13883 27838 58184 52125 75296 05214 63.5 63.4 63.3 63.2 63.1 27.0 27.1 27.2 21.3 21.4 0.45399 0.45554 0.45709 0.45864 0.46019 04997 39547 49072 33516 79270 58694 95544 84315 97847 83852 0.89100 0.89021 0.88941 0.88861 0.88781 65241 88368 28046 11127 63732 91298 72326 54949 53851 36401 63.0 62.9 62.8 62.7 62.6 27.5 27.6 21.7 27.8 27.9 0.46174 0.46329 0.46484 0.46638 0.46792 86132 35034 60351 19862 20457 24620 66403 39891 98142 60573 0.88701 0.88620 0.88539 0.88458 0.88376 08331 78222 35792 31215 36257 54416 09752 15084 56300 88693 62.5 62.4 62.3 62.2 62.1 28.0 28.1 28.2 28.3 28.4 0.46947 0.47101 0.47255 0.47408 0.47562 15627 85891 18812 19410 07648 69054 82090 47116 42090 70275 0.88294 0.88212 0.88130 0.88047 0.87964 75928 58927 68660 17668 34520 64992 13535 09162 85728 66617 62.0 61.9 61.8 61.7 61.6 28.5 28.6 28.7 28.8 28.9 0.47715 0.47869 0.48022 0.48175 0.48328 87602 59608 18579 40607 34914 43189 36741 01715 23832 55002 0.87881 0.87798 0.87714 0.87630 0.87546 71126 61965 29754 27981 61637 05589 66800 43864 45270 00018 61.5 61.4 61. 3 61.2 61.1 29.0 29.1 29.2 29.3 29.4 0.48480 0.48633 0.48785 0.48938 0.49090 96202 46337 53804 23490 96591 38733 24517 48846 37536 15141 0.87461 0.87377 0.87292 0.87206 0.87121 97071 39396 22230 35465 20772 69810 92724 32121 38111 20189 61.0 60.9 60.8 60.7 60.6 29.5 29.6 29.7 29.8 29.9 0.49242 0.49394 0.49545 0.49697 0.49848 35601 03467 18665 84231 86684 32408 39610 27555 17397 53830 0.87035 56959 39900 0.86949 49295 05219 0.86863 15144 38191 0.86776 54533 68928 Oi86689 67489 35603 60.5 60.4 60.3 60.2 60.1 0.50000 00000 00000 cose * (-7)2 [ 5I 0.86602 54037 84439 60.0 e 30.0 9o"-s *See page n. sin e 5 [C-7)41 ELEMENTARY CIRCULAR SINES ZRANSCENDENTAL AND COSINES 195 FUNCTIONS TO TENTHS OF A DEGREE sin 0 cos 0 Table 4.10 9o”-s 30.0° 30.1 30.2 30.3 30.4 0.50000 0.50151 0.50301 0.50452 0.50603 00000 00000 07371 59457 99466 30235 76238 15019 37641 21164 0.86602 54037 84439 0.86515 14205 69704 Oi86427 48019 53705 0.86339 55506 06772 0.86251 36692 07257 60.0' 59.9 59.8 59.7 59.6 30.5 30.6 30.7 30.8 30.9 0.50753 0.50904 0.51054 0.51204 0.51354 83629 60704 14157 50371 29179 11606 28648 70572 12520 58170 0.86162 0.86074 0.85985 0.85895 0.85806 91604 41526 20270 03944 22715 96873 98969 30664 49057 23645 59.5 59.4 59.3 59.2 59.1 31. 0 31.1 31.2 31.3 31.4 0.51503 0.51653 0.51802 0.51951 0.52100 80749 10054 33288 66642 70093 73130 91118 79509 96318 40576 0.85716 0.85626 0.85536 0.85445 0.85355 73007 02112 70846 00328 42601 60507 88301 32807 07972 75327 59.0 58.9 58.8 58.7 58.6 31.5 31.6 31.7 31.8 31.9 0.52249 0.52398 0.52547 0.52695 0.52843 85647 15949 59059 70079 16510 72268 57954 96678 83347 22347 0.85264 0.85172 0.85081 0.84989 0.84897 01643 54092 69341 43048 11094 24051 26929 86864 16876 29141 58.5 58.4 58.3 58.2 58.1 32.0 32.1 32.2 32.3 32.4 0.52991 0.53139 0.53287 0.53435 0.53582 92642 33205 85795 18083 62760 70730 23493 89826 67949 78997 0.84804 0.84712 0.84619 0.84526 0.84432 80961 56426 19213 82137 31661 27564 18332 21856 79255 02015 58.0 57.9 57.8 51.7 57.6 32.5 32.6 32.7 32.8 32.9 0.53729 0.53877 0.54024 0.54170 0.54317 96083 46824 07850 06863 03204 77655 82102 82740 44499 50671 0.84339 14458 12886 0.84245 23970 07148 0.84151 07819 45306 0.84056 66034 95684 Oi83961 98645 34413 57.5 57.4 57.3 57.2 57.1 33.0 33.1 33.2 33. 3 33.4 0.54463 0.54610 0.54756 0.54902 0.55048 90350 15027 19610 14429 32234 92550 28179 98132 07400 84996 0.83867 0.83771 0.83676 0.83580 0.83484 05679 45424 87166 20439 43134 58962 73613 68270 78632 63407 57.0 56.9 56.8 56.7 56.6 33.5 33.6 33.7 33.8 33.9 0.55193 0.55339 0.55484 0.55629 0.55774 69853 12058 15492 43344 44274 47999 56155 00305 51089 79690 0.83388 58220 67168 0.83292 12407 10099 0.83195 41221 30483 0.83098 44692 74328 0:SSOOl 22850 95367 56.5 56.4 56.3 56.2 56.1 34.0 34.1 34.2 34.3 34.4 0.55919 0.56063 0.56208 0.56352 0.56496 29034 70747 89945 63242 33778 52131 60489 37571 70034 24938 0.82903 0.82806 0.82708 0.82609 0.82511 75725 55042 03346 22494 05742 74562 82944 95764 34982 78295 56.0 55.9 55.8 55.7 55.6 34.5 34.6 34.7 34.8 34.9 0.56640 0.56784 0.56927 0.57071 0.57214 62369 24833 37450 53101 95234 30844 35676 84432 58734 45516 0.82412 0.82313 Oi82214 0.82114 0.82015 61886 22016 63685 34442 40410 30737 92091 33704 18758 73772 55.5 55.4 55.3 55.2 55.1 35.0 9o"- e 0.57357 64363 51046 cos e 0.81915 20442 88992 55. 0 e sin 0 196 ELEMENTARY Table 4.10 CIRCULAR TRANSCENDENTAL SINES AND COSINES FUNCTIONS TO TENTHS sin e I9 OF A DEGREE cos e 900-e 35.0° 35.1 35.2 35.3 35.4 0.57357 0.57500 0.57643 0.57785 0.57928 64363 51046 52520 43279 23161 69793 76243 83505 11723 42679 0.81915 0.81814 0.81714 0.81613 0.81512 20442 88992 97174 25023 48983 35129 75900 80160 77957 28554 55.0° 54.9 54.8 54.7 54.6 35.5 35.6 35.7 35.8 35.9 0.58070 29557 10940 0.58212 29701 57289 0.58354 12113 56118 0.58495 76749 87215 Oi58637 23567 35789 0.81411 0.81310 0.81208 0.81106 0.81004 55183 56319 07610 47028 35268 91806 38189 89327 16404 45796 54.5 54.4 54.3 54.2 54.1 36.0 36.1 36.2 36.3 36.4 0.58778 0.58919 0.59060 0.59201 0.59341 52522 92473 63573 53342 56676 19925 31787 99220 88866 03701 0.80901 0.80798 0.80696 0.80592 0.80489 69943 74947 98838 98031 03121 43802 82822 48516 37973 55914 54.0 53.9 53.8 53.7 53.6 36.5 36.6 36.7 36.8 36.9 0.59482 0.59622 0.59762 0.59902 0.60042 27867 51341 48749 65616 51469 75521 35985 15586 02253 25884 0.80385 0.80281 0.80177 0.80073 0.79968 68606 17217 74751 91115 56442 43754 13709 48733 46584 87091 E3.5 53.4 53.3 53.2 53.1 37.0 37.1 37.2 37.3 37.4 0.60181 0.60320 0.60459 0.60598 0.60737 50231 52048 79877 45282 91148 62375 84002 65711 58397 23287 0.79863 0.79758 0.79652 0.79547 0.79441 55100 47293 39288 25229 99180 24196 34808 54896 46205 35418 53.0 52.9 52.8 52.7 52.6 37.5 37.6 37.7 37.8 37.9 0.60876 0.61014 0.61152 0.61290 0.61428 14290 08721 51639 01268 70401 85831 70536 52976 52000 98943 0.79335 0.79228 0.79122 0.79015 0.78908 33402 91235 96433 55191 35329 67490 50123 75690 40848 34691 52.5 52.4 38.0 38.1 38.2 38.3 38.4 0.61566 0.61703 0.61840 0.61977 0.62114 14753 25658 58751 40749 83953 57554 90317 95140 77802 78310 0.78801 0.78693 0.78585 0.78477 0.78369 07536 06722 50219 61337 68931 75402 63705 33083 34573 25840 52.0 51.9 51.8 51.7 51.6 38.5 38.6 38. 7 38.8 38.9 0.62251 0.62387 0.62524 0.62660 0.62796 46366 37620 95967 09386 26563 35705 38113 64461 30576 49338 0.78260 0.78152 0.78043 0.77933 0.77824 81568 52414 04724 18819 04073 38330 79649 31474 31485 26021 51.5 51.4 51.3 51.2 51.1 39.0 39.1 39.2 39.3 39.4 0.62932 03910 49837 Oi63067 58074 31286 0.63202 93026 64851 Oi63338 08726 27550 0.63473 05132 02268 0.77714 0.77604 0.77494 0.77384 0.77273 59614 56971 64070 66546 44887 04180 02097 26506 35734 97351 51. 0 50.9 50.8 50.7 50.6 39.5 39.6 39.7 39.8 39.9 0.63607 0.63742 0.63876 0.64010 0.64144 0.77162 0.77051 0.76939 0.76828 0.76716 45833 87720 32427 75789 95550 46895 35235 93523 51518 15300 50.5 50.4 50.3 50.2 50.1 0.76604 44431 18978 50.0 e 40.0 go"- e 82202 77764 39897 48690 78175 15598 96994 84955 96315 69158 0.64278 76096 86539 cose * 5 [(-7)2 1 sin 9 C [I -5713 1 z 52:l \ ELEMENTARY TRANSCENDENTAL FUNCTIONS CIRCULAR SINES AND e COSINES TO TENTHS OF A DEGREE sin 0 cos e 197 Table 4.10 90”-e 40.0° 40.1 40.2 4.0.3 40.4 0.64278 0.64412 0.64545 0.64678 0.64811 76096 86539 36297 61387 76877 23951 97795 10460 99010 63131 0.76604 0.76492 0.76379 0.76266 0.76153 44431 18978 14009 18432 60286 34642 83296 95688 83075 36737 50.0° 49.9 49.8 49.7 49.6 40.5 40.6 40.7 40.8 40.9 0.64944 0.65077 0.65209 0.65342 0.65474 80483 30184 42172 65851 84038 30392 06039 90105 08137 17340 0.76040 0.75927 0.75813 0.75699 0.75585 59656 00031 13073 34881 43361 97652 50556 51756 34691 67640 49.5 49.4 49. 3 49.2 49.1 41.0 41.1 41.2 41.3 41.4 0.65605 0.65737 0.65868 0.66000 0.66131 90289 90507 52457 94096 94601 18680 16679 60937 18653 23652 0.75470 0.75356 0.75241 0.75126 0.75011 95802 22772 33923 01638 49088 95724 41335 03511 10696 30460 49.0 48.9 48.8 48.7 48.6 41.5 41.6 41.7 41.8 41.9 0.66262 00482 15737 0.66392 62126 52242 0.66523 03546 54361 0.66653 24702 49452 Oi66783 25554 71047 0.74895 0.74779 0.74663 0.74547 0.74431 57207 89002 80904 98532 81822 85391 59996 82862 15462 31154 48.5 48.4 48.3 48.2 48.1 42.0 42.1 42.2 42.3 42.4 0.66913 0.67042 0.67172 0.67301 0.67430 06063 58858 66189 58799 05893 22990 25135 09773 23875 83723 0.74314 0.74197 0.74080 0.73963 0.73845 48254 77394 58409 75616 45962 86750 10949 78610 53406 25884 48.0 47.9 47.8 47.7 47.6 42.5 42.6 42.7 42.8 42.9 0.67559 0.67687 0.67815 0.67944 0.68072 02076 15660 59696 82661 96698 68071 13042 61517 08689 58918 0.73727 0.73609 0.73491 0.73372 0.73254 73368 10124 70871 19734 45951 49960 98645 02876 28987 87379 47.5 47.4 47.3 47.2 47.1 43.0 43.1 43.2 43.3 43.4 0.68199 0.68327 0.68454 0.68581 0.68708 83600 62499 37736 80799 71059 28689 83529 27376 75108 04423 0.73135 0.73016 0.72896 0.72777 0.72657 37016 19170 22766 20752 86274 21412 27576 57210 46709 70976 47.0 46.9 46.8 46.7 46.6 43.5 43.6 43.7 43.8 43.9 0.68835 0.68961 0.69088 0.69214 0.69340 45756 93754 95437 35670 24110 76858 31738 70407 18282 75813 0.72537 0.72417 0.72296 0.72176 0.72055 43710 12288 18614 37468 71459 09568 02280 98362 11116 80330 46.5 46.4 46.3 46.2 46.1 44.0 44.1 44.2 44.3 44.4 0.69465 0.69591 0.69716 0.69841 0.69966 83704 58997 27965 92314 51028 54565 52854 31006 33405 13365 0.71933 0.71812 0.71691 0.71569 0.71447 98003 38651 62977 63189 06076 50483 27337 03736 26796 32803 46.0 45.9 45.8 45.7 45.6 44.5 44.6 44.7 44.8 44.9 0.70090 0.70215 0.70339 0.70463 0.70587 92642 99851 30529 95162 47028 10504 42099 63595 15706 78681 0.71325 0.71202 0.71079 0.70957 0.70833 04491 54182 60459 90996 94738 72992 07365 36521 98377 24529 45.5 45.4 45.3 45.2 45.1 45. 0 9o"-s 0.70710 67811 86548 cose * (-7)3 5 0.70710 67811 86548 45.0 e [1 ‘See page Il. sin 0 [(-yj7)31 198 ELEMENTARY Table 4.11 CIRCULAR TRANSCENDENTAL FUNCTIONS TANGENTS, COTANGENTS, SECANTS AND COSECANTS TO FIVE TENTHS OF A DEGREE 90”-e 0.00000 0.00872 0.01745 0.02618 0.03492 tan 0 cot e 00000 00000 68677 90759 114.58865 07293 09608 50649 28217 57.28996 16307 59424 59215 69187 38.18845 92970 25609 07694 91747 28.63625 32829 15603 set 9 csc9 1.00000 000 1.00003 808 114.5;301 348 1.00015 233 57.29868 850 1.00034 279 38.20155 001 1.00060 954 28.65370 835 0.04366 0.05240 0.06116 0.06992 0.07870 09429 08512 77792 83041 26201 50484 68119 43510 17068,24618 22.90376 19.08113 16.34985 14.30066 12.70620 55484 31198 66877 28211 54760 99672 62567 11928 47361 74704 1.00095 1.00137 1.00186 1.00244 1.00309 269 235 869 190 220 22.92558 19.10732 16.38040 14.33558 12.74549 563 261 824 703 484 87.5 87.0 86.5 86.0 85.5 0.08748 0.09628 0.10510 0.11393 0.12278 86635 25924 90481 97538 42352 65676 56083 01645 45609 02904 11.43005 10.38539 9.51436 8.77688 8.14434 23027 61343 70801 38159 44542 22585 73568 69956 64279 74594 1.00381 1.00462 1.00550 1.00646 1.00750 984 509 828 973 983 11.47371 10.43343 9.56677 8.83367 8.20550 325 052 223 147 905 85.0 84.5 84.0 83.5 83.0 0.13165 0.14054 0.14945 0.15838 0.16734 24975 87396 08347 02391 10013 49128 44403 24536 26090 81419 7.59575 7.11536 6.69115 6.31375 5.97576 41127 25150 97223 84209 62383 17409 15146 75043 43644 33065 1.00862 1.00982 1.01110 1.01246 1.01390 896 757 613 513 510 7.66129 7.18529 6.76546 6.39245 6.05885 758 653 908 322 796 82.5 82.0 81.5 81. 0 80.5 10. 0 10.5 11. 0 11.5 12.0 0.17632 69807 08465 0.18533 90449 31534 0.19438 03091 37718 0.20345 22994 23699 0121255 65616 70022 5.67128 5.39551 5.14455 4.91515 4.70463 18196 17709 71743 19137 40159 70310 70310 71205 01094 78454 1.01542 1.01703 1.01871 1.02048 1.02234 661 027 670 657 059 5.75877 5.48740 5.24084 5.01585 4.80973 049 427 307 174 435 80.0 79.5 79.0 78.5 78.0 12.5 13.0 13.5 14.0 14.5 0.22169 46626 42940 0.23086 81911 25563 0.24007 87590 80116 0.24932 80028 43180 0125861 75843 55890 4.51070 4.33147 4.16529 4.01078 3.86671 85036 62057 58742 84155 97700 90417 09335 35844 30948 98738 1.02427 1.02630 1.02841 1.03061 1.03290 951 411 519 363 031 4.62022 4.44541 4.28365 4.13356 3.99392 632 148 757 550 916 77.5 77.0 76.5 76.0 75.5 15.0 15.5 16. 0 16.5 17.0 0.26794 0.27732 0.28674 0.29621 0.30573 91924 31122 45440 59838 53857 58808 34949 62080 06814 58660 3.73205 3.60588 5.48741 3.37594 3.27085 08075 68877 35087 60874 44438 40408 34225 91246 26184 84141 1.03527 1.03774 1.04029 1.04294 1.04569 618 221 944 891 176 3.86370 3.74197 3.62795 3.52093 3.42030 331 754 528 652 362 75.0 74.5 74.0 17.5 18.0 18.5 19.0 19.5 0.31529 0.32491 0.33459 0.34432 0.35411 87888 78983 96962 32906 53195 02073 76132 89665 85725 30698 3.17159 3.07768 2.98868 2.90421 2.82391 48023 63212 35371 75253 49627 42893 08776 75823 28856 00801 1.04852 1.05146 1.05449 1.05762 1.06084 913 222 231 068 870 3.32550 3.23606 3.15154 3.07155 2.99574 952 798 530 349 431 72.5 72.0 71.5 71. 0 70.5 20.0 20.5 21.0 21.5 22.0 0.36397 0.37388 0.38386 0.39391 0.40402 02342 66202 46794 84804 40350 35416 04756 14942 62258 35157 2.74747 2.67462 2.60508 2.53864 2.47508 74194 54622 14939 26824 90646 93801 78956 64307 68534 16296 1.06417 1.06760 1.07114 1.07478 1.07853 777 936 499 624 474 2.92380 2.85545 2.79042 2.72850 2.66946 440 095 811 383 716 70.0 69.5 69.0 68.5 68.0 2.61312 593 set 0 67.5 e 3-Z 3:5 ::: E t-50 7:o i:'o E 9:5 22.5 90°-0 0.41421 35623 73095 cot e [c-y1 2.41421 35623 73095 tan 0 1.08W0220 1 [(-;I1 90.0° 89.5 89.0 88.5 88.0 :33-z . ELEMENTARY TRANSCENDENTAL FUNCTIONS 199 CIRCULAR TANGENTS, COTANGENTS, SECANTS AND COSECANTS TO FIVE TENTHS OF A DEGREE tan 8 set H csc e cot e Table 4.11 e 22.5O 23.0 23.5 24. 0 24.5 0.41421 0.42447 0.43481 0.44522 0.45572 35623 73095 48162 09604 23749 60933 86853 08536 62555 32584 2.41421 2.35585 2.29984 2.24603 2.19429 35623 73095 23658 23753 25472 36257 67739 04216 97311 65038 1.08239 1.08636 1.09044 1.09463 1.09894 220 038 110 628 787 2.61312 2.55930 2.50784 2.45859 2.41142 593 467 285 334 102 9o”-s 67.5' 67.0 66.5 66.0 65.5 25.0 25.5 26.0 26.5 27.0 0.46630 0.47697 0.48773 0.49858 0.50952 76581 54998 55326 98160 25885 65861 16080 53431 54494 94429 2.14450 69205 09558 2109654 35990 88174 2.05030 38415 79296 2.00568 97082 59020 1.96261 05055 05150 1.10337 1.10792 1.11260 1.11740 1.12232 792 854 194 038 624 2.36620 2.32282 2.28117 2.24115 2.20268 158 050 203 845 926 65.0 64.5 64.0 63.5 63.0 27.5 28.0 28.5 29. 0 29.5 0.52056 0.53170 0.54295 0.55430 0.56577 70505 51746 94316 61479 56996 38437 90514 52769 27781 87770 1.92098 1.88072 1.84177 1.80404 1.76749 21269 71166 64653 46332 08860 33458 77552 71424 40162 42891 1.12738 1.13257 1.13789 1.14335 1.14895 195 005 318 407 554 2.16568 2.13005 2.09573 2.06266 2.03077 057 447 853 534 204 62.5 62.0 61.5 61.0 60.5 30.0 30.5 31.0 31.5 32.0 0.57735 0.58904 0.60086 0.61280 0.62486 02691 89626 50164 20551 06190 27560 07881 39932 93519 09327 1.73205 1.69766 1.66427 1.63185 1.60033 08075 68877 31193 26089 94823 50518 16871 28789 45290 41050 1.15470 1.16059 1.16663 1.17282 1.17917 054 210 340 770 840 2.00000 1.97029 1.94160 1.91388 1.88707 000 441 403 086 991 60.0 59.5 59.0 58.5 58.0 32.5 33.0 33.5 34.0 34.5 0.63707 0.64940 0.66188 0.67450 0.68728 02608 07493 75931 97510 55611 95691 85168 42426 09586 01613 1.56968 1.53986 1.51083 1.48256 1.45500 55771 17490 49638 14583 51936 14901 09685 12740 90286 72445 1.18568 1.19236 1.19920 1.20621 1.21340 905 329 494 795 641 1.86115 1.83607 1.81180 1.78829 1.76551 900 846 103 165 728 57.5 57. 0 56.5 56.0 55.5 35.0 35.5 36.0 36.5 37.0 0.70020 0.71329 0.72654 0.73996 0.75355 75382 09710 30678 97005 25280 05361 10750 ,28487 40501 02794 1.42814 1.40194 1.37638 1.35142 1.32704 80067 42114 82944 76336 19204 71173 24379 45808 48216 20410 1.22077 1.22832 1.23606 1.24400 1.25213 459 691 798 257 566 1.74344 1.72205 1.70130 1.68117 1.66164 680 082 162 299 014 55.0 54.5 54.0 53.5 53.0 37.5 38.0 38.5 39.0 39.5 0.76732 0.78128 0.79543 0.80978 0.82433 69879 78960 56265 06717 59166 67828 40331 95007 63858 17495 1.30322 1.27994 1.25717 1.23489 1.21309 53728 41206 16321 93079 22989 18954 71565 35051 70040 92932 1.26047 1.26901 1.27777 1.28675 1.29596 241 822 866 957 700 1.64267 1.62426 1.60638 1.58901 1.57213 963 925 793 573 369 52.5 52.0 51.5 51. 0 50.5 40.0 40.5 41.0 41.5 42.0 0.83909 0.85408 0.86928 0.88472 0.90040 96311 77280 06854 63466 67378 16226 52645 55944 40442 97840 1.19175 1.17084 1.15036 1.13029 1.11061 35925 94210 95661 12539 84072 21009 43863 61753 25148 29193 1.30540 1.31508 1.32501 1.33519 1.34563 729 700 299 242 273 1.55572 1.53976 1.52425 1.50916 1.49447 383 904 309 050 655 50.0 49.5 49.0 48.5 48.0 42.5 43. 0 43.5 44.0 44.5 0.91633 0.93251 0.94896 0.96568 0.98269 11740 17423 50861 37661 45667 14880 87748 07074 72631 15690 1.09130 1.07236 1.05378 1.03553 1.01760 85010 69271 87100 24682 01252 80962 03137 90569 73929 72125 1.35634 1.36732 1.37859 1.39016 1.40203 170 746 847 359 206 1.48018 1.46627 1.45273 1.43955 1.42671 723 919 967 654 819 47.5 47.0 46.5 46. 0 45.5 45.0 90°-e 1.00000 00000 00000 cot tJ [1 C-i)4 1.00000 o~ooy 00000 [1 C-f!4 [‘-4’31 L 1.41Q&356 -1 1.414&356 [1 c-:)3 45.0 e ELEMENTARY 200 Table 4.12 CIRCULAR sin 1: TRANSCENDENTAL FUNCTIONS FOR FUNCTIONS THE ARGUMENT co9 T T r, 2 ;x l-x taniz 0.00 0.01 0.02 0.03 0.04 0.00000 0.01570 0.03141 0.04710 0.06279 00000 73173 07590 64507 05195 00000 11820 78128 09642 29313 00000 1.00000 67575 0.99987 29384 0.99950 66090 0.99888 37607 0.99802 2" 00000 00000 66324 81660 65603 65731 98749 61969 67284 28271 0.05 0.06 0.07 0.08 0.09 0.07845 0.09410 0.10973 0.12533 0.14090 90957 83133 43110 32335 12319 27844 18514 91045 64304 37582 94503 31847 26802 24537 66116 0.99691 0.99556 0.99396 0.99211 0.99002 73337 19646 09554 47013 36577 33127 03080 55179 14477 16557 97620 01290 68775 83105 56725 0.07870 0.09452 0.11040 0.12632 0.14232 17068 78311 10278 93784 10757 24618 79282 15818 46108 02942 44806 04901 94497 17478 94229 0.95 0.94 0.93 0.92 0.91 0.10 0.11 0.12 0.13 0.14 0.15643 0.17192 0.18738 0.20278 0.21814 44650 91002 13145 72953 32413 40230 79409 85724 56512 96542 86901 54661 63054 48344 55202 0.98768 0.98510 0.98228 0.97922 0.97591 83405 93261 72507 28106 67619 95137 54773 28688 21765 38747 72619 91802 68108 78086 39896 0.15838 0.17452 0.19076 0.20709 0.22352 44403 79388 02022 00444 64828 24536 94365 18566 27938 97149 29384 08461 74856 70402 10184 0.90 0.89 0.88 0.15 0.16 0.17 0.18 0.19 0.23344 0.24868 0.26387 0.27899 0.29404 53638 98871 30499 11060 03252 55905 64854 65372 39229 32303 41177 78824 89696 25185 95777 0.97236 0.96858 0.96455 0.96029 0.95579 99203 31611 74184 36856 30147 97676 28631 57798 76943 98330 60183 11949 09366 07175 12664 0.24007 0.25675 0.27356 0.29052 0.30764 87590 63603 90430 68567 01696 80116 67726 82237 31916 59898 03926 78332 23655 45432 29067 0.85 0. 84 0.83 0. 82 0.81 0.20 0.21 0.22 0.23 0.24 0.30901 0.32391 0.33873 0.35347 0.36812 69943 74181 79202 48437 45526 74947 98149 45291 79257 84677 42410 41440 38122 12472 95915 0.95105 0.94608 0.94088 0.93544 0.92977 65162 53588 07689 40308 64858 95153 27545 54225 29867 88251 57211 31853 47232 32518 40366 0.32491 0.34237 0.36002 0.37786 0.39592 96962 65257 21530 85117 80087 32906 28683 95756 75820 97721 32615 05965 62634 93670 26049 0.80 0.79 0.78 0.77 0.76 0.25 0.26 E; 0:29 0.38268 0.39714 0.41151 0.42577 0.43993 34323 78906 43586 92915 91698 65089 34780 05108 65072 55915 77173 61375 77405 64886 14083 0.92387 0.91775 0.91140 0.90482 0.89802 95325 46256 32766 70524 75757 11286 83981 35445 66019 60615 75613 14114 24821 52771 63093 0.41421 0.43273 0.45151 0.47056 0.48989 35623 86422 73130 42812 49450 73095 47425 86983 12251 22477 04880 93197 28945 49308 05270 0.75 0.74 0.73 0.72 0.71 0.30 0.31 0.32 0.33 0.34 0.45399 0.46792 0.48175 0.49545 0.50904 04997 98142 36741 86684 14157 39546 60573 01715 32407 50371 79156 37723 27498 53805 30028 0.89100 0.88376 0.87630 0.86863 0.86074 65241 56300 66800 15144 20270 88367 88693 43863 38191 03943 86236 42432 58731 24777 63716 0.50952 0.52947 0.54975 0.57038 0.59139 54494 27451 46521 99296 83513 94428 82014 92770 73294 99471 81051 63252 07429 88698 09817 0.70 0.69 0.68 0.67 0.66 0.35 0.36 0.37 00% . 0.52249 0.53582 0.54902 0.56208 0.57500 85647 67949 28179 33778 52520 15948 78996 98131 52130 43278 86499 61827 74352 60010 56590 0.85264 0.84432 0.83580 0.82708 0.81814 01643 79255 73613 05742 97174 54092 02015 68270 74561 25023 22152 07855 25847 82492 43213 0.61280 0.63461 0.65687 0.67959 0.70281 07881 92975 72224 92982 17712 39931 44148 01279 24526 40357 99664 10071 37691 52184 33761 0. 65 0. 64 0.63 0.62 0.61 0.40 0.41 0.42 0.43 0.44 0.58778 0.60042 0.61290 0.62524 0.63742 52522 02253 70536 26563 39897 92473 25884 52976 35705 48689 12917 04976 49336 17290 71017 0.80901 0.79968 0.79015 0.78043 0.77051 69943 46584 50123 04073 32427 74947 87090 75690 38329 75789 42410 53868 36516 73585 23080 0.72654 0.75082 0.77567 0.80115 0.82727 25280 12380 95110 10705 19459 05360 38764 49613 58751 72475 88589 68575 10378 23382 63403 0.60 0.59 0.58 0.57 0.56 0.45 0.46 0.47 0.48 0.49 0.64944 0.66131 0.67301 0.68454 0.69591 80483 18653 25135 71059 27965 30183 23651 09773 28688 92314 65572 87657 33872 67373 32549 0.76040 0.75011 0.73963 0.72896 0.71812 59656 10696 10949 86274 62977 00030 30459 78609 21411 63188 93817 54151 69747 52314 83037 0.85408 0.88161 0.90992 0.93906 0.96906 06854 85923 99881 25058 74171 63466 63189 77737 17492 93793 63752 11465 46579 35255 27618 0.55 0.54 0.53 0.52 0.51 0.50 0.70710 67811 86547 52440 1.00000 00000 00000 00000 0.50 l-x cos ;x (-62 [ ii3 1 00000 59864 55700 97264 56195 0.00000 0.01570 0.03142 0.04715 0.06291 00000 92553 62660 88028 46672 00000 23664 43351 77480 53649 00000 91632 14782 47448 75722 1.00 0.99 0.98 0.97 0.96 0.70710 67811 86547 52440 sin g2 [(-i$3 1 cot ;x [(J4jl 1 00% . 2 ELEMENTARY CIRCULAR TRANSCENDENTAL FUNCTIONS FOR 201 FUNCTIONS THE ARGUMENT Table fx csc ;x X 4.12 1-X 1 00000 1'00012 1:00049 1.00111 1.00197 00000 33827 36832 13587 71730 00000 39761 37144 85243 71142 00000 81169 42400 76109 10978 63 66459 31:83622 21.22851 15.92597 53060:0564 52090 97622 50958 16816 11099 08654 58546 95566 17580 59358 1.00 0.99 0.98 0.97 0.96 0.05 12.70620 47361 74704 64602 0.06 10.57889 49934 05635 52417 0.07 9.05788 66862 38928 19329 0.08 7.91581 50883 05826 84427 0.09 7.02636 62290 41380 19848 1.00309 1.00445 1.00607 1.00794 1.01007 21984 78193 57361 79708 68726 82825 57019 86291 09297 13784 50283 51480 90575 28943 19104 12.74549 10.62605 9.11292 7.97872 7.09717 48431 37962 00161 97555 00264 82374 83115 49841 59476 69225 28619 99865 72675 60149 38129 0.95 0.94 0.93 0.92 0.91 0.10 0.11 0.12 0.13 0.14 6.31375 5.72974 5.24218 4.82881 4.47374 15146 16467 35811 73521 28292 75043 24314 13176 92759 11554 09898 86192 73758 97818 62415 1.01246 1.01511 1.01803 1.02121 1.02467 51257 57576 21481 80406 75534 88002 62501 91042 26567 55900 93136 87437 38259 47910 33566 6.39245 5.81635 5.33671 4.93127 4.58414 32214 10329 14122 53949 38570 99661 24944 92458 49859 27373 54704 03199 78659 96253 56913 0.90 0.89 0.88 0.87 0.86 0.15 0.16 0.17 0.18 0.19 4.16529 3789474 3.65538 3:44202 3.25055 97700 28549 43546 25766 08012 90417 29859 52259 69218 99836 20387 33474 73004 62809 37634 1.02841 1.03243 1.03674 1.04134 1.04625 51936 58734 49162 80947 16303 65208 17339 32016 70681 39647 54585 88710 53065 14007 78848 4.28365 4.02107 3.78970 3.58434 3.40089 75697 22333 11465 36523 40753 31185 75967 59780 72161 61802 03924 50952 81919 57038 31848 0. 85 0.84 0. 83 0.82 0. 81 0.20 0.21 0.22 0.23 0.24 3.07768 2.92076 2.77760 2.64642 2.52571 35371 09892 68539 32102 16894 75253 98816 14974 86631 47304 40257 40048 88865 86514 99451 1.05146 1.05698 1.06283 1.06901 1.07552 22242 70790 39243 10439 73070 38267 93232 36113 98926 22247 21205 61183 96396 01199 78234 3.23606 3.08720 2.95213 2.82905 2.71647 79774 66268 47928 56388 18916 99789 08416 09339 91501 65871 69641 38088 97327 64260 74307 0. 80 0.79 0.78 0.77 0.76 0.25 2.41421 35623 73095 04880 0.26 2.31086 36538 82410 63708 0.27 2.21475 44978 13361 51875 0.28 2.12510 81731 57202 76115 0..29 2.04125 39671 21703 26026 1.08239 1.08961 1.09720 1.10518 1.11355 22002 58646 91341 35787 15511 92393 48705 29537 56399 90413 96880 30888 26252 59380 37268 2.61312 2.51795 2.43004 2.34863 2.27304 59297 36983 88648 46560 15214 52753 10349 55296 54351 61957 05571 34110 52041 86300 72361 0.75 0.74 0.73 0.72 0.71 0.30 0.31 0.32 0.33 0.34 1.96261 1.88867 1.81899 1.75318 1.69090 05055 13416 32472 66324 76557 05150 31067 81066 72237 85011 58230 67620 27571 08332 24674 1.12232 1.13152 1.14115 1.15123 1.16178 62376 17133 30035 61494 82810 34360 97749 92241 81376 72765 80715 42882 17245 51287 98515 2.20268 2.13707 2.07574 2.01833 1.96447 92645 26325 96076 18280 66988 85266 27611 48793 89559 67248 62156 85837 05903 43676 48330 0.70 0.69 0.68 0. 67 0.66 0.35 0.36 0.37 0.38 0.39 1.63185 1.57574 1.52235 1.47145 1.42285 16871 78599 45068 53158 60774 28789 68651 96131 19969 31870 61767 08688 24085 04283 59031 1.17282 1.18437 1.19644 1.20907 1.22227 76966 39497 79450 20434 01770 14008 36918 89806 06541 86068 94955 17500 17366 15436 14117 1.91388 1.86627 1.82141 1.77909 1.73911 08554 47167 79214 54854 45497 30942 00567 74081 79867 30640 72280 54120 38479 33350 74960 0.65 0.64 0.63 0.62 0.61 0.40 0.41 0.42 0.43 0.44 1.37638 1.33187 1.28919 1.24820 1.20879 19204 49515 22317 40363 23504 71173 02597 85066 53049 09609 53820 59439 67042 43751 13115 1.23606 1.25049 1.26557 1.28134 1.29783 79774 29154 44560 42308 62271 99789 09784 72090 20677 84727 69641 85573 15648 31999 12712 1.70130 1.66550 1.63156 1.59937 1.56881 16167 01910 87575 90408 45035 04079 65749 13749 68062 05365 86436 08074 73007 88301 75750 0.60 0.59 0.58 0.57 0.56 0.45 0.46 0.47 0.48 0.49 1.17084 1.13427 1.09898 1.06489 1.03191 95661 73492 56505 18403 99492 12539 55405 36301 24791 80495 22520 46422 56382 86700 57182 1.31508 1.33313 1.35202 1.37180 1.39251 69998 59054 53634 11480 27141 90784 50172 40027 64918 49012 80424 40410 12805 28453 49662 1.53976 1.51214 1.48585 1.46081 1.43696 90432 58610 64735 98491 16493 22366 31226 81717 22513 57094 30748 40092 76608 12750 20394 0.55 0.54 0.53 0.52 0.51 0.50 l-x 1.00000 00000 00000 00000 tan;x 1.41421 35623 73095 04880 0.50 0.00 0.01 0.02 0.03 0.04 63.65674 31.82051 21.20494 15.89454 1162; 59537 87896 48438 71580 73958 88751 65303 99500 03934 52283 44576 1.41421 35623 73095 04880 cscTx 2 SW;X X 202 ELEMENTARY Table 4.13 TRANSCENDENTAL HARMONIC 2r7 sin -; 0.86602 0.86602 0.86602 0.00000 76097 77530 54038 01433 0.50000 0.86602 1.00000 0.86602 0.50000 0.00000 00000 54038 00000 54038 00000 00000 0.40673 0.74314 0.95105 0.99452 0.86602 0.58778 0.20791 66431 48255 65163 18954 54038 52523 16908 0.34202 0.64278 0.86602 0.98480 0.98480 0.86602 0.64278 0.34202 0.00000 01433 76097 54038 77530 77530 54038 76097 01433 00000 0.29475 0.56332 0.78183 0.93087 0.99720 0.91492 0.86602 0.68017 0.43388 0.14904 51744 00580 14825 37486 37972 79122 54038 27378 37391 22662 0.25881 0.50000 0.70710 0.86602 0.96592 1.00000 0.96592 0.86602 0.70710 0.50000 0.25881 0.00000 9045; 00000 67812 54038 58263 00000 58263 54038 67812 00000 90451 00000 s=3 -0.50000 54038 54038 00000 0.64278 0.98480 0.86602 0.34202 ANALYSIS (30s 2s 54038 84 00000 1.00000 0.00000 00000 00000 +o.ooooo -1.00000 00000 00000 0.95105 0.58778 00000 00000 00000 0.78183 0.97492 0.43388 14824 79122 37391 s=7 +0.62348 -0.22252 -0.90096 98019 09340 88679 0.70710 1.00000 0.70710 0.00000 67812 00000 67812 00000 44431 81777 00000 26208 0.58778 0.95105 0.95105 0.58778 0.00000 52523 65163 65163 52523 00000 s=lO 0.80901 to.30901 -0.30901 -0.80901 -1.00000 69944 69944 69944 69944 00000 0.54064 0.90963 0.98982 0.75574 0.28173 ~43 0.70710 +o.ooooo -0.70710 -1.00000 67012 00000 67812 00000 08174 19953 14419 95743 25568 0.84125 +0.41541 -0.14231 -0.65486 -0.95949 35328 50130 48383 07340 29736 37391 14825 79122 79122 14825 37391 00000 s=14 0.90096 0;62348 +0.22252 -0.22252 -0.62348 -0.90096 -1.00000 88679 98019 09340 09340 98019 88679 00000 95325 67812 34324 00000 34324 67812 95325 00000 0 36124 0:67369 0.89516 0.99573 0.96182 0.79801 0.52643 0.18374 16662s= 56436 32913 41763 56432 72273 21629 95178 =17 0.93247 0.73900 0.44573 +0.09226 -0.27366 -0.60263 -0.85021 -0.98297 22294 89172 03558 a3595 29901 46364 71357 30997 72417 05094 81581 54872 93455 54247 15716 37512 13034 0.30901 0.58778 0.80901 0.95105 1.00000 0.95105 0.80901 0.58778 0.30901 0.00000 69944 52523 69944 65163 00000 65163 69944 52523 69944 00000 29736 35328 07340 50130 48383 48383 50130 07340 35328 29736 00000 0.26979 0.51958 0.73083 oiaa788 0.97908 67711 39500 59643 52184 40877 87692 09221 98930 s=ll s=12 0.86602 0.50000 +o.ooooo -0.50000 -0.86602 -1.00000 54038 00000 00000 00000 54038 00000 s=15 s=16 -6.92307 0.70710 0.38268 +o.ooooo -0.38268 -0.70710 -0.92387 -1.00000 54576 06064 69944 a4633 00000 69944 76007 0.38268 0.70710 0.92387 1.00000 0.92387 0.70710 0.38268 0.00000 34324 67812 95325 00000 95325 67812 34324 00000 0.93969 0.76604 0.50000 +0.17364 -0.17364 -0.50000 -0.76604 -0.93969 -1.00000 26208 44431 00000 81777 81777 00000 44431 26208 00000 0 32469 0:61421 0.83716 0.96940 0.99658 0.91577 0.73572 0.47594 0.16459 94692? 27127 64782 02659 44930 33266 39107 73930 45903 -21 0.95557 0.82623 0.62348 0.36534 +0.07473 -0.22252 -0.50000 -0.73305 -0.90096 -0.98883 28058 a7743 98019 10244 00936 09340 00000 18718 88679 08262 0.28173 0.54064 0.75574 0.90963 0.98982 0.98982 0.90963 0.75574 0.54064 0.28173 0.00000 25568 08174 95743 19953 14419 14419 19953 95743 08174 25568 00000 58263 54038 67812 00000 90451 00000 90451 00000 67812 54038 58263 00000 0.24868 0.48175 0.68454 0.84432 0.95105 0.99802 ii98228 0.90482 0.77051 98872 36741 71059 79255 65163 67284 72507 70525 32428 52523 45527 32336 0.91354 0.66913 +0.30901 -0.10452 -0.50000 -0.80901 -0.97814 s=18 SC 8 0.96592 0.86602 0.70710 0.50000 0.25881 + 0.00000 -0.25881 -0.50000 -0.70710 -0.86602 -0.96592 -1.00000 69944 69944 0.43388 0.78183 0.97492 0.97492 0.78183 0.43388 0.00000 s=g 0.76604 +0.17364 -0.50000 -0.93969 s=5 +0.30901 -0.80901 65163 52523 s=6 +0.50000 -0.50000 -1.00000 FUNCTIONS 94581 0:78914 0.54694 +0.24548 -0.08257 -0.40169 -0.67728 -0.87947 -0.98636 s=22 0.95949 0.84125 0.65486 0.41541 +0.14231 -0.14231 -0.41541 -0.65486 -0.84125 -0.95949 -1.00000 b-=25 =24 0.58178 0.36812 0.12533 0.96858 0.87630 0.72896 0.53582 0.30901 +0.06279 -0.18738 -0.42577 -0.63742 -0.80901 -0.92977 -0.99211 31611 66801 86274 67950 69944 05196 13146 92916 39898 69944 64859 47013 0.99166 0.94226 0.81696 0.63108 0.39840 0.13616 s=20 0.95105 0.80901 0.50778 0.30901 +o.ooooo -0.30901 -0.58778 -0.80901 -0.95105 -1.00000 65163 69944 52523 69944 00000 69944 52523 69944 65163 00000 s==23 0. 96291 72874 0. a5441 94046 0. 68255 31432 0. 46006 50378 + 0. 20345 60131 -0. 06824 24134 33487 96122 10": 57668 03221 19443 77571 12907 10898 1; 91721 13015 66491 -0: 99068 59460 ELEMENTARY INVERSE TRANSCENDENTAL CIRCULAR SINES 0.o”o0 0.00000 00000 00 0.001 0.002 0.003 0.004 0.00100 0.00200 0.00300 0.00400 00001 67 00013 33 00045 00 00106 67 arctan z 0.00000 00000 00 0.00099 0.00199 0.00299 0.00399 0.005 0.006 0.007 0.008 0.009 0.00500 0.00600 0.00700 0.00800 0.00900 00208 34 00360 01 00571 68 00853 36 01215 04 0.010 0.011 0.012 0.013 0.014 0.01000 0.01100 0.01200 0.01300 0.01400 0.015 0.016 0.017 0.018 0.019 arcsin 1: 203 FWCTIONS AND Table TANGENTS 4.14 arctan L 99996 67 99973 33 99910 00 99786 67 0.;50 0.051 0.052 0.053 0.054 0 OSO:i%?5:8 06 0:05102 21344 17 0.05202 34632 28 0.05302 48442 51 0.05402 62784 97 0.04995 0.05095 0.05195 0.05295 0.05394 83957 22 58518 77 32065 61 04578 05 76036 42 0.00499 0.00599 0.00699 0.00799 0.00899 99583 34 99280 02 98856 70 98293 40 97570 12 0.055 0.056 0.057 0.058 0.059 0.05502 0.05602 0.05703 0.05803 0.05903 77669 81 93107 15 09107 14 25679 92 42835 64 0.05494 0.05594 0.05693 0.05793 0.05893 46421 07 15712 34 83890 60 50936 23 16829 64 01666 74 02218 45 02880 19 03661 95 04573 74 0.00999 0.01099 0.01199 0.01299 0.01399 96666 87 95563 66 94240 50 92577 41 90'954 41 0.060 0.061 0.062 0.063 0.064 0.06003 0.06103 0.06203 0.06304 0.06404 605'84 45 789.36 52 97902 01 17491 09 37713 94 0.05992 0.06092 0.06192 0.06291 0.06391 81551 21 45081 38 07400 58 68489 26 28327 89 0.01500 0.01600 0.01700 0.01800 0.01900 05625 57 06827 45 08189 40 09721 42 11433 52 0.01499 0.01599 0.01699 0.01799 0.01899 88751 52 86348 76 83626 17 80!563 78 77141 62 0.065 0.066 0.067 0.068 0.069 0.06504 58580 75 0.06604 80101 69 0.06705 02286 97 0.06805 25146 79 0~06905 48691 36 0.06490 0.06590 0.06690 0.06789 0.06889 86896 93 44176 90 00148 29 54791 63 08087 46 0.020 0.021 0.022 0.023 0.024 0.02000 0.02100 0.02200 0.02300 0.02400 13335 73 15438 06 17750 53 20283 16 23045 97 0.01999 73339 73 0.02099 691138 17 0.02199 64516 97 0.02299 59456 20 OiO2399 53935 92 0.070 0.071 0.072 0.073 0.074 0.07005 72930 88 0.07105 97875 58 0.07206 23535 68 0.07306 49921 42 OiO7406 77043 03 0.06988 0.07088 0.07187 0.07287 0.07386 60016 35 10558 85 59695 56 07407 09 53674 06 0.025 0.026 0.027 0.028 0.029 0.02500 0.02600 0.02700 0.02800 0.02900 26048 99 29302 25 32815 77 36599 58 40663 72 0.02499 0.02599 0.02699 0.02799 0.02899 47936 19 41437 08 34418 68 26861 07 18744 33 0.075 0.076 or 077 0.078 0.079 0.07507 04910 77 0.07607 33534 87 OiO7707 62925 62 0.07807 93093 26 0.07908 24048 07 0.07485 98477 11 0.07585 41796 89 OiO7684 83614 08 0.07784 23909 37 0.07883 62663 48 0.030 0.031 0.032 0.033 0.034 0.03000 0.03100 0.03200 0.03300 0.03400 45018 23 49673 15 54638 51 59924 37 65540 77 0.02999 0.03099 0.03198 0.03298 0.03398 10048 57 00753 89 30840 39 80288 21 69077 46 0.080 0.081 0.082 0.083 0.084 0.08008 0.08108 0.08209 0.08309 0.08409 55800 34 88360 35 21738 40 55944 79 90989 83 0.07982 0.08082 0.08181 0.08281 0.08380 99857 12 35471 05 69486 04 01882 86 32642 31 0.035 0.036 0.037 0.038 0.039 0.03500 0.03600 0.03700 0.03800 0.03900 71497 75 77805 38 84473 72 91512 81 98932 73 0.03498 0.03598 0.03698 0.03798 0.03898 57188 29 44600 82 31295 22 17251 64 02450 25 0.085 ii* E 0:oss 0.089 0.08510 0.08610 0.08711 0.08811 0.08911 26883 84 63637 15 01260 09 39763 00 79156 23 0.08479 0.08578 0.08678 0.08777 0.08876 61745 23 89172 45 14904 84 38923 27 61208 65 0.040 0.041 0.042 0.043 0.044 0.04001 0.04101 0.04201 0.04301 0.04401 06743 54 14955 31 23578 12 32622 04 42097 16 0.03997 0.04097 0.04197 0.04297 0.04397 86871 23 70494 77 53301 05 35270 30 16382 71 0.090 0.091 0.092 0.093 0.094 0.09012 0.09112 0.09213 0.09313 0.09413 19450 15 60655 11 02781 49 45839 68 89840 07 0.08975 0.09075 0.09174 0.09273 0.09372 81741 90 00503 96 17475 79 32638 38 45972 74 0.045 0.046 0.047 0.048 0.049 0.04501 0.04601 0.04701 0.04801 0.04901 52013 56 62381 33 73210 57 84511 37 96293 83 0.04496 0.04596 0.04696 0.04796 0.04896 96618 52 75957 97 54381 30 31868 77 08400 65 0.095 0.096 0.097 0.098 0.099 '0.09514 0.09614 0.09715 0.09815 0.09916 34793 06 80709 05 27598 48 75471 75 24339 32 0.09471 0.09570 0.09669 0.09768 0.09867 57459 88 67080 87 74816 76 80648 65 84557 66 0.050 0.05002 08568 06 0.04995 83957 22 0.100 0.10016 74211 62 [1 0.09966 86524 91 (9’16 For use and extension of the table see Examples 21-25. For other inverse functions see 4.4 and 4.3.45. ;=1.57079 63267 95 Compilation of arcsin :c from National Bureau of Standards, Table of arcsin z. Columbia Univ. Press, New York, N.Y., 1945 (with permission). 204 ELEMENTARY Table 4.14 X INVERSE arcsin z TRANSCENDENTAL CIRCULAR FUNCTIONS SINES AND TANGENTS arctan x arcsin 2 X arctan r 0.100 0.101 0.102 0.103 0.104 0.10016 0.10117 0.10217 0.10318 0.10418 74211 25099 77012 29961 83957 62 11 25 53 41 0.09966 0.10065 0.10164 0.10263 0.10362 86524 86531 84558 80587 74599 91 58 83 89 97 0.150 0.151 0.152 0.153 0.154 0.15056 0.15157 0.15259 0.15360 0.15461 82727 97940 14716 33066 53001 77 40 20 23 61 0.14888 0.14986 0.15084 0.15182 0.15279 99476 77989 53616 26338 96139 09 58 21 59 37 0.105 0.106 0.107 0.108 0.109 0.10519 0.10619 0.10720 0.10821 0.10921 39010 95131 52329 10617 70003 40 00 72 08 62 0.10461 0.10560 0.10659 0.10758 0.10857 66576 56498 44346 30103 13750 33 23 99 93 39 0.155 0.156 0.157 0.158 0.159 0.15562 0.15663 0.15765 0.15866 0.15967 74533 97672 22431 48819 76848 44 86 01 05 15 0.15377 0.15475 0.15572 0.15670 0.15768 63001 26906 87838 45780 00713 20 78 86 19 58 0.110 0.111 0.112 0.113 0.114 0.11022 0.11122 0.11223 0.11324 0.11424 30499 92116 54863 18752 83793 88 41 77 55 32 0.10955 0.11054 0.11153 0.11252 0.11350 95267 74637 51840 26859 99674 74 38 74 25 40 0.160 0.161 0.162 0.163 0.164 0.16069 0.16170 0.16271 0.16373 0.16474 06529 37874 70893 05599 42001 52 35 88 34 99 0.15865 0.15963 0.16060 0.16157 0.16255 52621 01487 47294 90024 29661 86 91 61 91 78 0.115 0.116 0.117 0.118 0.119 0.11525 0.11626 0.11726 0.11827 0.11928 49996 17373 85933 55688 26648 68 23 61 42 32 0.11449 0.11548 0.11647 0.11745 0.11844 70267 38620 04714 68531 30052 67 60 73 63 90 0.165 Oiibk 0.167 0.168 0.169 0.16575 Oii6677 0.16778 0.16880 0.16981 80113 19943 61505 04810 49868 10 96 87 17 19 0.16352 0.16449 0.16547 0.16644 0.16741 66188 99587 29841 56935 80850 21 25 97 49 93 0.120 0.121 0.122 0.123 0.124 0.12028 0.12129 0.12230 0.12331 0.12431 98823 72225 46865 22751 99897 95 97 07 92 22 0.11942 0.12041 0.12140 0.12238 0.12337 89260 46135 00659 52814 02582 18 12 40 72 82 0.170 0.171 0.172 0.173 0.174 0.17082 0.17184 0.17285 0.17387 0.17489 96691 45290 95678 47864 01862 29 84 23 87 19 0.16839 0.16936 0.17033 0.17130 0.17227 01571 19080 33360 44396 52169 48 34 78 07 54 0.125 oIi2k 0.127 0.128 0.129 0.12532 OIi2633 0.12734 0.12835 0.12936 78311 58006 38990 21277 04875 68 02 98 29 72 0.12435 0:125>3 0.12632 0.12730 0.12829 49945 94884 37381 77418 14977 47 45 58 71 71 i* E 0:177 0.178 0.179 0.17590 0.17692 0.17793 0.17895 0.17996 57681 15334 74832 36187 99410 64 66 75 40 13 0.17324 0.17421 0.17518 0.17615 0.17712 56664 57864 55752 50312 41528 52 43 68 74 10 0.130 0.131 0.132 0.133 0.134 0.13036 0.13137 0.13238 0.13339 0.13440 89797 76052 63651 52606 42926 03 01 45 16 95 0.12927 0.13025 0.13124 0.13222 0.13320 50040 82588 12605 40070 64968 48 96 10 89 35 0.135 0.136 0.137 0.138 0.139 0.13541 0.13642 0.13743 0.13844 Oil3945 34624 27710 22194 18087 15401 67 15 25 85 83 0.13418 0.13517 0.13615 0.13713 0.13811 87279 06986 24071 38516 50303 0.140 0.141 0.142 0.143 0.144 0.14046 0.14147 0.14248 0.14349 0.14450 14147 14334 15975 19079 23659 10 56 13 77 42 0.13909 0.14007 0.14105 0.14203 0.14301 0.145 Oil46 0.147 0.148 0.149 0.14551 Oil4652 0.14753 0.14854 0.14955 29725 37287 46358 56947 69067 04 64 19 71 22 0.14399 Oil4497 0.14595 0.14693 0.14791 0.150 0.15056 82727 77 II(792 1 0.18098 64512 47 0.17809 29382 31 0:182 ;;;; 0.183 0.184 0.18200 31505 20 0.18302 00402 97 0.18403 71212 76 0.18505 43949 25 0.17906 13858 59 0.18002 94941 94 0.18099 72613 91 0.18196 46859 59 52 49 35 25 34 0.185 0.186 0.187 0.188 0.189 0.18607 0.18708 0.18810 0.18912 0.19014 18623 95246 73830 54387 36928 31 57 71 40 36 0.18293 0.18389 0.18486 0.18583 0.18679 17662 85005 48874 09250 66119 35 94 16 85 87 59414 65832 69539 70518 68749 82 92 90 03 65 0.190 0.191 0.192 0.193 0.194 0.19116 0.19218 0.19319 0.19421 0.19523 21465 08009 96574 87169 79808 31 99 17 63 18 0.18776 0.18872 0.18969 0.19065 0.19161 19465 69270 15520 58198 97288 14 59 22 05 15 64217 56902 46789 33858 18093 09 74 00 33 19 0.195 0.196 0.197 0.198 0.199 0.19625 0.19727 0.19829 0.19931 0.20033 74501 71261 70100 71030 74061 64 85 69 03 80 0.19258 0.19354 0.19450 0.19547 0.19643 32774 64641 92873 17453 38367 60 55 18 71 38 0.14888 99476 09 0.200 0.20135 79207 90 [c-y4 1 [ C-f)3 1 ;=1.57079 63267 95 0.19739 55598 50 [c-j15 1 ELEMENTARY TRANSCENDENTAL 205 F’UNCTIONS INVERSE CIRCULAR SINES AND TANGENTS X arcsin x arctan x X arcsin x Table 4.14 arctan x 0.200 0.201 0.202 0.203 0.204 0.20135 0.20237 0.20339 0.20442 0.20544 79207 90 86480 31 95890 97 07451 90 21175 10 0.19739 0.19835 0.19931 0.20027 0.20123 55598 50 69131 40 78950 44 85040 06 87384 69 0.250 0.251 0.252 0.253 0.254 0.25268 0.25371 0.25474 0.25577 0.25681 02551 42 31886 28 63988 49 98871 33 36548 08 0.24497 0.24591 0.24686 0.24780 0.24873 86631 27 96179 19 01284 51 01933 77 98113 53 0.205 0.206 0.207 0.208 0.209 0.20646 0.20748 0.20850 0.20952 0.21055 37072 61 55156 48 75438 81 97931 68 22647 22 0.20219 0.20315 0.20411 0.20507 0.20603 85968 83 80777 01 71793 81 59003 83 42391 73 0.255 0.256 0.257 0.258 0.259 0.25784 0.25888 0.25991 0.26095 0.26198 77032 07 20336 66 66475 22 15461 18 67307 97 0.24967 0.25061 0.25155 0.25249 0.25343 89810 38 77010 99 59702 05 37870 29 11502 51 0.210 0.211 0.212 0.213 0.214 0.21157 0.21259 0.21362 0.21464 0.21566 49597 58 78794 93 10251 46 43979 39 79990 96 0.20699 0.20794 0.20890 0.20986 0.21082 21942 20 97639 97 69469 83 37416 57 01465 06 0.260 0.261 0.262 0.263 0.264 0.26302 0.26405 0.26509 0.26613 0.26716 22029 08 79638 02 40148 31 03573 53 69927 28 0.25436 0.25530 0.25624 0.25717 0.25811 80585 53 45106 23 05051 53 60408 40 11163 83 0.215 0.216 0.217 0.218 0.219 0.21669 0.21771 0.21874 0.21976 0.22078 18298 42 58914 06 01850 19 47119 15 94733 28 0.21177 0.21273 0.21368 0.21464 0.21559 61600 20 17806 92 70070 19 18375 04 62706 53 0.265 0.266 0.267 0.268 0.269 0.26820 0.26924 0.27027 0.27131 0.27235 39223 20 11474 95 86696 22 64900 75 46102 31 0.25904 0.25997 0.26091 0.26184 0.26277 57304 89 98818 68 35692 33 67913 04 95468 05 0.220 0.221 0.222 0.223 0.224 0.22181 0.22283 0.22386 0.22489 0.22591 44704 97 97046 62 51770 66 08889 55 68415 75 0.21655 0.21750 0.21845 0.21941 0.22036 03049 76 39389 87 71712 05 00001 53 24243 57 0.270 0.271 0.272 0.273 0.274 0.27339 0.27443 0.27547 0.27651 0.27754 30314 67 17551 69 07827 21 01155 13 97549 38 0.26371 0.26464 0.26557 0.26650 0.26743 18344 62 36530 10 50011 84 58777 27 62813 84 0.225 0.226 0.227 0.228 0.229 0.22694 0.22796 0.22899 0.23002 0.23105 30361 79 94740 17 61563 45 30844 22 02595 07 0.22131 0.22226 0.22321 0.22416 0.22511 44423 48 60526 61 72538 37 80444 19 84229 53 0.275 0.276 0.277 0.278 0.279 0.27858 0.27962 0.28067 0.28171 0.28275 97023 92 99592 75 05269 90 14069 43 26005 45 0.26836 0.26929 0.27022 0.27115 0.27208 62109 06 56650 49 46425 71 31422 39 11628 19 0.230 0.231 0.232 0.233 0.234 0.23207 0.23310 0.23413 0.23516 0.23618 76828 63 53557 56 32794 53 14552 26 98843 48 0.22606 0.22701 0.22796 0.22891 0.22986 83879 94 79380 96 70718 22 57877 34 40844 03 0.280 0.281 0.282 0.283 0.284 0.28379 0.28483 0.28587 0.28692 0.28796 41092 08 59343 51 80773 93 05397 58 33228 75 0.27300 0.27393 0.27486 0.27578 0.27671 87030 87 57618 19 23377 99 84298 14 40366 55 0.235 0.236 0.237 0.238 0.239 0.23721 0.13824 0.23927 0.24030 0.24133 85680 94 75077 44 67045 78 61598 80 58749 37 0.23081 0.23175 0.23270 0.23365 0.23459 19604 03 94143 10 64447 07 30501 80 92293 19 0.285 0.286 0.287 0.288 0.289 0.28900 0.29004 0.29109 0.29213 0.29318 64281 74 98570 89 36110 61 76915 30 20999 43 0.27763 0.27856 0.27948 0.28041 0.28133 91571 20 37900 08 79341 26 15882 83 47512 95 0.240 0.241 0.242 0.243 0.244 0.24236 0.24339 0.24442 0.24545 0.24648 58510 39 60894 77 65915 47 73585 45 83917 73 0.23554 0.23649 0.23743 0.23837 0.23932 49807 21 03029 83 51947 10 96545 10 36809 95 0.290 0.291 0.292 0.293 0.294 0.29422 68377 49 Oi29527 19064 01 0.29631 73073 57 0.29736 30420 76 0.29840 91120 25 0.28225 0.28317 0.28410 0.28502 0.28594 74219 81 95991 65 12816 76 24683 46 31580 14 0.245 0,246 0.247 0.248 0.249 0.24751 0.24855 0.24958 0.25061 0.25164 96925 34 12621 33 31018 81 52130 88 75970 69 0.24026 0.24121 0.24215 0.24309 0.24403 72727 81 04284 90 31467 47 54261 82 72654 29 0.295 0.296 0.297 Oi298 0.299 0.29945 0.30050 0.30154 0.30259 0.30364 0.28686 0.28778 0.28870 0.28962 0.29053 33495 23 30417 18 22334 53 09235 83 91109 69 0.250 0.25268 02551 42 0.24497 86631 27 0.300 0.30469 26540 15 [y4 1 55186 70 22634 85 93479 45 67735 30 45417 24 [C-j)6 1 P'1.57079 6326795 [F-f)4 1 2 0.29145 67944 78 [C-i)‘31 206 ELEMENTARY Table x INVERSE 4.14 arcsin x TRANSCENDENTAL CIRCULAR SINES nrctan x AND FUNCTIONS TANGENTS arcsin x 5 arctan x 0.300 0.301 0.302 0.303 0.304 0.30469 0.30574 0.30678 0.30783 0.30888 26540 15 11118 95 99168 60 90704 09 85740 46 0.29145 0.29237 0.29329 0.29420 0.29512 67944 78 39729 79 06453 47 68104 62 24672 09 0.350 0.351 0.352 0.353 0.354 0.35757 0.35863 0.35970 0.36077 0.36184 11036 46 88378 55 69995 85 55905 70 46125 51 0.33667 0.33756 0.33845 0.33934 0.34023 48193 a7 54100 58 54442 85 49211 al 38398 61 0.305 0.306 0.307 0.308 0.309 0.30993 0.31098 0.31203 0.31309 0.31414 a4292 78 a6376 19 92005 83 01196 91 13964 68 0.29603 0.29695 0.29786 0.29877 0.29969 76144 75 22511 55 63761 46 99883 52 30866 80 0.355 0.356 0.357 0.358 0.359 0.36291 0.36398 0.36505 0.36612 0.36719 40672 71 39564 82 42819 39 50454 05 62486 46 0.34112 0.34200 0.34289 0.34378 0.34467 21994 49 99990 70 72378 56 39149 42 00294 69 0.310 0.311 0.312 0.313 0.314 0.31519 0.31624 0.31729 0.31835 0.31940 30324 41 50291 43 73881 12 01108 aa 31990 la 0.30060 0.30151 0.30242 0.30334 0.30425 56700 42 77373 55 92875 41 03195 25 08322 38 0.360 0.361 0.362 0.363 0.364 0.36826 0.36933 0.37041 0.37148 0.37255 78934 37 99815 54 25147 84 54949 16 89237 46 0.34555 0.34644 0.34732 0.34820 0.34909 55805 a2 05674 30 49891 68 a8449 54 21339 52 0.315 0.316 0.317 0.318 0.319 0.32045 0.32151 0.32256 0.32361 0.32467 66540 50 04775 38 46710 42 92361 24 41743 51 0.30516 0.30607 0.30697 0.30788 0.30879 08246 16 02955 99 92441 31 76691 62 55696 46 0.365 0.366 0.367 0.368 0.369 0.37363 0.37470 0.37578 0.37685 0.37793 28030 75 71347 12 19204 71 71621 69 28616 34 0.34997 0.35085 0.35173 0.35261 0.35350 48553 30 70082 60 a5919 21 96054 93 00481 64 0.320 0.321 0.322 0.323 0.324 0.32572 0.32678 0.32784 0.32889 0.32995 94872 95 51765 31 12436 42 76902 11 45178 29 0.30970 0.31060 0.31151 0.31242 0.31332 29445 42 97928 14 61134 29 19053 60 71675 a4 0.370 0.371 0.372 0.373 0.374 0.37900 0.38008 0.38116 0.38224 0.38331 90206 96 56411 93 27249 69 02738 73 82897 61 0.35437 0.35525 0.35613 0.35701 0.35789 99191 23 92175 68 79426 98 60937 la 36698 38 0.325 0.326 0.327 0.328 0.329 0.33101 0.33206 0.33312 0.33418 0.33524 17280 a9 93225 91 73029 38 56707 38 44276 04 0.31423 0.31513 0.31603 0.31694 0.31784 la990 a4 60988 47 97658 63 28991 30 54976 47 0.375 0.376 0.377 0.378 0.379 0.38439 0.38547 0.38655 0.38763 0.38871 67744 96 57299 45 51579 a3 50604 92 54393 57 0.35877 0.35964 0.36052 0.36139 0.36227 06702 71 70942 35 29409 56 82096 58 28995 76 0.330 0.331 0.332 0.333 0.334 0.33630 0.33736 0.33842 0.33948 0.34054 35751 54 31150 09 30487 98 33781 50 41047 05 0.31874 0.31964 0.32055 0.32145 0.32235 75604 21 90864 60 00747 al 05244 03 04343 49 0.380 0.381 0.382 0.383 0.384 0.38979 0.39087 0.39195 0.39304 0.39412 62964 74 76337 42 94530 68 17563 64 45455 51 0.36314 0.36402 0.36489 0.36576 0.36663 70099 46 05400 09 34890 12 58562 04 76408 40 0.335 0.336 0.337 0.338 0.339 0.34160 0.34266 0.34372 0.34479 0.34585 52301 02 67559 88 86840 15 10158 39 37531 21 0.32324 0.32414 0.32504 0.32594 0.32684 98036 48 86313 34 69164 46 46580 25 la551 19 0.385 0.386 0.387 0.388 0.389 0.39520 0.39629 0.39737 0.39846 0.39954 78225 54 15893 06 58477 48 05998 24 58474 a9 0.36750 0.36837 0.36924 0.37011 0.37098 88421 al 94594 90 94920 36 a9390 92 77999 35 0.340 0.341 0.342 0.343 0.344 0.34691 0.34798 0.34904 0.35010 0.35117 68975 27 04507 29 44144 03 87902 30 35798 98 0.32773 0.32863 0.32953 0.33042 0.33131 85067 al 46120 66 01700 37 51797 60 96403 04 0.390 0.391 0.392 0.393 0.394 0.40063 0.40171 0.40280 0.40389 0.40497 15927 01 78374 28 45836 44 la333 27 95884 67 0.37185 0.37272 0.37359 0.37445 0.37532 60738 49 37601 la 08580 36 73668 96 32860 01 0.345 0.346 0.347 0.348 0.349 0.35223 0.35330 0.35437 0.35543 0.35650 87850 97 44075 25 04488 a4 69108 al 37952 29 0.33221 0.33310 0.33399 0.33489 0.33578 35507 47 69101 67 97176 49 19722 a3 36731 63 0.395 0.396 0.397 0.398 Oi399 0.40606 0.40715 0.40824 0.40933 0.41042 78510 57 66231 00 59066 02 57035 al 60160 60 0.37618 0.37705 0.37791 0.37878 0.37964 a6146 53 33521 62 74978 43 10510 12 40109 93 0.350 0.35757 11036 46 0.33667 48193 a7 0.400 0.41151 68460 67 cc-:)8)51 1 -f)6 1 C [ c-t)7 1 ;=I.57079 63267 95 0.38050 63771 12 [c-pa 1 ELEMENTARY INVERSE X arcsin x TRANSCENDENTAL CIRCULAR SINES arctan x 207 FUNCTIONS AND TANGENTS Table X arcsin z 4.14 arctan z 0.400 0.401 0.402 0.403 0.404 0.41151 0.41260 0.41370 0.41479 0.41588 68460 67 81956 42 00668 29 24616 80 53822 54 0.38050 0.38136 0.38222 0.38308 0.38394 63771 12 81487 02 93250 97 99056 39 98896 72 0.450 0.451 0.452 0.453 0.454 0.46676 53390 47 0.46788 54404 09 0.46900 61761 03 0.47012 75486 20 Oi47124 95604 59 0.42285 0.42368 0.42451 0.42534 0.42617 39261 33 52156 87 58823 89 59257 92 53454 56 0.405 0.406 0.407 0.408 0.409 0.41697 0.41807 0.41916 0.42026 0.42135 88306 20 28088 50 73190 29 23632 45 79435 96 0.38480 0.38566 0.38652 0.38738 0.38824 92765 46 80656 14 62562 34 38477 69 08395 85 0.455 0.456 0.457 0.458 0.459 0.47237 0.47349 0.47461 0.47574 0.47686 22141 29 55121 50 94570 53 40513 79 92976 80 0.42700 0.42783 0.42865 0.42948 0.43031 41409 43 23118 21 98576 60 67780 36 30725 28 0.410 0.411 0.412 0.413 0.414 0.42245 0.42355 0.42464 0.42574 0.42684 40621 87 07211 31 79225 49 56685 70 39613 30 0.38909 0.38995 0.39080 0.39166 0.39251 72310 55 30215 54 82104 62 27971 64 67810 48 0.460 0.461 0.462 0.463 0.464 0.47799 0.47912 0.48024 0.48137 0.48250 51985 19 17564 68 89741 12 68540 46 53988 75 0.43113 0.43196 0.43278 0.43361 0.43443 87407 19 37821 96 81965 51 19833 80 51422 81 0.415 0.416 0.417 0.418 0.419 0.42794 0.42904 0.43014 0.43124 0.43234 28029 74 21956 53 21415 30 26427 72 37015 57 0.39337 0.39422 0.39507 0.39592 0.39677 01615 09 29379 43 51097 52 66763 44 76371 29 0.465 0.466 0.467 0.468 0.469 0.48363 0.48476 0.48589 0.48702 0.48815 46112 18 44937 02 50489 67 62796 64 81884 55 0.43525 0.43607 0.43690 0.43772 0.43854 76728 58 95747 19 08474 74 14907 40 15041 36 0.420 0.421 0.422 0.423 0.424 0.43344 0.43454 0.43565 0.43675 0.43785 53200 70 75005 03 02450 60 35559 49 74353 90 0.39762 0.39847 0.39932 0.40017 0.40102 79915 22 77389 43 68788 14 54105 66 33336 29 0.470 0.471 0.472 0.473 0.474 0.48929 0.49042 0.49155 0.49269 0.49382 07780 14 40510 26 80101 88 26582 08 79978 07 0.43936 0.44017 0.44099 0.44181 0.44263 08872 85 96398 14 77613 55 52515 43 21100 17 0.425 0.426 0.427 0.428 0.429 0.43896 0.44006 0.44117 0.44227 0.44338 18856 10 69088 44 25073 36 86833 39 54391 16 0.40187 0.40271 0.40356 0.40440 0.40525 06474 40 73514 42 34450 79 89278 00 37990 60 0.475 0.476 0.477 0.478 0.479 0.49496 0.49610 0.49723 0.49837 0.49951 40317 17 07626 82 81934 59 63268 16 51655 34 0.44344 0.44426 0.44507 0.44589 0.44670 83364 20 39303 99 88916 06 32196 95 69143 24 0.430 0.431 0.432 0.433 0.434 0.44449 0.44560 0.44670 0.44781 0.44892 27769 36 06990 78 92078 31 83054 92 79943 67 0.40609 0.40694 0.40778 0.40862 0.40946 80583 18 17050 34 47386 77 71587 18 89646 31 0.480 0.481 0.482 0.483 0.484 0.50065 0.50179 0.50293 0.50407 0.50522 47124 05 49702 34 59418 39 76300 52 00377 13 0.44751 0.44833 0.44914 0.44995 0.45076 99751 57 24018 60 41941 03 53515 61 58739 11 0.435 0.436 0.437 0.438 0.439 0.45003 0.45114 0.45226 0.45337 0.45448 82767 71 91550 28 06314 71 27084 44 53882 99 0.41031 0.41115 0.41199 0.41283 0.41366 01558 96 07319 97 06924 22 00366 64 87642 17 0.485 0.486 0.487 0.488 0.489 0.50636 0.50750 0.50865 0.50979 0.51094 31676 79 70228 19 16060 14 69201 57 29681 57 0.45157 0.45238 0.45319 0.45400 0.45480 57608 36 50120 20 36271 55 16059 33 89480 51 0.440 0.441 0.442 0.443 0.444 0.45559 0.45671 0.45782 0.45894 0.46005 86733 96 25661 07 70688 11 21838 99 79137 71 0.41450 0.41534 0.41618 0.41701 0.41785 68745 85 43672 70 12417 83 74976 36 31343 48 0.490 0.491 0.492 0.493 0.494 0.51208 0.51323 0.51438 0.51553 0.51668 97529 34 72774 22 55445 69 45573 34 43186 93 0.45561 0.45642 0.45722 0.45803 0.45883 56532 11 17211 17 71514 78 19440 06 60984 16 0.445 0.446 0.447 0.448 0.449 0.46117 0.46229 0.46340 0.46452 0.46564 42608 35 12275 10 88162 25 70294 19 58695 40 0.41868 0.41952 0.42035 0.42118 0.42202 81514 38 25484 34 63248 66 94802 67 20141 75 0.495 0.496 0.497 0.498 0.499 0.51783 0.51898 0.52013 0.52129 0.52244 48316 32 60991 55 81242 77 09100 26 44594 47 0.45963 0.46044 0.46124 0.46204 0.46284 96144 30 24917 71 47301 65 63293 45 72890 44 0.450 0.46676 53390 47 0.42285 39261 33 0.500 0.52359 87755 98 1 -J)8 C1 [c-y1 ;=1.57079 6326795 [c-y1 [c-y1 0.46364 76090 01 ELEMENTARY 208 Table 4.14 INVERSE arcsin x X TRANSCENDENTAL CIRCULAR SINES arctan 2 AND FUNCTIONS TANGENTS arcsin x arctan 2 0.550 0.551 0.552 0.553 0.554 0.58236 42378 69 0.58356 20792 89 0.50284 32109 28 0.50361 06410 37 0.58476 08688 0.50437 74226 2 76090 01 98 0.46364 51 0.46444 72889 58 91 ' 0.46524 63286 62 20 0.46604 47278 61 24863 09 54 0.46684 0.500 0.501 0.502 0.503 0.504 0.52359 0.52475 0.52590 0.52706 0.52822 87755 38615 97203 63552 37691 0.505 0.506 0.507 0.508 0.509 0.52938 0.53054 0.53170 0.53286 0.53402 19653 22 09468 69 07169 56 12787 56 26354 61 0.46763 0.46843 0.46923 0.47002 0.47082 96037 63 60799 83 19147 34 71077 82 16589 00 0.555 0.556 0.557 0.558 0.559 0.58836 0.58956 0.59076 0.59197 0.59317 29661 37 55882 10 91785 32 37411 92 92803 04 0.50667 0.50743 0.50820 0.50896 0.50972 38759 68 80629 53 16011 02 44903 52 67306 43 0.510 0.511 0.512 0.513 0.514 0.53518 0.53634 0.53751 0.53867 0.53984 47902 76 77464 20 15071 30 55678 62 88344 48 14584 38 34396 20 47777 82 0.560 0.561 14552 69 0.47161 0.47240 0.47320 0.47399 0.47478 0.59438 58000 01 0.59559 33044 41 0.59680 17978 05 0.59801 12842 95 0.59922 17681 37 0.51048 0.51124 0.51200 0.51276 0.51352 83219 17 92641 21 95572 04 92011 19 81958 22 0.515 0.516 0.517 0.518 0.519 0.54100 0.54217 0.54334 0.54451 0.54568 76492 0.47557 54727 17 0.60043 32535 81 0.60164 57448 99 0.60285 92463 89 0.60407 37623 71 0.60528 92971 89 0.51428 0.51504 0.51580 0.51655 0.51731 65412 69 42374 25 12842 52 76817 18 34297 96 0.520 0.521 0.522 0.523 0.524 0.54685 0.54802 0.54919 0.55036 0.55154 09506 0.60650 58552 13 0.60772 34408 36 0.61016 17125 74 0.61138 24076 01 0.51806 85284 57 0.51882 29776 79 0.51957 67774 41 0.52032 99277 27 0.52108 24285 22 0.525 0.526 0.527 0.528 0.529 0.577 0.578 0.579 0.61260 41480 49 69384 37 0.61505 07833 09 0.61627 56872 37 0.61750 16548 17 0.52183 0.52258 0.52333 0.52408 0.52483 42798 14 54815 96 60338 62 59366 09 51898 38 37935 60756 57 0.562 0.563 0.564 33 0.58596 06104 84 0.58716 13082 43 73 0.50514 35557 57 0.50590 90402 12 0.47636 55242 0.47715 49320 97 0.47794 36961 45 0.47873 18161 73 0.565 0.566 0.567 0.568 0.569 21007 28 40885 61 69176 11 05913 07 0.47951 0.48030 0.48109 0.48187 0.48266 92919 93 61234 17 23102 64 78523 54 27495 12 0.570 0.571 0.572 0.573 0.574 0.55271 0.55389 0.55506 0.55624 0.55742 51130 97 04864 46 67148 37 38017 69 17507 59 0.48344 0.48423 0.48501 0.48579 0.48657 70015 67 0.575 06083 0.576 0.530 0.531 0.532 0.533 0.534 0.55860 0.55978 0.56096 0.56214 0.56332 05653 43 02490 72 08055 18 22382 69 45509 33 0.48735 0.48813 0.48891 0.48969 0.49047 85795 05 89575 18 86893 19 77747 65 62137 12 0.580 0.581 0.582 0.583 0.584 0.61872 86906 72 0.61995 67994 52 0.62118 59858 34 0.62241 62545 21 0.62364 76102 44 0.52558 0.52633 0.535 0.536 0.537 0.538 0.539 0.56450 77471 34 0.56569 18305 17 0.49125 0.49203 0.49280 0.49358 0.49435 40060 25 11515 68 76502 10 35018 23 87062 83 0.585 0.586 0.587 0.588 0.589 0.62488 00577 61 0.62611 36018 60 0.62734 82473 54 0.62858 39990 0.52931 70697 19 0.53006 17765 76 0.53080 58340 23 0.53154 92420 86 0.53229 20007 93 0.540 0.541 0.542 0.543 0.544 0.57043 0.57162 0.57281 0.57400 0.57519 71094 00 56840 08 51680 58 55653 28 68796 15 0.49513 32634 68 0.49590 71732 62 0.590 0.591 0.592 0.593 0.594 0.63105 0.63229 0.63353 0.63477 88407 78 79405 66 81bb2 50 95228 17 20152 84 0.53303 0.53377 0.53451 0.53525 0.53599 41101 77 55702 73 63811 18 0.545 0.546 0.547 0.548 0.549 0.57638 0.57758 0.57877 0.57997 0.58116 91147 36 22745 29 63628 51 13835 79 73406 12 0.49899 49185 66 31328 39 06980 90 76143 74 38817 48 0.550 0.58236 42378 69 0.56687 49 46608 96 24935 25 11504 67 06350 69 68047 96 44 0.56806 26734 97 0.56924 94404 76 C-l)1 [I 1 0.49668 22 50 35696 94 58854 40 75554 29 04355 48 0.49745 30502 17 0.49822 50171 59 0.49976 63362 70074 0.60894 20584 75 0.61382 87 0.62982 08619 28 0.63602 52 17477 57 0.52707 90524 63 0.52782 57076 82 0.52857 17134 28 65427 53 60552 20 71 0.595 50 0.596 0.63851 0.63726 56487 00 04281 42 0.63975 63587 17 0.64100 34455 66 0.64225 16938 57 0.53673 0.53747 0.53821 0.53894 0.53968 0.600 0.64350 11087 93 0.54041 95002 71 0.50053 70305 98 0.50130 64056 22 0.50207 51324 28 c(-y 1 f= 0.50284 32109 28 0.597 0.598 0.599 [(-;I2 1 1.57079 63267 95 [c-y1 ELEMENTARY INVERSE X arcsin 5 TRANSCENDENTAL CIRCULAR arctan x 209 FUNCTIONS SINES AND TANGENTS X Table 4.14 arcsin x arctan 22 0.600 0.601 0: 602 0.603 0.604 0.64350 0.64475 0.64600 0.64725 0.64851 11087 93 16956 07 34595 63 64059 60 05401 26 0.54041 0.54115 0.54188 0.54262 0.54335 95002 71 44700 04 87910 15 24633 69 54871 37 0.650 0.651 0.652 0.653 0.654 0.70758 0.70890 0.71021 0.71153 0.71285 44367 25 10818 82 92154 53 88447 93 99773 14 0.57637 0.57707 0.57777 0.57848 0.57918 52205 91 78870 95 99113 37 12935 07 20337 94 0.605 0.606 0.607 0.608 0.609 0.64976 0.65102 0.65228 0.65353 0.65479 58674 24 23932 51 01230 34 90622 38 92163 58 0.54408 0.54481 0.54555 0.54628 0.54701 78623 92 95892 10 06676 70 10978 51 08798 38 0.655 0.656 0.657 0.658 0.659 0.71418 0.71550 0.71683 0.71815 0.71948 26204 76 67817 97 24688 45 96892 45 84506 75 0.57988 0.58058 0.58128 0.58197 0.58267 21323 94 15895 01 04053 13 85800 31 61138 57 0.610 0.611 0.612 0.613 0.614 0.65606 0.65732 0.65858 0.65985 0.66111 05909 25 31915 05 70237 00 20931 44 84055 09 0.54774 0.54846 0.54919 0.54992 0.55065 00137 16 84995 75 63375 05 35276 01 00699 59 0.660 0.661 0.662 0.663 0.664 0.72081 0.72215 0.72348 0.72481 0.72615 87608 70 06276 21 40587 76 90622 40 56459 74 0.58337 0.58406 0.58476 0.58545 0.58615 30069 94 92596 49 48720 31 98443 49 41768 17 0.615 0.616 0.617 0.618 0.619 0.66238 0.66365 0.66492 0.66619 0.66746 59665 02 47818 67 48573 84 61988 69 88121 78 0.55137 0.55210 0.55282 0.55354 0.55427 59646 79 12118 61 58116 10 97640 33 30692 38 0.665 0.666 0.667 0.668 0.669 0.72749 0.72883 0.73017 0.73151 0.73286 38180 01 35864 02 49593 16 79449 44 25515 49 0.58684 0.58754 0.58823 0.58892 0.58961 78696 50 09230 63 33372 77 51125 11 62489 89 0.620 0.621 0.622 0.623 0.624 0.66874 0.67001 0.67129 0.67257 0.67385 27032 02 78778 71 43421 53 21020 54 11636 20 0.55499 0.55571 0.55643 0.55715 0.55787 57273 39 77384 48 91026 82 98201 62 98910 07 0.670 0.671 0.672 0.673 0.674 0.73420 0.73555 0.73690 0.73825 0.73961 87874 53 66610 44 61807 69 73551 41 01927 39 0.59030 0.59099 0.59168 0.59237 0.59306 67469 35 66065 77 58281 44 44118 66 23579 77 0.625 0.626 0.627 0.628 0.629 0.67513 0.67641 0.67769 0.67898 0.68026 15329 37 32161 29 62193 62 05488 41 62108 12 0.55859 0.55931 0.56003 0.56075 0.56147 93153 44 80932 97 62249 97 37105 74 05501 63 0.675 0.676 0.677 0.678 0.679 0.74096 0.74232 0.74367 0.74503 0.74639 47022 03 08922 43 87716 32 83492 13 96338 96 0.59374 0.59443 0.59512 0.59580 0.59649 96667 11 63383 05 23729 99 77710 32 25326 49 0.630 0.631 0.632 0.633 0.634 0.68155 0.68284 0.68413 0.68542 0.68671 32115 63 15574 24 12547 66 23100 04 47295 93 0.56218 0.56290 0.56361 0.56433 0.56504 67439 00 22919 24 71943 75 14513 97 50631 37 0.680 0.681 0.682 0.683 0.684 0.74776 0.74912 0.75049 0.75186 0.75323 26346 60 73605 52 38206 91 20242 68 19805 42 0.59717 0.597&6 0.59854 0.59922 0.59990 66580 93 01476 11 30014 52 52198 66 68031 06 0.635 0.636 0.637 0.638 0.639 0.68800 0.68930 0.69060 0.69189 0.69319 85200 35 36878 74 02396 97 81821 37 75218 73 0.56575 0.56647 0.56718 0.56789 0.56860 80297 42 03513 63 20281 53 30602 67 34478 63 0.685 0.686 0.687 0.688 0.689 0.75460 0.75597 0.75735 0.75872 0.76010 36988 49 71885 95 24592 63 95204 10 83816 68 0.60058 0.60126 0.60194 0.60262 0.60330 77514 26 80650 81 77443 31 67894 35 52006 54 0.640 0.641 0.642 0.643 0.644 0.69449 0.69580 0.69710 0.69840 0.69971 82656 27 04201 68 39923 13 89889 23 54169 09 0.56931 0.57002 0.57073 0.57143 0.57214 31911 01 22901 42 07451 52 85562 98 57237 47 0.690 0.691 0.692 01693 0.694 0.76148 0.76287 0.76425 0.76564 0.76703 90527 48 15434 36 58636 00 20231 84 00322 15 0.60398 0.60466 0.60533 0.60601 0.60668 29782 53 01224 96 66336 52 25119 88 77577 76 0.645 0.646 0.647 0.648 0.649 0.70102 0.70233 0.70364 0.70495 0.70626 32832 27 25948 84 33589 34 55824 80 92726 76 0.57285 0.57355 0.57426 0.57496 0.57567 22476 73 81282 48 33656 48 79600 51 19116 38 0.695 0.696 0.697 0.698 0.699 0.76841 0.76981 0.77120 0.77260 0.77399 99008 00 16391 29 52574 75 07661 95 81757 30 0.60736 0.60803 0.60870 0.60938 0.61005 23712 89 63528 01 97025 88 24209 28 45081 01 0.650 0.70758 44367 25 (-;I2 [ 1 (-JP 0.700 0.7753;-WY;66 11 [ I- ;=1.57079 6326795 [ 5 1 0.57637 52205 91 0.61072 59643 89 [c-y3 1 210 ELEMENTARY INVERSE Table 4.14 X arcsin x TRANSCENDENTAL CIRCULAR SINES arctan x FUNCTIONS AND TANGENTS arcsin z X arctan x 0.700 0.701 0.702 0.703 0.704 0.77539 0.77679 0.77820 0.77960 0.78101 74966 11 87394 52 19149 57 70339 20 41072 23 0.61072 0.61139 0.61206 0.61273 0.61340 59643 89 67900 75 69854 44 65507 83 54863 79 0.750 0.751 0.752 0.753 0.754 0.84806 0.84957 0.85109 0.85260 0.85413 20789 81 52355 56 10007 70 93916 63 04254 45 0.64350 0.64414 0.64477 0.64541 0.64605 11087 93 08016 53 98804 75 83456 20 61974 52 0.705 0.706 0.707 0.708 0.709 0.78242 0.78383 0.78524 0.78666 0.78807 31458 43 41608 47 71633 95 21647 44 91762 45 0.61407 37925 25 Or61474 14695 10 0.61540 85176 29 Oi61607 49371 78 0.61674 07284 52 0.755 0.756 0.757 0.758 0.759 0.85565 0.85718 0.85870 0.86024 0.86177 41195 04 04914 02 95588 84 13398 74 58524 85 0.64669 0.64733 0.64796 0.64860 0.64923 34363 37 00626 40 60767 30 14789 75 62697 45 0.710 0.711 0.712 0.713 0.714 0.78949 0.79091 0.79234 0.79376 0.79519 82093 46 92755 96 23866 39 75542 24 47901 99 0.61740 0.61807 0.61873 0.61939 0.62006 58917 52 04273 76 43356 27 76168 09 02712 26 0.760 0.761 0.762 0.763 0.764 0.86331 0.86485 0.86639 0.86794 0.86949 31150 16 31459 55 59639 86 15879 89 00370 42 0.64987 04494 12 Oi65050 40183 48 0.65113 69769 28 0.65176 93255 25 0.65240 10645 18 0.715 0.716 0.717 0.718 0.719 0.79662 41065 16 0.79805 55152 32 0.79948 90285 08 Oi80092 46586 13 0.80236 24179 26 0.62072 0.62138 0.62204 0;62270 0.62336 22991 86 37009 97 44769 70 46274 14 41526 45 0.765 0.766 0.767 0.768 0.769 0.87104 0.87259 0.87415 0.87571 0.87727 13304 26 54876 26 25283 38 24724 65 53401 29 0.65303 0.65366 0.65429 0.65492 0.65555 21942 83 27151 99 26276 46 19320 05 06286 59 0.720 0.721 0.722 0.723 0.724 0.80380 0.80524 0.80668 0.80813 0.80958 23189 33 43742 33 85965 35 49986 66 35935 64 0.62402 0.62468 0.62533 0.62599 0.62665 30529 77 13287 26 89802 10 60077 48 24116 63 0.770 0.771 0.772 0.773 0.774 0.87884 0.88040 0.88198 0.88355 0.88513 11516 69 99276 42 16888 33 64562 55 42511 51 0.65617 0.65680 0.65743 0.65805 0.65868 87179 91 62003 87 30762 31 93459 11 50098 15 0.725 0.726 0.727 0.728 0.729 0.81103 0.81248 0.81394 0.81540 0.81685 43942 88 74140 11 26660 28 01637 58 99207 37 0.62730 0.62796 0.62861 0.62927 0.62992 81922 76 33499 11 78848 95 17975 54 50882 17 0.775 0.776 0.777 0.778 0.779 0.88671 0.88829 0.88988 0.89147 0.89306 50950 00 90095 19 60166 70 61386 58 93979 43 0.65931 0.65993 0.66055 0.66118 0.66180 00683 33 45218 55 83707 72 16154 79 42563 67 0.730 or731 0.732 0.733 0.734 0.81832 0.81978 0.82125 0.82272 0.82419 19506 32 62672 31 28844 52 18163 44 30770 85 0.63057 77572 15 Oi63122 98048 79 0.63188 12315 41 0.63253 20375 38 0.63318 22232 04 0.780 0.781 0.782 0.783 0.784 0.89466 0.89626 0.89786 0.89947 0.90108 58172 34 54195 03 82279 83 42661 72 35578 41 0.66242 0.66304 0.66366 0.66428 0.66490 62938 33 77282 73 85600 83 87896 62 84174 09 0.735 0.736 0.737 0.738 0.739 0.82566 0.82714 0.82862 0.83010 0.83158 66809 86 26424 94 09761 92 16968 01 48191 83 0.63383 0.63448 0.63512 0.63577 0.63642 17888 78 07348 99 90616 06 67693 42 38584 50 0.785 0.786 0.787 0.788 0.789 0.90269 61270 38 Oi90431 19980 87 0.90593 11956 01 0.90755 37444 80 0.90917 96699 17 0.66552 74437 26 Oi66614 58690 12 0.66676 36936 71 0.66738 09181 07 0.66799 75427 24 0.740 0.741 0.742 0.743 0.744 0.83307 0.83455 0.83604 0.83754 0.83903 03583 42 83294 24 87477 24 16286 83 69878 93 0.63707 03292 76 Oi63771 61821 64 0.63836 14174 63 Oi6i900 60355 21 0.63965 00366 89 0.790 0.791 0.792 0.793 0.794 0.91080 89974 07 0.91244 17527 48 0.91407 79620 46 Oi91571 76517 23 0.91736 08485 19 0.66861 35679 28 Oi66922 89941 25 0.66984 38217 24 Oi67045 80511'32 0.67107 16827 61 0.745 0.746 0.747 0.748 0.749 0.84053 48410 98 Oi84203 52041 95 0.84353 80932 39 0.84504 35244 42 0.84655 15141 77 0.64029 34213 19 Oi64093 61897 63 0.64157 83423 76 0.64221 98795 14 0.64286 08015 33 0.795 0.796 0.797 0.798 0.799 0.91900 75795 02 Oi92065 78720 67 0.92231 17539 49 0.92396 92532 24 0.92563 03983 15 0.67168 47170 20 Oi67229 71543 22 0.67290 89950 79 0.67352 02397 05 0.67413 08886 15 0.750 0.84806 20789 81 0.64350 11087 93 0.800 0.92729 52180 02 0.67474 09422 24 ;=1.57079 63267 95 [ c-:15 1 [I(-f)8 1 211 ELEMENTARY TRANSCENDENTAL FUNCTIONS INVERSE X arcsin x CIRCULAR SINES arctan x AND TANGENTS Table arcsin x X 4.14 arctan x 0.800 O.SOl 0.802 0.303 0.804 52180 02 37414 22 59980 83 20178 64 18310 25 0.67474 0.67535 0.67595 0.67656 0.67717 09422 24 04009 49 92652 08 75354 19 52120 01 0.850 0.851 0.852 0.853 0.854 1.01598 1.01788 1.01979 1.02170 1.02362 52938 15 65272 25 36361 62 66824 41 57289 29 0.70449 0.70507 0.70565 0.70623 0.70681 40642 42 43293 58 40219 63 31425 16 16914 73 0.805 0.806 0.807 0.808 0.809 0.93567 0.93736 0.93905 0.94074 0.94244 54682 12 29604 66 43392 28 96363 49 88840 95 0.67778 22953 77 0.67838 87859 65 0.67899 46841 90 0.67959 99904 74 Oi68020 47052 41 0.855 0.856 0.857 0.858 0.859 1.02555 1.02748 1.02941 1.03136 1.03331 08395 76 20794 40 95147 10 32127 41 32420 77 0.70738 0.70796 0.70854 0.70912 0.70969 96692 96 70764 42 39133 73 01805 50 58784 34 0.810 0.811 0.812 0.813 0.814 0.94415 0.94585 0.94757 0.94928 0.95100 21151 54 93626 48 06601 38 60416 29 55415 87 0.68080 0.68141 0.68201 0.68261 0.68321 88289 16 23619 25 53046 96 76576 55 94212 31 0.860 0.861 0.862 0.863 0.864 1.03526 1.03723 1.03920 1.04117 1.04316 96724 81 25749 68 20218 39 80867 05 08445 30 0.71027 0.71084 0.71141 0.71199 0.71256 10074 87 55681 72 95609 52 29862 92 58446 55 0.815 0.816 0.817 0.818 0.819 \ 0.92729 0.92896 0.93063 0.93231 0.93399 0.95272 0.95445 0.95618 0.95792 0.95966 91949 40 70370 88 91039 18 54318 04 60576 23 0.68382 0.68442 0.68502 0.68562 0.68621 05958 54 11819 54 11799 62 05903 10 94134 31 0.865 0.866 0.867 0.868 0.869 1.04515 1.04714 1.04915 1.05116 1.05317 03716 61 67458 63 00463 62 03538 76 77506 61 0.71313 0.71370 0.71428 0.71485 0.71542 81365 07 98623 14 10225 41 16176 56 16481 25 0.820 0.821 0.822 0.823 0.824 0.96141 0.96316 0.96491 0.96667 0.96843 10187 64 03531 36 40991 79 22958 76 49827 60 0.68681 0.68741 0.68801 0.68860 0.68920 76497 59 52997 28 23637 73 88423 31 47358 39 0.870 0.871 0.872 0.873 0.874 1.05520 1.05723 1.05927 1.0613! 1.0633 23205 49 41489 91 33231 01 99317 03 40653 78 0.71599 0.71656 0.71712 0.71769 0.71826 11144 16 00169 99 83563 41 61329 12 33471 82 0.825 0.826 0.827 0.828 0.829 0.97020 0.97197 0.97375 0.97553 0.97731 21999 29 39880 56 03884 00 14428 17 71937 77 0.68980 0.69039 0.69098 0.69158 0.69217 00447 34 47694 55 89104 41 24681 33 54429 71 0.875 0.876 0.877 0.878 0.879 1.06543 1.06750 1.06958 1.07166 1.07376 58165 11 52793 43 25500 24 77266 67 09094 07 0.71882 0.71939 0.71996 0.72052 0.72109 99996 22 60907 02 16208 94 65906 70 10005 03 0.830 0.831 0.832 0.833 0.834 0.97910 0.98090 0.98270 0.98450 0.98631 76843 68 29583 19 30600 05 80344 64 79274 13 0.69276 0.69335 0.69395 0.69454 0.69513 78353 97 96458 54 08747 85 15226 33 15898 44 0.880 0.881 0.882 Oi883 0.884 1.07586 1.07797 1.08008 liO8221 1.08435 22004 54 17041 59 95270 75 57780 22 05681 59 0.72165 0.72221 0.72278 0.72334 0.72390 48508 65 81422 30 08750 71 30498 64 46670 83 0.835 0.836 0.837 0.838 0.839 0.98813 0.98995 0.99177 0.99360 0.99544 27852 56 26551 06 75847 95 76228 94 28187 22 0.69572 10768 63 0;69630 99841 36 0.69689 83121 11 0169748 60612 34 0.69807 32319 55 0.885 OI886 0.887 0.888 0.889 1.08649 1.08864 1.09080 1.09297 1.09515 40110 49 62227 36 73218 22 74295 43 66698 56 0.72446 0.72502 0.72558 0.72614 0.72670 57272 04 62307 01 61780 53 55697 34 44062 23 0.840 0.841 0.842 0.843 0.844 0.99728 0.99912 1.00097 1.00283 1.00469 32223 72 88847 18 98574 39 61930 35 79448 46 0.69865 0.69924 0.69983 0.70041 0.70100 98247 21 58399 85 12781 94 61398 02 04252 59 0.890 0.891 0.892 0.893 0.894 1.09734 1.09954 1.10175 1.10396 1.10619 51695 23 30581 99 04685 30 75362 43 44002 56 0.72726 0.72782 0.72837 0.72893 0.72949 26879 97 04155 34 75893 12 42098 11 02775 09 0.845 0.846 0.847 0.848 Oi849 1.00656 1.00843 1.01031 1.01220 1.01408 51670 67 79147 75 62439 41 02114 56 98751 50 0.70158 0;70216 0.70274 0.70333 0.70391 41350 19 72695 35 98292 60 18146 49 32261 58 0.895 0.896 0.897 0.898 0.899 1.10843 1.11067 1.11293 1.11520 1.11748 12027 75 80894 12 52092 94 27151 85 07636 13 0.73004 0.73060 0.73115 0.73170 0.73226 57928 87 07564 24 51686 02 90299 00 23408 01 0.850 1.01598 52938 15 (-[)7 0.70447~4'$42 42 0.900 1.11976 95149 99 [ 1 [ 4 1 .[ ;=1.57079 63267 95 I,-6)1 6 1 0.73281 51017 87 1 -l)7 1 C 212 Table ELEMENTARY 4,.14 TRANSCENDENTAL INVERSE arcsin J 0.9”00 1.11976 95149 99 arctan .t 87 38 38 70 lb CIRCULAR SINES O.&O 0.951 0.952 0.953 0.954 1.25323 1.25645 1.25970 1.26298 1.26630 AND TANGENTS arcsin.t f (2) 0.901 0.902 0.903 0.904 1.12206 1.12437 1.12670 1.12903 91337 97886 lb524 49026 93 21 29 45 0.905 0.906 0.907 0.908 0.909 1.13137 1.13373 1.13610 1.13848 1.14087 97213 62953 48166 99 54823 84946 E zl 73557 73612 73666 73721 73776 06748 01464 90715 74506 52841 0.955 0.956 0.957 0.958 0.959 1.26965 1.27304 1.27647 1.27994 1.28345 97812 97667 76222 46878 23838 42 20 92 88 00 0.76238 0.76290 0.76342 0.76395 0.76447 43244 70690 92916 09927 21729 37 08 23 81 78 1.00378 1.00370 1.00361 1.00353 1.00344 84851 34492 84523 34944 85754 78 58 57 39 69 0.910 0.911 0.912 0.913 0.914 1.14328 40618 1.14570 23976 2 1.14813 37219 91 1.15057 82610 10 1.15303 62474 12 73831 73885 73940 73995 74049 25725 93163 55160 11721 62850 0.960 0.961 0.962 0.963 0.964 1.28700 1.29059 1.29423 1.29792 1.30165 22175 57917 48124 10987 65939 87 69 14 43 20 0.76499 0.76551 0.76603 0.76655 0.76707 28327 29724 25927 16941 02769 11 78 75 02 55 1.00336 1.00327 1.00319 1.00310 1.00302 36954 88542 40518 92883 45635 10 28 88 53 89 0.915 0.916 0.917 0.918 0.919 1.15550 1.15799 1.16049 1.16300 1.16553 79206 35274 33215 75647 65266 90 19 50 25 04 0.74104 08553 83 0.74158 48835 32 0.74212 83700 10 0.74267 13153 04 0.74321 37199 05 0.965 0.966 0.967 0.968 0.969 1.30544 33771 97 1.30928 36776 35 1.31317 98896 52 1.31713 45907 19 1.32115 05615 54 0.76758 0.76810 0.76862 0.76913 0.76965 83418 58892 29196 94335 54315 33 33 53 92 49 1.00293 1.00285 1.00277 1.00268 1.00260 98775 52302 06215 60515 15201 61 33 71 39 02 0.920 0.921 0.922 0.923 0.924 1.16808 1.17063 1.17321 1.17580 1.17841 04852 97273 45487 52550 21615 14 lb 95 71 31 0.74375 0.74429 0.74483 0.74537 0.74591 99 76 25 35 97 0.970 0.971 0.972 0.973 0.974 1.32523 08092 80 1.32937 85940 93 1.33359 74601 02 1.33789 12711 79 1.34226 42528 47 0.77017 0.77068 0.77120 0.77171 0.77222 09140 58815 03345 42735 76990 20 06 05 14 34 1.00251 1.00243 1.00234 1.00226 1.00217 70272 25728 81570 37796 94406 25 74 13 07 23 0.925 0.926 0.927 0.928 0.929 1.18103 1.18367 1.18633 1.18900 1.19170 55939 58892 33953 84725 14936 97' 09 44 71 35 0.74645 68203 00 0.74699 54537 6174753 35503 92 0.74807 11107 62 0.74860 81353 36 0.975 0.976 0.977 0.978 0.979 1.34672 1.35126 1.35590 1.36064 1.36549 10414 67425 69996 80777 69629 93 45 85 70 42 0.77274 0.77325 0.77376 0.77427 0.77478 06115 30116 48996 62761 71417 63 01 45 95 51 1.00209 51400 25 1.00201 08777 78 1.00192 66538 49 1.00184 24682 01 1.00175 83208 02 0.930 0.931 0.932 0.933 0.934 1.19441 1.19714 1.19989 1.20266 1.20544 28444 29249 21492 09472 97647 77 00 75 92 69 0.74914 0.74968 0.75021 0.75075 0.75128 46246 Ob 05790 63 59991 99 08855 06 52384 76 0.980 0.981 0.982 0.983 0.984 1.37046 1.37555 1.38077 1.38614 1.39167 14844 04644 39033 32129 15119 72 29 32 70 16 0.77529 0.77580 0.77631 0.77682 0.77733 74968 73418 66774 55040 38220 12 77 45 17 91 1.00167 42116 16 1.00159 01406 08 1.00150 61077 45 1.00142 21129 93 1.00133 81563 16 0.935 0.936 0.937 0.938 0.939 1.20825 1.21108 1.21394 1.21681 1.21971 90645 93272 10524 47598 09898 07 10 70 22 74 0.75181 0.75235 0.75288 0.75341 0.75394 90586 23463 51022 73268 90205 03 79 96 49 30 0.985 0.986 0.987 0.988 0.989 1.39737 1.40326 1.40937 1.41572 1.42233 40056 84832 59766 16538 60557 99 96 46 31 98 0.77784 16321 67 0.77834 a9347 44 0.77885 57303 23 0.77936 20194 04 0.77986 78024 85 1.00125 1.00117 1.00108 1.00100 1.00091 0.940 0.941 0.942 0.943 0.944 1.22263 1.22557 1.22854 1.23153 1.23455 03055 32932 05645 27575 05382 22 59 81 05 02 0.75448 0.75501 0.75554 0.75607 0.75659 01838 08172 09212 04964 95431 34 55 86 22 57 0.990 0.991 0.992 1.42925 1.43653 1.44422 1.45240 1.46119 68534 14207 07408 56012 69689 70 77 32 67 63 0.78037 0.78087 0.78138 0.78188 0.78238 30800 78526 21207 58848 91453 67 49 32 15 98 1.00083 52139 33 1.00075 15228 31 1.00066 78695 32 1.00058 42540 02 1.00050 06762 08 0.945 0,946 0.947 0,948 0.949 1.23759 1.24066 1.24376 1.24689 1.25004 46027 56791 45292 19509 87811 74 62 24 90 06 0.75712 0.75765 0.75818 0.75871 0.75923 80619 60534 35179 04559 68681 86 05 08 90 48 0.995 0.996 0.997 0.998 0.999 1.47075 46131 83 1.48132 37665 90 1.49331 72818 71 1.50754 02279.20 1.52607 12396 26 0.78289 0.78339 0.78389 0.78439 0.78489 19029 41580 59111 71627 79133 81 64 47 31 14 1.00041 1.00033 1.00025 1.00016 1.00008 0.950 1.25323 58975 03 0.75976 27548 76 1.000 1.57079 0.78539 81633 97 c1 55842 69089 76944 79411 76495 35 58975 03 42223 06 47250 03 arcian z 0.73281 0.73336 0.73391 0.73447 0.73502 (-i)4 For arctan ,v,.~>l seeExample 51017 73133 a9759 00900 06562 FUNCTIONS 64000 67 84259 28 0.75976 27548 76 0.76028 81166 70 0.76081 29540 28 0.76133 72674 43 0.76186 10574 14 1.00421 1.00412 1.09404 1.00395 1.00387 42513 90197 38274 86742 35601 02 55 04 15 52 63267 95 71361 36336 01689 67417 33520 $=1.570796326795 80 52 98 82 72 15 91 01 11 89 1.000000000000 (-!)5 [1 22. arcsin.x=;-[2(1-:t)]tJ(x) 42376 03570 65143 27096 89428 _ ELEMENTARY TRANSCENDENTAL HYPERBOLIC sinh x 0.00000 0000 FUNCTIONS Table FUNCTIONS cash I(: 1.00000 0000 tanh x 0.00000 000 4.15 coth x 00 0.01000 0.02000 0.03000 0.04001 ' 0167 1333 4500 0668 1.00005 1.00020 1.00045 1.00080 0000 0007 0034 0107 0.00999 0.01999 0.02999 0.03997 967 733 100 868 0.05002 0.06003 0.07005 0.08008 0.09012 0836 6006 7181 5361 1549 1.00125 1.00180 1.00245 1.00320 1.00405 0260 0540 1001 1707 2734 0.04995 0.05992 0.06988 0.07982 0.08975 838 810 589 977 779 20.01666 16.68666 14.30904 12.52665 11.14109 0.10016 0.11022 0.12028 0.13036 0.14045 6750 1968 8207 6476 7782 1.00500 1.00605 1.00720 1.00846 1.00981 4168 6103 8644 1907 6017 0.09966 0.10955 0.11942 0.12927 0.13909 800 847 730 258 245 10.03331 11 9.12754 62 8.37329 50 7.73559 23 7118946 29 0.15 0. 16 0.17 0.18 0.19 0.15056 0.16068 0.17082 0.18097 0.19114 3133 3541 0017 3576 5232 1.01127 1.01282 1.01448 1.01624 1.01810 1110 7330 4834 3787 4366 0.14888 0.15864 0.16838 0.17808 0.18774 503 850 105 087 621 6.71659 6.30324 5.93891 5.61542 5.32633 18 25 07 64 93 0.20 0.21 0.22 0.23 0.24 0.20133 0.21154 0.22177 0.23203 0.24231 6003 6907 8966 3204 0645 1.02006 1.02213 1.02429 1.02656 1.02893 6756 1153 7764 6806 8506 0.19737 0.20696 0.21651 0.22602 0.23549 532 650 806 835 575 5.06648 4.83169 4.61855 4.42422 4.24636 96 98 23 37 11 0.25 0.26 0.27 0.28 0.29 0.25261 0.26293 0.27329 0.28367 0.29408 2317 9250 2478 3035 1960 1.03141 1.03399 1.03667 1.03945 1.04234 3100 0836 1973 6777 5528 0.24491 0.25429 0.26362 0.27290 0.28213 866 553 484 508 481 4.08298 3.93243 3.79326 3.66427 3.54440 82 24 93 77 49 0.30 0.31 0.32 0. 33 0.34 0.30452 0.31498 0.32548 0.33602 0.34658 0293 9079 9364 2198 8634 1.04533 1.04843 1.05163 1.05494 1.05835 8514 6035 8401 5931 8957 0.29131 0.30043 0.30950 0.31852 0.32747 261 710 692 078 740 3.43273 3.32848 3.23094 3.13951 3.05364 84 38 55 26 59 0.35 0.36 0.37 0.38 0.39 0.35718 0.36782 0.37850 0.38921 0.39996 9729 6544 0142 1590 1960 1.06187 1.06550 1.06923 1.07307 1.07701 7819 2870 4473 2999 8834 0.33637 0.34521 0.35399 0.36270 0.37136 554 403 171 747 023 2.97286 2.89675 2.82492 2.75704 2.69280 77 36 49 28 32 0.40 0.41 0.42 0.43 0.44 0.41075 0.42158 0.43245 0.44337 0.45433 2326 3767 7368 4214 5399 1.08107 1.08523 1.08950 1.09388 1.09837 2372 4018 4188 3309 1820 0.37994 0.38847 0.39693 0.40532 0.41364 896 268 043 131 444 2.63193 2.57418 2.51933 2.46717 2.41753 24 36 32 85 52 0.45 0.46 0.47 0. 48 0.49 0.46534 0.47639 0.48749 0.49864 0.50984 2017 5170 5962 5505 4913 1.10297 1.10767 1.11249 1.11742 1.12247 0169 8815 8231 8897 1307 0.42189 0.43008 0.43819 0.44624 0.45421 901 421 932 361 643 2.37023 2.32512 2.28206 2.24092 2.20159 55 60 66 84 36 0.05 0.06 0. 07 0.08 0.09 0.52109 5305 1.12762 5965 0.46211 716 100.00333 33 50.00666 65 33.34333 27 25.01333 19 39 19 00 53 49 2.16395 34 For coth 2, x 2 .l use 4.5.67. Compilation of tanh x and coth x from National Bureau of Standards, Table of circular and hyperbolic tangents and cotangents for radian arguments, 2d printing. Columbia Univ. Press, New York, N.Y., 1947 (with permission). 214 ELEMENTARY FUNCTIONS HYPERBOLIC Table 4.15 2 TRANSCENDENTAL cash x sinh x FUNCTIONS tanhx 0.50 0.51 0.52 0.53 0.54 0.52109 0.53239 0.54375 0.55516 0.56662 5305 7808 3551 3669 9305 1.12762 1.13289 1.13827 1.14376 1.14937 5965 3387 4099 0.55 0.57 0. 58 0.59 0.57815 0.58973 0.60137 0.61307 0.62483 1604 1718 0806 0032 0565 1.15510 1.16094 1.16689 1.17296 1.17915 1414 0782 6245 8399 7850 0.60 0.61 0.62 0.63 0.64 0.b3665 0.64854 0.66049 0.67250 0.68459 3582 0265 1802 9389 4228 1.18546 1.19189 1.19843 1.20510 1.21188 5218 1134 6240 1190 6652 0.53704 0.54412 0.55112 0.55805 0.65 0.66 0. 67 7526 0500 4371 0370 9732 1.21879 1.22582 1.23297 1.24024 1.24764 3303 1834 2949 0.68 0.69 0.69674 0.70897 0.72126 0.73363 0.74606 0.70 0.71 0.72 0.73 0.74 0.75858 0.77117 0.78384 0.79658 0.80941 3702 3531 0477 5809 0799 0.75 0.76 0.77 0.78 0.79 8639 7557 0.46211 0.46994 0.47770 0.48538 0.49298 coth x 716 520 001 109 797 2.06023 68 2.02844 71 1.99792 1.96859 1.94039 1.91326 1.88716 13 14 39 98 42 957 710 803 222 55 955 1.86202 1.83780 1.81446 1.79194 1.77022 997 341 988 940 200 1.74926 1.72901 1.70946 1.69056 1.67229 10 5801 0.57166 0.57836 0.58497 0.59151 0.59798 1.25516 1.26281 1.27059 1.27849 1.28652 9006 7728 2733 4799 4715 0.60436 0.61067 0.61690 0.62306 0.62914 778 1.65462 1.63752 1.62098 1.60496 1.58945 lb 73 0.82231 6732 0.83530 4897 0.84837 6593 0.86153 3127 0.87477 5815 1.29468 1.30297 1.31138 1.31993 1.32862 3285 1324 9661 9138 0611 0.63514 895 0.64107 696 0.80 0.81 0.82 0.83 0.84 0.88810 0.90152 0.91503 0.92863 0.94232 5982 4960 4092 4727 8227 1.33743 1.34638 1.35546 1.36468 1.37403 4946 3026 5746 4013 8750 0.66403 677 0.66959 026 0.67506 0.68047 987 0. 85 0.86 0.87 0. 88 0.89 0.95611 0.96999 0.98397 0.99805 1.01223 5960 9306 9652 8397 6949 1.38353 1.39316 1.40293 1.41284 1.42289 0.90 0.91 0.92 0.93 0.94 1.02651 1.04089 1.05538 1.06997 1.08467 6726 0.95 0.96 0.97 0. 98 0.99 1.09948 1.11440 1.12943 I.14457 1.15982 1.00 1.17520 1194 0. 56 7362 0.50052 021 0.50797 743 0.51535 928 2.16395 34 2.12790 77 2.09336 40 0.52266 543 0.52989 561 0.56489 0.64692 683 930 535 516 945 0.65270 671 0.65840 904 59 04 70 62 67 05 16 11 38 81 83 1.57443'38 1.55987 51 1.54576 36 1.53208 17 1.51881 27 601 0.68580 906 1.50594 1.49345 1.48132 1.46955 1.45813 07 06 81 95 18 0892 1388 1201 1309 2702 0.69106 0.69625 0.70137 0.70641 0.71139 947 413 932 373 1.44703 1.43624 1.42577 1.41558 1.40569 25 99 6791 1.43308 1.44342 1.45390 1.46453 1.47530 6385 3379 4716 1444 0.71629 0.72113 0.72589 0.73059 0.73522 787 225 742 390 225 1.39606 iii8670 1.37760 1.36874 1.36013 73 82 51 95 29 4318 1794 0711 2572 8891 1.48622 1.49729 1.50851 1.51988 1.53140 5341 4680 3749 0.73978 0.74427 0.74870 0.75306 0.75736 305 1.35174 1.34358 1.33564 1.32790 1.32037 76 9155 5674 7734 [c-y1 4627 3670 5582 1.54308 0635 C-i’2 [ 1 767 687 429 591 232 0.76159 416 [c-y9 1 26 98 13 60 08 50 20 1.31303 53 c-y [ 1 ' ELEMENTARY TRANSCENDENTAL HYPERBOLIC X sinh x FUNCTIONS cash z 215 FUNCTIONS Table tanhx 4.15 cothx 1.00 1.01 1.02 1.03 1.04 1.17520 1.19069 1.20629 1.22202 1.23788 1194 1018 9912 9437 1166 1.54308 1.55490 1.56689 1.57903 1.59133 0635 9997 4852 6398 5848 0.76159 0.76576 0.76986 0.77390 0.77788 416 202 654 834 807 1.31303 1.30588 1.29892 1.29214 1.28553 53 87 64 27 20 1. 05 1.06 1.07 1.08 1. 09 1.25385 1.26995 1.28618 1.30254 1.31902 6684 7589 5491 2013 8789 1.60379 1.61641 1.62919 1.64213 1.65524 4434 3400 4009 7538 5283 0.78180 0.78566 0.78946 0.79319 0.79687 636 386 122 910 814 1.27908 1.27280 1.26668 1.26071 1.25489 91 90 67 75 70 1.10 1.11 1.12 1.13 1.14 1.33564 1.35239 1.36928 1.38631 1.40347 7470 9717 7204 1622 4672 1.66851 1.68195 1.69556 1.70934 1.72329 8554 8678 6999 4878 3694 0.80049 0.80406 0.80756 0.81101 0.81441 902 239 892 926 409 1.24922 1.24368 1.23828 1.23301 1.22787 08 46 44 63 66 1.15 1.16 1.17 1.18 1.19 1.42077 1.43822 1.45581 1.47354 1.49142 8070 3548 2849 7732 9972 1.73741 1.75170 1.76617 1.78082 1.79565 4840 9728 9790 6471 1236 0.81775 0.82103 0.82427 0.82745 0.83057 408 988 217 161 887 1.22286 1.21796 1.21319 1.20852 1.20397 15 76 15 99 96 1.20 1.21 1.22 1.23 1.24 1.50946 1.52764 1.54597 1.56446 1.58311 1355 3687 8783 8479 4623 1.81065 1.82584 1.84120 1.85676 1.87249 5567 0966 8950 1057 8841 0.83365 0.83667 0.83965 0.84257 0.84545 461 949 418 933 560 1.19953 1.19520 1.19096 1.18683 1.18279 75 08 65 19 42 1.25 1.26 1.27 1.28 1.29 1.60191 1.62088 1.64001 1.65930 1.67875 9080 3730 0470 1213 7886 1.88842 1.90453 1.92084 1.93733 1.95402 3877 7757 2092 8513 8669 0.84828 0.85106 0.85379 0.85648 0.85912 364 411 765 492 654 1.17885 1.17499 1.17123 1.16756 1.16397 10 96 77 29 29 1.30 1.31 1.32 1.33 1.34 1.69838 1.71817 1.73814 1.75828 1.77859 2437 6828 3038 3063 8918 1.97041 1.98799 2.00527 2.02276 2.04044 4230 6884 8340 0324 4587 0.86172 0.86427 0.86678 0.86924 0.87167 316 541 393 933 225 1.16046 1.15703 1.15369 1.15041 1.14722 55 86 01 79 02 1.35 1.36 1.37 1.38 1.39 1.79909 1.81976 1.84062 1.86166 1.88288 2635 6262 1868 1537 7374 2.05833 2.07642 2.09472 2.11324 2.13196 2896 7039 8828 0090 2679 0.87405 0.87639 0.87869 0.88095 0.88317 329 307 219 127 089 1.14409 1.14104 1.13805 1.13513 1.13228 50 05 50 66 37 1.40 1.41 1.42 1.43 1.44 1.90430 1.92590 1.94770 1.96969 1.99188 1501 6060 3212 5135 4029 2.15089 2.17004 2.18941 2.20900 2.22881 8465 9344 7229 4057 1788 0.88535 0.88749 0.88959 0.89166 0.89369 165 413 892 660 773 1.12949 1.12676 1.12410 1.12149 1.11894 47 80 21 54 66 1.45 1.46 1.47 1.48 1.49 2.01427 2.03686 2.05965 2.08265 2.10586 2114 1627 4828 3996 1432 2.24884 2.26909 2.28958 2.31029 2.33123 2402 7902 0313 1685 4087 0.89569 0.89765 0.89957 0.90146 0.90332 287 260 745 799 474 1.11645 1.11401 1.11163 1.10930 1.10702 41 67 30 17 16 1.50 2.12927 9455 C-j)3 [ 1 2.35240 9615 (-[)3 [ 1 0.90514 825 [c-y1 1.10479 14 C-i’2 [ 1 216 ELEMENTARY Table x 4.15 TRANSCENDENTAL HYPERBOLIC sinh x FUNCTIONS FUNCTIONS cash x tanhx coth x 1.50 1.51 1.52 1.53 1.54 2.12927 2.15291 2.17675 2.20082 2.22510 9455 0408 6654 0577 4585 2.35240 2.37382 2.39546 2.41735 2.43948 9615 0386 8541 6245 5686 0.90514 0.90693 0.90869 0.91042 0.91212 a25 905 766 459 037 1.10479 1.10260 1.10047 1.09838 1.09634 14 99 60 86 65 1.55 1.56 1.57 1.58 1.59 2.24961 2.27434 2.29930 2.32449 2.34991 1104 2587 1506 0357 1658 2.46185 2.48447 2.50734 2.53046 2.55383 9078 8658 6688 5455 7270 0.91378 0.91542 0.91702 0.91860 0.92014 549 046 576 189 933 1.09434 1.09239 1.09048 1.08861 1.08678 87 42 19 09 01 1.60 1.61 1. 62 1. 63 1.64 2.37556 2.40146 2.42759 2145397 2.48059 7953 1807 5809 2572 4735 2.57746 2.60134 2.62549 2.64990 2.67457 4471 9421 4508 2146 4777 0.92166 0.92316 0.92462 0.92606 0.92747 855 003 422 158 257 1.08498 1.08323 1.08152 1.07984 1.07819 87 58 04 18 90 1.65 1.66 1.67 1.68 1.69 2.50746 2.53458 2.56196 2.58959 2.61748 4959 5932 0366 0998 0591 2.69951 2.72472 2.75020 2.77596 2.80200 4868 4912 7431 4974 0115 0.92885 0.93021 0.93155 0.93286 0.93414 762 718 168 155 721 1.07659 1.07501 1.07347 1.07197 1.07049 13 78 77 04 51 1. 70 1. 71 1.72 1. 73 1.74 2.64563 2.67404 2.70273 2.73168 2.76091 1934 7843 1158 4749 1511 2.82831 2.85491 2.88179 2.90896 2.93643 5458 3635 7306 9159 1912 0.93540 0.93664 0.93786 0.93905 0.94022 907 754 303 593 664 1.06905 1.06763 1.06625 1.06489 1.06357 10 75 38 93 34 1.75 1.76 1.77 1.78 1.79 2.79041 2.82019 2.85026 2.88060 2.91124 4366 6265 0186 9136 6148 2.96418 2.99224 3.02059 3.04924 3.07820 8310 1129 3175 7283 6318 0.94137 0.94250 0.94360 0.94469 0.94576 554 301 942 516 057 1.06227 1.06100 1.05976 1.05854 1.05735 53 46 05 25 01 1.80 1.81 1.82 1.83 1.84 2.94217 2.97339 3.00491 3.03673 3.06886 4288 6648 6349 6545 0417 3.10747 3.13705 3.16694 3.19715 3.22767 3176 0785 2100 0113 7844 0.94680 0.94783 0.94883 0.94982 0.95079 601 la5 a42 608 514 1.05618 1.05503 1.05392 1.05282 1.05175 26 95 02 43 13 1.85 1.86 1.87 1.88 1.89 3.10129 3.13403 3.16708 3.20045 3.23414 1178 2071 6369 7378 8436 3.25852 3.28970 3.32121 3.35304 3.38522 8344 4701 0031 7484 0245 0.95174 0.95267 0.95359 0.95449 0.95537 596 884 412 211 312 1.05070 1.04967 1.04866 1.04767 1.04671 05 17 42 76 15 1.90 1.91 1. 92 1.93 1.94 3.26816 3.30250 3.33717 3.37218 3.40752 2912 4206 5754 1022 3510 3.41773 3.45058 3.48378 3.51732 3.55122 1531 4593 2716 9220 7460 0.95623 0.95708 0.95791 0.95873 0.95953 746 542 731 341 401 1.04576 1.04483 1.04393 1.04304 1.04217 53 88 14 28 25 1.95 1.96 1.97 1.98 1.99 3.44320 3.47923 3.51560 3.55233 3.58941 6754 4322 9816 6874 9168 3.58548 3.62009 3.65506 3.69040 3.72611 0826 2743 6672 6111 4594 0.96031 0.96108 0.96184 0.96258 0.96331 939 983 561 698 422 1.04132 1.04048 1.03966 1.03886 1.03808 02 55 79 72 29 2. 00 3.62686 0408 C-55)4 [ 1 3.76219 5691 C-f)5 [ 1 0.96402 758 (-J)4 11 1.03731 47 C-i’” 11 ELEMENTARY TRANSCENDENTAL HYPERBOLIC 2 sinh x 217 FUNCTIONS FUNCTIONS cash x Table 4.15 coth x tanh x 3.62686 4.02185 4.45710 4.93696 5.46622 0408 6742 5171 1806 9214 3.76219 4.14431 4.56790 5.03722 5.55694 5691 3170 8329 0649 7167 0.96402 0.97045 0.97574 0.98009 0.98367 75801 19366 31300 63963 48577 1.03731 47207 1.03044 77350 1.02485 98932 1.02030 78022 1.01659 60756 6.05020 6.69473 7.40626 8.19191 9.05956 4481 2228 3106 8354 1075 6.13228 6.76900 7.47346 8.25272 9.11458 9480 5807 8619 8417 4295 0.98661 0.98902 0.99100 0.99263 0.99396 42982 74022 74537 15202 31674 I.01356 1.01109 1.00907 1.00742 1.00607 73098 43314 41460 31773 34973 10.01787 11.07645 12.24588 13.53787 14.96536 4927 1040 3997 7877 3389 10.06766 11.12150 12.28664 13.57476 14.99873 1996 0242 6201 1044 6659 0.99505 0.99594 0.99668 0.99728 0.99777 47537 93592 23978 29601 49279 1.00496 1.00406 1.00332 1.00272 1.00223 98233 71152 86453 44423 00341 16.54262 18.28545 20.21129 22.33940 24.69110 7288 5361 0417 6861 3597 16.57282 18.31277 20.23601 22.36177 24.71134 4671 9083 3943 7633 5508 0.99817 0.99850 0.99877 0.99899 0.99918 78976 79423 82413 95978 08657 1.00182 1.00149 1.00122 1.00100 1.00081 54285 42872 32532 14040 98059 27.28991 30.16185 33.33566 36.84311 40.71929 7197 7461 7732 2570 5663 27.30823 30.17843 33.35066 36.85668 40.73157 2836 0136 3309 1129 3002 0.99932 0.99945 0.99955 0.99963 0.99969 92997 08437 03665 18562 85793 1.00067 1.00054 1.00044 1.00036 1.00030 11504 94581 98358 82794 15116 45.00301 49.73713 54.96903 60.75109 67.14116 1152 1903 8588 3886 6551 45.01412 49.74718 54.97813 60.75932 67.14861 0149 3739 3865 3633 3134 0.99975 0.99979 0.99983 0.99986 0.99988 32108 79416 45656 45517 91030 1.00024 1.00020 1.00016 1.00013 1.00011 68501 20992 54618 54666 09093 74.20321 82.00790 90.63336 100.16590 110.70094 0578 5277 2655 9190 9812 74.20994 82.01400 90.63887 100.17090 110.70546 8525 2023 9220 0784 6393 0.99990 0.99992 0.99993 0.99995 0.99995 92043 56621 91369 01692 92018 1.00009 1.00007 1.00006 1.00004 1.00004 08040 43434 08668 98333 07998 55-i 5: 8 5.9 122.34392 135.21135 149.43202 165.14826 182.51736 2746 4781 7501 6177 4210 122.34800 135.21505 149.43537 165.15129 182.52010 9518 2645 3466 3732 3655 0.99996 0.99997 0.99997 0.99998 0.99998 65972 26520 76093 16680 49910 1.00003 1.00002 1.00002 1.00001 1.00001 34040 73488 23912 83323 50092 6. 0 201.71315 7370 201.71563 6122 0.99998 77117 1.00001 22885 ;*t 2:2 2.3 2.4 2.5 $2 2; 3. 0 ;=: 3:3 3. 4 E 3:7 Z t-10 4: 2 t5 4. 5 4.6 4. 7 44:: 5. 0 ;*: 5: 3 5.4 5.5 c c -y1 c1 c-p 218 Table ELEMENTARY 4.15 TRANSCENDENTAL HYPERBOLIC FUNCTIONS FUNCTIONS sinh x 201.71315 7370 222.92776 3607 246.37350 5831 272.28503 6911 300.92168 8157 coshx 201.71563 6122 222.93000 6475 246.37553 5262 272.28687 3215 300.92334 9715 tanhx 0.99998 77117 0.99998 99391 0.99999 17629 0.99999 32560 0.99999 44785 cothx 1.00001 22885 1.00001 00610 1.00000 82372 1.00000 67441 1.00000 55216 332.57006 367.54691 406.20229 448.92308 496.13685 4803 4437 7128 8938 3910 332.57156 367.54827 406.20352 448.92420 496.13786 8242 4805 8040 2713 1695 0.99999 0.99999 0.99999 0.99999 0.99999 54794 62988 69697 75190 79687 1.00000 1.00000 1.00000 1.00000 1.00000 45207 37012 30303 24810 20313 548.31612 605.98312 669.71500 740.14962 817.99190 3273 4694 8904 6023 9372 548.31703 605.98394 669.71575 740.15030 817.99252 5155 9799 5490 1562 0624 0.99999 0.99999 0.99999 0.99999 0.99999 83369 86384 88852 90873 92527 1.00000 1.00000 1.00000 1.00000 1.00000 16631 13616 11148 09127 07473 904.02093 999.09769 1104.17376 1220.30078 1348.64097 0686 7326 9530 3945 8762 904.02148 999.09819 1104.17422 1220.30119 1348.64134 3770 7778 2357 3680 9506 0.99999 0.99999 0.99999 0.99999 0.99999 93882 94991 95899 96642 97251 1.00000 1.00000 1.00000 1.00000 1.00000 06118 05009 04101 03358 02749 1490.47882 1647.23388 1820.47501 2011.93607 2223.53326 5790 5872 6339 2653 1416 1490.47916 1647.23418 1820.47529 2011.93632 2223.53348 1252 9411 0993 1170 6284 0.99999 0.99999 0.99999 0.99999 0.99999 97749 98157 98491 98765 98989 1.00000 1.00000 1.00000 1.00000 1.00000 02251 01843 01509 01235 01011 2457.38431 2715.82970 3001.45602 3317.12192 3665.98670 8415 3629 5338 7772 1384 2457.38452 2715.82988 3001.45619 3317.12207 3665.98683 1884 7734 1923 8505 7772 0.99999 0.99999 0.99999 0.99999 0.99999 99172 99322 99445 99546 99628 1.00000 1.00000 1.00000 1.00000 1.00000 00828 00678 00555 00454 00372 4051.54190 4477.64629 4948.56447 5469.00955 6044.19032 2083 5908 8852 8370 3746 4051.54202 4477.64640 4948.56457 5469.00964 6044.19040 5493 7574 9892 9795 6471 0.99999 0.99999 0.99999 0.99999 0.99999 99695 99751 99796 99833 99863 1.00000 1.00000 1.00000 1.00000 1.00000 00305 00249 00204 00167 00137 6679.86337 7382.39074 8158.80356 9016.87243 9965.18519 7405 8924 8366 6188 4028 6679.86345 7382.39081 8158.80362 9016.87249 9965.18524 2257 6653 9649 1640 4202 0.99999 0.99999 0.99999 0.99999 0.99999 99888 99908 99925 99939 99950 1.00000 1.00000 1.00000 1.00000 1.00000 00112 00092 00075 00061 00050 11013.23287 4703 Forx>>O,sinhx-coshx-i 11013.23292 0103 ez. Forx>lO, tanhx-l-2e-22, 0.99999 99959 * (-58)5 1.00000 00041 (-;I7 1 [cothx-1+2e-2zto 10D. 1 1 ELEMENTARY EXPONENTIAL AND X erz 0. 00 0. 01 1.00000 00000 1.03191 46153 1.06484 77733 0.02 0. 03 0. 04 1.09883 19803 1.13390 07803 TRANSCENDENTAL HYPERBOLIC e-“2 FUNCTIONS 1.00000 00000 219 FUNCTIONS FOR sinh zr 0.00000 00000 0.03142 10945 THE ARGUMENT cash ti ti Table 4.16 tanh ti 0.06287 0.09438 0.12599 32029 73698 47010 1.00000 1.00049 liOO197 1.00444 1.00790 59992 41813 98355 76792 32120 0.15772 0.18961 0.22168 0.25398 0.28652 63942 37699 83022 16502 56886 1.01236 1.01781 1.02427 1.03174 1.04023 23933 79512 81377 93294 89006 0.15580 0.18629 0.21643 0.24616 0.27544 03292 43856 36952 60434 21974 61929 55730 15776 92833 76928 0.96907 0.93910 0.91005 0.88191 24263 13674 72407 13783 00000 35208 45704 46105 60793 0.00000 0.06274 0.09396 0.12500 63906 00000 0.03140 55952 93000 97111 0.05 0.06 0. 07 0.08 0. 09 1.17008 87875 1.20743 17210 1.24596 64399 1.28573 09795 1.32676 45892 0.85463 0.82820 0.80258 0.77776 0.75371 0.10 0.11 0.12 0.13 0.14 1.36910 1.41280 1.45789 1.50441 1.55243 77706 23184 13610 94029 23694 0.73040 0.70781 0.68592 0.66470 0.64415 26910 31080 21659 82576 04440 0.31935 0.35249 0.38598 0.41985 0.45414 25398 46052 45975 55727 09627 1.04975 52308 1.06030 77132 1.07190 67634 2.08456 1.09829 14067 38303 0.30421 0.33244 0.36009 0.38711 0.41349 0.15 0.16 0.17 0.18 0.19 1.60197 1.65310 1.70586 1.76030 1.81648 76513 41518 23348 42750 37088 0.62422 0.60492 0.58621 0.56808 0.55051 84336 25628 37756 36059 41583 0.48887 0.52409 0.55982 0.59611 0.63298 46088 07945 42796 03346 47753 1.11310 1.12901 1.14603 1.16419 1.18349 30425 33573 80552 39405 89335 0.43919 0.46420 0.48848 0.51203 0.53484 97777 24748 66406 69673 18637 0.20 0.21 0.22 0.23 0.24 1.87445 60876 1.93427 86325 1.99601 03910 0.67048 0.70864 0.74750 0.78710 0.82747 39982 50169 54976 37973 90013 20893 22948 72203 80911 85988 93958 47001 92177 1.20397 2.05971 2.12544 0.53348 0.51698 0.50099 0.48550 0.47048 1.29796 82190 0.55689 0.57818 0.59872 0.61849 0.63751 33069 66683 05188 64181 86920 0.25 0.26 0. 27 0.28 0.29 2.19328 2.26327 2.33550 2.41004 2.48696 00507 77398 93782 62616 19609 0.45593 0.44183 0.42817 0.41492 0.40209 81278 70677 21192 97945 70227 0.86867 0.91072 0.95366 0.99755 09615 03361 86295 82336 1.32460 1.35255 1.38184 1.41248 1.44452 90893 74038 07487 80280 94918 0.65579 0.67333 0.69014 0.70624 0.72164 42026 21140 36583 19035 15276 0.30 0. 31 0. 32 0.33 0.34 2.56633 2.64823 2.73275 2.81996 2.90996 23952 59064 33366 81081 63054 0.38966 0.37760 0.36593 0.35461 0.34364 11374 98638 13069 39395 65907 1.08833 1.13531 1.18341 1.23267 1.28315 56289 30213 10148 70843 98573 1.47799 67663 1.51292 28851 1.54934 23218 1.58729 10238 0.73635 0.75041 0.76381 0.77659 0.78876 85995 03695 50706 17313 00021 0.35 0.36 0. 37 0.38 0. 39 3.00283 3.09867 3.19756 3.29961 3.40491 67606 11407 40381 30643 89460 0.33301 0.32271 0.31273 0.30306 0.29369 84355 89833 80681 58385 27474 1.33490 1.38797 1.44241 1.49827 1.55561 91626 60787 29850 36129 30993 1.66792 1.71069 1.75515 1.80133 75980 50620 10531 94514 0.80033 0.81135 0.82181 0.83175 0.84118 99933 21279 70068 52873 75743 0.40 0.41 0.42 0.43 0. 44 3.51358 3.62572 3.74143 3.86084 3.98405 56243 03579 38283 02496 74810 0.28460 0.27580 0.26727 0.25901 0.25100 95433 72607 72113 09757 03946 1.61448 1.67495 1.73707 1.80091 1.86652 80405 65486 83085 46370 85432 43239 57589 17947 19762 54241 0. 45 0.46 0. 47 0.48 0.49 4.11120 4.24241 4.37780 4.51752 4.66170 71429 0.24323 0.23571 0.22842 0.22136 0.21451 75614 48138 47266 01040 39731 2.00334 2.07469 2.14808 2.22359 07899 62194 93236 71557 61950 0.50 4.81047 47373 97717 58864 09873 73810 (-;I6 c 1 0.20787 95764 r( -p) 11 L 0J 1.04243 24691 li22563 36157 1.24850 48934 1.27260 84975 1.62680 64481 1.84930 58467 1.89909 75838 1.95076 38093 2.00435 2.05992 2.11752 55198 56127 89378 0.85013 0.85861 0.86665 0.87426 0.88146 99617 25226 28912 35071 2.17722 2.23906 2.30311 2.36944 2.43810 23522 47756 72491 29952 74802 0.88828 0.89472 0.90081 0.90657 0.91201 89023 2.50917 84787 0.91715 1.93398 47907 2.30129 [1 (d’3 [c-y1 Circular and 23357 Compiled from British Association for the Advancement of Science, Mathematical Tables, vol. 1. hyperbolic functions, exponential, sine and cosine integrals, factorial function and allied functions, Hermitian probability functions, 3d ed. Cambridge Univ. Press, Cambridge, England, 1951 (with permission). Known errors have been corrected. 220 Table ELEMENTARY EXPONENTIAL 4.16 AND TRANSCENDENTAL HYPERBOLIC FUNCTIONS FUNCTIONS FOR THE ARGUMENT m 0.50 0.51 0. 52 0.53 0. 54 4.81047 4.96400 5.12242 5.28590 5.45460 erz 73810 19160 61276 63869 40558 0.55 0.56 0.57 0.58 0. 59 5.62868 5.80832 5.99369 6.18497 6.38237 56460 29831 33767 97951 10460 0.17766 0.17216 0.16684 0.16168 0.15668 13694 67343 20350 20156 15832 2.72551 2.81807 2.91342 3.01164 3.11284 21383 81244 56709 88897 47314 2.90317 2.99024 3.08026 3.17333 3.26952 35077 48587 77058 09054 63146 0.93880 0.94242 0.94583 0.94904 0.95207 44259 38675 52160 97460 82009 0.60 0. 61 0.62 0. 63 0. 64 6.58606 6.79625 7.01315 7.23697 7.46794 19627 35967 34158 55091 07985 0.15183 0.14713 0.14258 0.13817 0.13390 58020 98890 92093 92710 57214 3.21711 3.32455 3.43528 3.54939 3.66701 30804 68538 21032 81191 75386 3.36894 3.47169 3.57787 3.68757 3.80092 88823 67428 13125 73901 32600 0.95493 0.95761 0.96014 0.96252 0.96477 08086 72978 69151 84417 02118 0.65 0. 66 0.67 0. 68 0.69 7.70627 7.95222 8.20601 8.46790 8.73815 72563 01304 21768 38986 37941 0.12976 0.12575 0.12186 0.11809 0.11444 43423 10461 18713 29793 06500 3.78825 3.91323 4.04207 4.17490 4.31185 64570 45422 51527 54597 65720 3.91802 4.03898 4.16393 4.29299 4.42629 07993 55883 70240 84390 72220 0.96688 0.96886 0.97073 0.97249 0.97414 01293 56859 39783 17255 52857 0.70 0.71 0.72 0.73 0. 74 9.01702 9.30480 9.60176 9.90819 10.22441 86109 36103 28381 94054 57779 0.11090 0.10747 0.10414 0.10092 0.09780 12784 13709 75422 65114 50993 4.45306 4.59866 4.74880 4.90363 5.06330 36663 61197 76480 64470 53393 4.56396 4.70613 4.85295 5.00456 5.16111 49447 74906 51901 29584 04386 0.97570 0.97716 0.97853 0.97983 0.98104 06726 35718 93563 31019 96015 0.75 0. 76 0.77 0. 78 0. 79 10.55072 10.88744 11.23491 11.59347 11.96347 40742 63743 50371 30285 42604 0.09478 0.09184 0.08900 0.08625 0.08358 02248 89025 82388 54299 77587 5.22797 5.39779 5.57295 5.75360 5.93994 19247 87359 33992 87993 32508 5.32275 5.48964 5.66196 5.83986 6.02353 21495 76384 16379 42292 10095 0.98219 0.98326 0.98427 0.98522 0.98612 33800 87071 96111 98912 31297 0.80 0. 81 0. 82 0.83 0.84 12.34528 12.73927 13.14584 13.56539 13.99832 39392 89270 81133 27988 70916 0.08100 0.07849 0.07606 0.07371 0.07143 25922 73785 96451 69955 71077 6.13214 6.33039 6.53488 6.74583 6.96344 06735 07743 92341 79017 49919 6.21314 6.40888 6.61095 6.81955 7.03488 32657 81528 88792 48972 20996 0.98696 0.98775 0.98849 Oi98919 0.98984 27033 17946 34022 b3509 53014 0.85 0.86 0.87 0. 88 0.89 14.44507 14.90608 15.38180 15.87271 16.37928 83157 74333 94795 40119 55735 0.06922 0.06708 0.06501 0.06300 0.06105 77313 66855 18571 11981 27239 7.18792 7.41950 7.65839 7.90485 8.15911 52922 03739 88112 64069 64248 7.25715 7.48658 7.72341 7.96785 8.22016 30235 70594 06683 76050 91487 0.99046 0.99103 0.99158 0.99209 0.99257 07591 90830 24938 30818 28142 0.90 0. 91 0.92 0.93 0.94 16.90202 17.44144 17.99808 18.57248 19.16521 41717 57711 28034 46925 83968 0.05916 0.05733 0.05556 0.05384 0.05217 45113 46965 14735 30919 78557 8.42142 8.69205 8.97126 9.25932 9.55652 98302 55373 06650 08003 02706 8.48059 8.74939 9.02682 9.31316 9.60869 43415 02338 21384 38922 81263 0.99302 0.99344 0.99384 0.99421 0.99456 35419 70066 48468 86036 97268 0.95 0.96 0.97 0.98 0. 99 19.77686 20.40804 21.05935 21.73145 22.42500 89693 01345 48847 60946 71560 0.05056 0.04900 0.04748 0.04601 0.04459 41212 02956 48354 62446 30738 9.86315 10.17951 10.50593 10.84271 11.19020 24240 99195 50247 99250 70411 9.91371 10.22852 10.55341 10.88873 11.23480 65453 02151 98601 61696 01149 0.99489 0.99520 0.99550 0.99577 0.99603 95797 94443 05263 39591 08084 1. 00 23.14069 26328 t-t)3 0.04321 39183 (-;I3 11.54873 (-y93573 [1 11.59195 X [ 1 95764 03654 99944 23136 13637 sinh 2.30129 2.38127 2.46360 2.54836 2.63563 TX 89023 57753 30666 20366 63461 cash TX 2.50917 84787 2.58272 61407 2.65882 30610 2.73754 43503 2.81896 77098 tanh 0.91715 0.92200 0.92657 0.93089 0.93496 ~fl: 23357 08803 65378 34251 50714 e-rz 0.20787 0.20145 0.19521 0.18918 0.18333 11 t-y32755 [1 0.99627 20762 (-[I4 11 ELEMENTARY TRANSCENDENTAL INVERSE 221 FUNCTIONS HYPERBOLIC Table FUNCTIONS arcsinh x 4.17 arctanh z arcsinh x 0.00000 0000 0.02 0.03 0. 04 9833 0.01999 8667 0.02999 5502 0.03998 9341 arctanh .E 0.00000 0000 0.01000 0.02000 0.03000 0.04002 0333 2667 9004 1353 o.x5o 0.51 0.52 0.53 0.54 0.48121 0.49013 0.49902 0.50788 0.51669 1825 8161 8444 2413 9824 0.54930 0.56272 0.57633 0.59014 0.60415 6144 9769 9754 5160 5603 0. 05 0. 06 0. 07 0. 08 0.09 0.04997 0.05996 0.06994 0.07991 0.08987 9190 4058 2959 4912 8941 0.05004 0.06007 0.07011 0.08017 0.09024 1729 2156 4671 1325 4188 0.55 0.56 0.57 0.58 0.59 0.52548 0.53422 0.54293 0.55159 0.56023 0448 4074 0505 9562 1077 0.61838 0.63283 0.64752 0.66246 0.67766 1313 3186 2844 2707 6068 0.10 0. 11 0. 12 0. 13 0.14 0.09983 0.10977 0.11971 0.12963 0.13954 4079 9366 3851 6590 6654 0.10033 0.11044 0.12058 0.13073 0.14092 5347 6915 1028 9850 5576 0.60 0.61 0.62 0.63 0.64 0.56882 0.57738 0.58589 0.59437 0.60282 4899 0892 8932 8911 0733 0.69314 0.70892 0.72500 0.74141 0.75817 7180 1359 5087 6144 3745 0.15 0.16 0.17 0.18 0.19 0.14944 0.15932 0.16919 0.17904 0.18887 3120 5080 1636 1904 5015 0.15114 0.16138 0.17166 0.18198 0.19233 0436 6696 6663 2689 7169 0.65 0.66 0.67 0.68 0.69 0.61122 0.61958 0.62791 0.63620 0.64445 4314 9584 6485 4970 5005 0.77529 0.79281 0.81074 0.82911 0.84795 8706 3631 3125 4038 5755 0.20 0.21 0.22 0.23 0.24 0.19869 0.20848 0.21826 0.22801 0.23775 0110 6350 2908 8972 3749 0.20273 0.21317 0.22365 0.23418 0.24477 2554 1346 6109 9466 4112 0.70 0.71 0.72 0.73 0.74 0.65266 0.66083 0.66897 0.67707 0.68512 6566 9641 4227 0332 7974 0.86730 0.88718 0.99764 0.92872 0.95047 0527 3863 4983 7364 9381 0.25 0.26 0:27 0.28 0.29 0.24746 0.25715 0.26682 0.27646 0.28608 6462 6349 2667 4691 1715 0.25541 0.26610 0.27686 0.28768 0.29856 2812 8407 3823 2072 6264 0.75 0.76 0.77 0.78 0.79 0.69314 0.70112 0.70907 0.71697 0.72484 7181 7988 0441 4594 0509 0.97295 0.99621 1.02032 1.04537 1.07143 5074 5082 7758 0548 1684 0.30 0.31 0.32 0.33 0.34 0.29567 0.30523 0.31477 0.32428 0.33376 3048 8020 5980 6295 8352 0.30951 0.32054 0.33164 0.34282 0.35409 9604 5409 7108 8254 2528 0.80 0.81 0.82 0.83 0.84 0.73266 0.74045 0.74820 0.75592 0.76359 8256 7912 9563 3300 9222 1.09861 1.12702 1.15681 1.18813 1.22117 2289 9026 7465 6404 3518 0.35 0.36 0.37 0. 38 0.39 0.34322 0.35264 0.36203 0.37140 0.38073 1555 5330 9121 2391 4624 0.36544 0.37688 0.38842 0.40005 0.41180 3754 5901 3100 9650 0034 0. 85 0.86 0.87 0.88 0.89 0.77123 0.77883 0.78640 0.79392 0.80141 7433 8046 1177 6950 5491 1.25615 1.29334 1.33307 1.37576 1.42192 2811 4672 9629 7657 5871 0. 40 0. 41 0.42 0.43 0. 44 0.39003 0.39930 0.40854 0.41774 0.42691 5320 4001 0208 3500 3454 0.42364 0.43561 0.44769 0.45989 0.47223 8930 1223 2023 6681 0804 0.90 0.91 0.92 0.93 0.94 0.80886 0.81628 0.82365 0.83100 0.83830 6936 1421 9091 0091 4575 1.47221 1.52752 1.58902 1.65839 1.73804 9490 4425 6915 0020 9345 0.45 0.46 0. 47 0. 48 0.49 0.43604 0.44515 0.45421 0.46325 0.47224 9669 1759 9359 2120 9713 0.48470 0.49731 0.51007 0.52298 0.53606 0279 1288 0337 4278 0337 0.95 0.96 0.97 0.98 0.99 0.84557 0.85280 0.86000 0.86716 0.87428 2697 4617 0498 0507 4812 1.83178 1.94591 2.09229 2.29755 2.64665 0823 0149 5720 9925 2412 0.50 0.48121 1825 0.54930 6144 1.00 0.88137 3587 co c.Lxoo0.00999 0 01 For use of the table see Examples 26-28. Qo(x) (Legendre Function-Second Kind)=arctanh x(/x!<l) =arccoth ~(1~1>1) Compiled from Harvard Computation Laboratory, Tables of inverse hyperbolic functions. Harvard Univ. Press, Cambrid@, Mass., 1949 (with permission). 222 ELEMENTARY Table 4.17 TRANSCENDENTAL INVERSE FUNCTIONS HYPERBOLIC FUNCTIONS 1.50 1.51 1.52 1.53 1.54 arcsinh x 1.19476 3217 1.20029 7449 1.20580 6263 1.21128 9840 1.21674 8362 arccosh x (S--l)+ 0.86081 788 0.85849 554 0.85618 806 0.85389 528 0.85161 706 968 798 099 841 994 1.55 1.56 1.57 1.58 1.59 1.22218 1.22759 1.23297 1.23833 1.24367 2008 0958 5390 5478 1400 0.84935 0.84710 0.84486 0.84264 0.84043 324 368 823 676 913 0.96794 0.96487 0.96182 0.95880 0.95579 529 415 625 131 904 1.60 1. 61 1.62 1.63 1.64 1.24898 1.25427 1.25953 1.26477 1.26999 3328 1436 5895 6877 4549 0.83824 0.83606 0.83389 0.83174 0.82960 520 483 788 424 376 6154 1729 5013 6208 5514 0.95281 0.94986 0.94692 0.94401 0.94111 918 146 561 139 853 1.65 1.66 1.67 1. 68 1.69 1.27518 1.28036 1.28550 1.29063 1.29573 9081 0639 9389 5495 9120 0.82747 0.82536 0.82326 0.82117 0.81909 632 179 005 097 443 1.01597 1.02235 1.02871 1.03503 1.04133 3134 9270 4123 7896 0792 0.93824 0.93539 0.93256 0.92975 0.92696 678 589 563 576 604 1.70 1.71 1. 72 1.73 1.74 1.30082 1.30587 1.31091 1.31593 1.32092 0427 9576 6727 2038 5666 0.81703 0.81497 0.81293 0.81091 0.80889 032 850 888 132 572 1.25 1.26 1.27 1.28 l-29 1.04759 1.05382 1.06002 1.06619 1.07233 3013 4760 6237 7645 9185 0.92419 0.92144 0.91871 0.91600 0.91331 624 613 550 411 175 1.75 1.76 1.77 1.78 1.79 1.32589 1.33084 1.33577 1.34068 1.34557 7767 8496 8006 6450 3978 0.80689 0.80489 0.80291 0.80095 0.79899 197 994 954 066 318 1.30 1. 31 1. 32 1.33 1. 34 1.07845 1.08453 1.09058 1.09661 1.10260 1059 3467 6610 0688 5899 0.91063 0.90798 0.90534 0.90272 0.90012 821 328 676 843 810 1.80 1.81 1.82 1.83 1.84 1.35044 1.35528 1.36011 1.36491 1.36970 0740 6886 2562 7914 3089 0.79704 0.79511 0.79318 0.79127 0.78937 701 203 816 527 328 1.35 1.36 1.37 1.38 1.39 1.10857 1.11451 1.12042 1.12630 1.13215 2442 0515 0317 2042 5887 0.89754 0.89498 0.89243 0.88990 0.88738 557 064 313 284 959 1.85 1.86 1.87 1.88 1.89 1.37446 1.37921 1.38393 1.38864 1.39333 8228 3477 8975 4863 1280 0.78748 0.78560 0.78373 0.78187 0.78002 209 160 170 231 334 1.40 1.41 1. 42 1.43 1.44 1.13798 1.14378 1.14955 1.15529 1.16101 2046 0715 2086 6351 3703 0.88489 0.88241 0.87995 0.87750 0.87507 320 348 026 336 261 1.90 1.91 1.92 1.93 1.94 1.39799 1.40264 1.40727 1.41188 1.41647 8365 6254 5083 4987 6099 0.77818 0.77635 0.77453 0.77272 0.77093 468 625 796 971 142 1. 45 1.46 1.47 1. 48 1. 49 1.16670 1.17236 1.17800 1.18361 1.18920 4331 8425 6174 7765 3384 0.87265 0.87025 0.86787 0.86550 0.86315 784 888 557 774 523 1.95 1.96 1.97 1.98 1.99 1.42104 1.42560 1.43013 1.43465 1.43915 8552 2476 8002 5259 4374 0.76914 0.76736 0.76559 0.76383 0.76208 300 437 544 612 633 1.50 1.19476 3217 t-f)4 2.00 1.44363 5475 (-i)3 0.76034 600 1. 00 1. 01 1. 02 1.03 1. 04 arcsinh x 0.88137 3587 0.88842 7007 0.89544 5249 0.90242 8496 0.90937 6928 arccosh x (227 1.00000 000 0.99667 995 0.99338 621 0.99011 848 0.98687 641 1.05 1.06 1.07 1. 08 1. 09 0.91629 0.92317 0.93001 0.93682 0.94360 0732 0094 5204 6251 3429 0.98365 0.98046 0.97730 0.97415 0.97103 1.10 1.11 1.12 1.13 1.14 0.95034 0.95705 0.96373 0.97037 0.97698 6930 6950 3684 7331 8088 1.15 1.16 1.17 1.18 1.19 0.98356 0.99011 0.99662 1.00310 1.00955 1.20 1.21 1.22 1.23 1.24 X [1 0.86081 788 (-!)3 1 c1 2 [I 1 [1 y2 ELEMENTARY TRANSCENDENTAL IUVERSE arccosh z-ln I 0.62381 07164 0.62685 90940 0.62981 77884 0.63268 90778 0.63547 51194 t-1 0.50 0.49 0.48 0.47 0.46 arcsinh 0.75048 0.74839 0.74632 0.74428 0.74228 z-111 z 82946 16011 48341 85962 34908 0.45 0.44 0.43 0.42 0.41 0.74031 0.73836 0.73646 0.73458 0.73274 01215 90921 10057 64641 60676 0.63817 0.64079 0.64334 0.64580 0.64819 0.40 0.39 0. 38 0.37 0. 36 0.73094 0.72917 0.72743 0.72573 0.72407 04145 01001 57167 78524 70912 223 FUNCTIONS HYPERBOLIC Table FUNCTIONS <Zi arcsinh z-111 z 0.70841 81861 0.70724 57326 0.70611 72820 0.70503 32895 0.70399 41963 2-I arccosh z-ln z 0.67714 27078 0.67842 57947 0.67965 18411 0.68082 14660 0.68193 52541 04288 23983 05002 51134 66000 0.68299 0.68399 0.68494 0.68584 0.68668 4.17 37571 74947 69555 25981 48518 z 0. 24 0.25 t 2 0.22 0.23 0.21 79566 95268 16670 61207 45429 : 0.19 0. 20 t 0.18 0.17 2 0.16 0.70300 0.70205 0.70115 0.70029 0.69948 0.65050 0.65274 0.65491 0.65701 0.65904 85051 95004 89477 81952 85249 3 0.15 3; 14 0.13 0.12 0.11 0.69872 0.69801 0.69734 0.69672 0.69615 53043 15527 56533 78946 85462 0.68747 0.68821 0.68889 0.68952 0.69010 41175 07683 51504 75836 83616 11555 72458 78974 41577 70226 3 0.10 0.09 0.08 0.07 0. 06 0.69563 0.69516 0.69474 0.69436 0.69404 78573 60572 33542 99357 59680 0.69063 0.69111 0.69154 0.69191 0.69224 77531 60018 33269 99235 59631 74382 63038 44732 27575 19258 :: 4 0.05 0.04 0.03 0.02 0. 01 0.69377 0.69354 0.69337 0.69324 0.69317 15954 69408 21047 71656 21796 0.69252 15938 0.69274 69403 0.69292 21046 0.69304 71656 0.69312 21796 4 0. 00 0.69314 71806 0.69314 0.35 0.34 0.33 0.32 0.31 0.72245 0.72086 0.71932 0.71781 0.71634 40117 91873 31846 65636 98766 0.66101 0.66290 0.66473 0.66650 0.66820 0.30 0.29 0.28 0.27 0.26 0.71492 0.71353 0.71219 0.71089 0.70963 36678 84725 48165 32154 41742 0.66984 0.67142 0.67294 0.67440 0.67580 0.25 0.70841 81861 C-56)5 [ 3 : : 3 : : 0.67714 27078 C-65) 1 [ I 0>=nearest ROOTS z,,, OF integer COL) r,, cash : 4.73004 7.85320 i 5 14.13716 10.99560 17.27875 to .I’. x,=1 46 07 55 78 96 For ~25, .J,,=; [2~+l]r ROOTS [1 C-56)6 a.,, OF cos z,‘ eoah II %I ii 4.69409 1.87510 41 11 3 z 7.85475 14.13716 10.99554 74 84 07 For )!>5, .z,,=: [2/t- 11~ x,,= - 1 * 71806 [1 t-y Table 4.18 224 ELEMENTARY Table -A 0.00 0.05 0.10 0.15 0.20 3.14159 2.99304 2.86277 2.75032 2.65366 0.25 0.30 0.35 0.40 0.45 TRANSCENDENTAL FUNCTIONS ROOTS x uOF tan X )I : pt, 4.19 Xl 26 27 12.5?637 12.02503 11.70268 11.52018 11.40863 15.7%#96 15.06247 14.73347 14.56638 14.46987 18.84956 18.11361 17.79083 17.64009 17.55621 21.99115 21.17717 20.86724 20.73148 20.65782 25.;;274 24.25156 23.95737 23.83468 23.76928 28 ;;433 27:33519 27.05755 26.94607 26.88740 11.33482 11.28284 11.24440 11.21491 11.19159 14.40797 14.36517 14.33391 14.31012 14.29142 17.50343 17.46732 17.44113 17.42129 17.40574 20.61203 20.58092 20.55844 20.54146 20.52818 23.72894 23.70166 23.68201 23.66719 23.65561 26.85142 26.82716 26.80971 26.79656 26.78631 8.09616 8.07544 8.05794 8.04298 8.03004 11.17271 11.15712 11.14403 11.13289 11.12330 14.27635 14.26395 14.25357 14.24475 14.23717 17.39324 17.38298 17.37439 17.36711 17.36086 20.51752 20.50877 20.50147 20.49528 20.48996 23.64632 23.63871 23.63235 23.62697 23.62235 26.77809 26.77135 26.76572 26.76096 26.75688 4.97428 4.95930 4.94592 4.93389 4.92303 8.01875 8.00881 7.99999 7.99212 7.98505 11.11496 11.10764 11.10116 11.09538 11.09021 14.23059 14.22482 14.21971 14.21517 14.21110 17.35543 17.35068 17.34648 17.34274 17.33939 20.48535 20.48131 20.47774 20.47457 20.47172 23.61834 23.61483 23.61173 23.60897 23.60651 26.75333 26.75023 26.74749 26.74506 26.74288 4.91318 7.97867 11.08554 14.20744 17.33638 20.46917 23.60428 26.74092 4.932318 7.97867 4.90375 7.97258 4.89425 7.96648 4.88468 7.96036 4.87504 7.95422 ll'r$554 11:08110 11.07665 11.07219 11.06773 14 i:744 14:20395 14.20046 14.19697 14.19347 17.37638 17.33351 17.33064 17.32777 17.32490 20 4?917 20:46673 20.46430 20.46187 20.45943 23.6?428 23.60217 23.60006 23.59795 23.59584 26 7:092 26:73905 26.73718 26.73532 26.73345 1.93974 1.92035 1.90036 1.87976 1.85852 4.86534 4.85557 4.84573 4.83583 4.82587 7.94807 7.94189 7.93571 7.92950 7.92329 11.06326 11.05879 11.05431 11.04982 11.04533 14.18997 14.18647 14.18296 14.17946 14.17594 17.32203 17.31915 17.31628 17.31340 17.31052 20.45700 20.45456 20.45212 20.44968 20.44724 23.59372 23.59161 23.58949 23.58738 23.58526 26.73159 26.72972 26.72785 26.72598 26.72411 -0.50 -0.45 -0.40 -0.35 -0.30 1.83660 1.81396 1.79058 1.76641 1.74140 4.81584 4.80575 4.79561 4.78540 4.77513 7.91705 7.91080 7.90454 7.89827 7.89198 11.04083 11.03633 11.03182 11.02730 11.02278 14.17243 14.16892 14.16540 14.16188 14.15835 17.30764 17.30476 17.30187 17.29899 17.29610 20.44480 20.44236 20.43992 20.43748 20.43503 23.58314 23.58102 23.57891 23.57679 23.57467 26.72225 26.72038 26.71851 26.71664 26.71477 -0.25 -0.20 -0.15 -0.10 -0.05 1.71551 1.68868 1.66087 1.63199 1.60200 4.76481 4.75443 4.74400 4.73351 4.72298 7.88567 7.87936 7.87303 7.86669 7.86034 11.01826 11.01373 11.00920 11.00466 11.00012 14.15483 14.15130 14.14777 14.14424 14.14070 17.29321 17.29033 17.28744 17.28454 17.28165 20.43259 20.43014 20.42769 20.42525 20.42280 23.57255 23.57043 23.56831 23.56619 23.56407 26.71290 26.71102 26.70915 26.70728 26.70541 0.00 0.05 0.10 0.15 0.20 1.57080 1.53830 1.50442 1.46904 1.43203 4.71239 4.70176 4.69108 4.68035 4.66958 7.85398 7.84761 7.84123 7.83484 7.82844 10.99557 10.99102 10.98647 10.98192 10.97736 14.13717 14.13363 14.13009 14.12655 14.12301 17.27875 17.27586 17.27297 17.27007 17.26718 20.42035 20.41790 20.41545 20.41300 20.41055 23.56194 23.55982 23.55770 23.55558 23.55345 26.70354 26.70166 26.69979 26.69792 26.69604 0.25 0.30 0.35 0.40 0.45 1.39325 1.35252 1.30965 1.26440 1.21649 4.65878 4.64793 4.63705 4.62614 4.61519 7.82203 7.81562 7.80919 7.80276 7.79633 10.97279 10.96823 10.96366 10.95909 10.95452 14.11946 14.11592 14.11237 14.10882 14.10527 17.26428 17.26138 17.25848 17.25558 17.25268 20.40810 20.40565 20.40320 20.40075 20.39829 23.55133 23.54921 23.54708 23.54496 23.54283 26.69417 26.69230 26.69042 26.68855 26.68668 0.50 0.55 0.60 0.65 0.70 1.16556 1.11118 1.05279 0.98966 0.92079 4.60422 4.59321 4.58219 4.57114 4.56007 7.78988 7.78344 7.77698 7.77053 7.76407 10.94994 10.94537 10.94079 10.93621 10.93163 14.10172 14.09817 14.09462 14.09107 14.08752 17.24978 17.24688 17.24398 17.24108 17.23817 20.39584 20.39339 20.39094 20.38848 20.38603 23.54071 23.53858 23.53646 23.53433 23.53221 26.68480 26.68293 26.68105 26.67918 26.67730 0.75 0.80 0.85 0.90 0.95 0.84473 0.75931 0.66086 0.54228 0.38537 4.54899 4.53789 4.52678 4.51566 4.50454 7.75760 7.75114 7.74467 7.73820 7.73172 10.92704 10.92246 10.91788 10.91329 10.90871 14.08396 14.08041 14.07686 14.07330 14.06975 17.23527 17.23237 17.22946 17.22656 17.22366 20.38357 20.38112 20.37867 20.37621 20.37376 23.53008 23.52796 23.52583 23.52370 23.52158 26.67543 26.67355 26.67168 26.66980 26.66793 1.00 0.00000 4.49341 7.72525 10.90412 14.06619 17.22075 20.37130 23.51945 26.66605 6.28~19 9.4iT78 2.57043 2.49840 2.43566 2.38064 2.33208 5.99209 5.76056 5.58578 5.45435 5.35403 5.27587 5.21370 5.16331 5.12176 9.00185 8.70831 8.51805 8.39135 8.30293 8.23845 8.18965 8.15156 8.12108 0.50 0.55 0.60 0.65 0.70 2.28893 2.25037 2.21571 2.18440 2.15598 5.08698 5.05750 5.03222 5.01031 4.99116 0.75 0.80 0.85 0.90 0.95 2.13008 2.10638 2.08460 2.06453 2.04597 1.00 2.02876 x-1 -1.00 -0.95 -0.90 -0.85 -0.80 2.02876 2.01194 1.99465 1.97687 1.95857 -0.75 -0.70 -0.65 -0.60 -0.55 Xl x3 For h-0, see Sof Table 10.6. jl. <x>=nearest integer to X. <A> 1: -1 1: : 1 ELEMENTARY TRANSCENDENTAL ROOTS x,, OF 225 FUNCTIONS cot xn =Xx,, Table 4.20 A x1 22 53 24 0.00 0.05 0.10 0.15 0.20 1.57080 1.49613 1.42887 1.36835 1.31384 4.71239 4.49148 4.30580 4.15504 4.03357 7.85398 7.49541 7.22811 7.04126 6.90960 10.99557 10.51167 10.20026 10.01222 9.89275 14,13717 13.54198 13.21418 13.03901 12.93522 17.27876 16.58639 16.25936 16.10053 16.01066 x7 20.42035 19.64394 19.32703 19.18401 19.10552 23.5*6194 22.71311 22.41085 22.28187 22.21256 26 70354 25:79232 25.50638 25.38952 25.32765 0.25 0.30 0.35 0.40 0.45 1.26459 1.21995 1.17933 1.14223 1.10820 3.93516 3.85460 3.78784 3.73184 3.68433 6.81401 6.74233 6.68698 6.64312 6.60761 9.81188 9.75407 9.71092 9.67758 9.65109 12.86775 12.82073 12.78621 12.75985 12.73907 15.95363 15.91443 15.88591 15.86426 15.84728 19.05645 19.02302 18.99882 18.98052 18.96619 22.16965 22.14058 22.11960 22.10377 22.09140 25.28961 25.26392 25.24544 25.23150 25.22062 0.50 0.55 0.60 0.65 0.70 1.07687 1.04794 1.02111 0.99617 0.97291 3.64360 3.60834 3.57756 3.55048 3.52649 6.57833 6.55380 6.53297 6.51508 6.49954 9.62956 9.61173 9.59673 9.58394 9.57292 12.72230 12.70847 12.69689 12.68704 12.67857 15.83361 15.82237 15.81297 15.80500 15.79814 18.95468 18.94523 18.93734 18.93065 18.92490 22.08147 22.07333 22.06653 22.06077 22.05583 25.21190 25.20475 25.19878 25.19373 25.18939 0.75 0.80 0.85 0.90 0.95 0.95116 0.93076 0.91158 0.89352 0.57647 3.50509 3.48590 3.46859 3.45292 3.43865 6.48593 6.47392 6.46324 6.45368 6.44508 9.56331 9.55486 9.54738 9.54072 9.53473 12.67121 12.66475 12.65904 1?.65395 12.64939 15.79219 15.78698 15.78237 15.77827 15.77459 18.91991 18.91554 18.91168 18.90825 18.90518 22.05154 22.04778 22.04447 22.04151 22.03887 25.18563 25.18234 25.17943 25.17684 25.17453 1.00 0.86033 3.42562 6.43730 9.52933 12.64529 15.77128 18.90241 22.03650 25.17245 X5 X6 x7 25 % 29 Xl 22 53 x4 1.00 0.95 0.90 0.85 0.80 0.86033 0.84426 0.82740 0.80968 0.79103 3.42562 3.41306 3.40034 3.38744 3.37438 6.43730 6.42987 6.42241 6.41492 6.40740 9.52933 9.52419 9.51904 9.51388 9.50871 12.64529 12.64138 12.63747 12.63355 12.62963 15.77128 15.76814 15.76499 15.76184 15.75868 18.90241 18.89978 18.89715 18.89451 18.89188 22.03650 22.03424 22.03197 22.02971 22.02745 25.17245 25.17047 25.16848 25.16650 25.16452 0.75 0.70 0.65 0.60 0.55 0.77136 0.75056 0.72851 0.70507 0.68006 3.36113 3.34772 3.33413 3.32037 3.30643 6.39984 6.39226 6.38464 6.37700 6.36932 9.50353 9.49834 9.49314 9.48793 9.48271 12.62570 12.62177 12.61784 12.61390 12.60996 15.75553 15.75237 15.74921 15.74605 15.74288 18.88924 18.88660 18.88396 18.88132 18.87868 22.02519 22.02292 22.02066 22.01839 22.01612 25.16254 25.16055 25.15857 25.15659 25.15460 0.50 0.45 0.40 0.35 0.30 0.65327 0.62444 0.59324 0.55922 0.52179 3.29231 3.27802 3.26355 3.24891 3.23409 6.36162 6.35389 6.34613 6.33835 6.33054 9.47749 9.47225 9.46700 9.46175 9.45649 12.60601 12.60206 12.59811 12.59415 12.59019 15.73972 15.73655 15.73338 15.73021 15.72704 18.87604 18.87339 18.87075 18.86810 18.86546 22.01386 22.01159 22.00932 22.00705 22.00478 25.15262 25.15063 25.14864 25.14666 25.14467 2 2 0.25 0.48009 3.21910 6.32270 0.20 0.43284 3.20393 6.31485 0.15 0.37788 3.18860 6.30696 0.10 0.31105 3.17310 6.29906 0.05 0.22176 3.15743 6.29113 9.45122 9.44595 9.44067 9.43538 9.43008 12.58623 12.58226 12.57829 12.57432 12.57035 15.72386 15.72068 15.71751 15.71433 15.71114 18.86281 18.86016 18.85751 18.85486 18.85221 22.00251 22.00024 21.99797 21.99569 21.99342 25.14268 25.14070 25.13871 25.13672 25.13473 4 0.00 0.00000 3.14159 6.28319 * [c-y] ['-;'l] 9.42478 12.56637 15.70796 18.84956 21.99115 25.13274 [(-p] ['-y] ['-;"] [(-p] [y;"] ~[(-y'! A-’ 3% 59 <A> =nearest integer to X. For h-l > .20, the maximum error in linear interpolation For A-l c .20, I--&+&2-&3+..: *see page n. 3 is (- 4)7; five-point interpolation gives 5D. <A> : : 1 : H 2 z 3 G :oo Co 5. Exponential Integral WALTER GAUTSCHI and Related l AND WILLIAM Functions I?. CAHILL 2 Contents Page Mathematical Properties .................... 5.1. Exponential Integral .................. 5.2. Sine and Cosine Integrals Numerical Methods ...................... ............. 233 233 .......................... 235 5.3. Use and Extension References Table of the Tables 5.1. Sine, Cosine and Exponential x-‘Si(x), x-2[Ci(x)-ln cc-‘[Ei(x)-ln Si(x), Ci(x), Si(x), Ci(x), Table ................ 5.2. Integrals (O<sl 10) . . . . 238 x-r] x-r], x-‘[E,(x)+ln x+-r], x=0(.01).5, 10s 10D; Ei(x), El(x), 9D; x=.5(.01)2 10D; xP Ei(x), xez El(x), 9D; x=2(.1)10 Sine, Cosine and Exponential Integrals for Large Arguments (101x1 a) . . . . . . . . . . . . . . . . . . . . . . . . . qf(x), 9D; x2g(x), 7D; ze-“Ei(x), 8D; xezEl(s), 10D f(x)=-si(x) cos x+Ci(x) sin 2, g(x)=-si(x) sin x-C;(x) co9 x x-‘=.l(-.005)0 Table 5.3. Si(rx), Table 5.4. 228 228 231 Sine and Cosine Integrals for Arguments Gin(m), x=0(.1)10, E,(x) TIZ (0 Ix 210) . . . 244 (0 5x 52) . . . . . . . . . . 245 7D Exponential 243 Integrals E2(x)--2 In 2, E,(x), n=3,4, 10, 20, x=0(.01).5 E,(x), 12=2, 3, 4, 10, 20, x=.5(.01)2, 7D Table 5.5. Exponential Integrals E,(x) for Large Arg,urnents (2 Ix _< 03) a (x+n)e”E,(x), n=2, 3, 4, 10, 20, x-‘=.5(-.05).1(-.01>0, 5D 248 Table 5.6. Exponential Integral for Complex Arguments (jzj<29) zezEl(z), z=X+iy, x=-19(1)20, y=O(1)20, 6D .. 249 (l.4 <5) . 251 Table 5.7. Exponential Integral for Small Complex Arguments ezEl(z), z=x+iy, x=-4(.5)-2, y=O(.2)1, 6D E,(z)+ln z, z=x+iy, x=-2(.5)2.5, y=O(.2)1, 6D The authors acknowledge the assistance of David S. Liepman in the preparation and checking of the tables, Robert L. Durrah for the computation of Table 5.2, and Alfred E. Beam for the computation of Table 5.6. r Guest worker, National Bureau of Standards, from the American Purdue University.) 2 National Bureau of Standards. (Presently NASA.) University. (Presently 227 5. Exponential Integral Mathematical 5.1. Exponential and Related Properties Explicit Integral Definitions 5.1.8 &(z) 5.1.2 5.1.3 Expressions for (Y,,(Z) and (l-l-z++ 22 a,(z)=n!z-*-le-” /3,,(z) . . . d-5) 5.1.9 (larg zl<?r> f ” e-’ r” edt Ei(x)=--J1 t dt=Tt 2 m Functions =n!2-n-1(e” [l-z+&. . . +(-l)“$] (xx9 --e-z (l+~+~+ 22 . . . +$I . (x>l) 5.1.4 E,(z)=jy e; (n=O, at (D %(4’ s 1,2, . . .; ~z>O) ‘(n=O, 1,2, . . .; 92>0) 5.1.5 t*e-z*dt 1 5.1.6 pn(z)=J:, t”e-“dt (n=O, 1, 2, . . . ) In 5.1.1 it is assumed that the path of integration excludes the origin and does not cross the negative real axis. Anal& continuation of the functions in 5.1.1, 5.1.2, and 5.1.4 for n>O yields multi-valued functions with branch points at z-0 and z= a.3 They are single-valued functions in the z-plane cut along the negative real axis.* The function ii(z), the logarithmic integral, has an additional branch point at z=I. Interrelations E1(-xfiO)=-Ei(x):)ii?r, -Ei(x)=~[~l(-x+iO)+E1(-x--iO)] [5.14], [5.16] use the entire 5.1. y=Ei(x) and y=&(x). i 5.1.7 * Some authors FIGURE (x>O) function : (I-e-g)dt/t 89 the basic function and denote IEm(z). We have Ein(z)=Er(z)+ln 2+7. it by 4 Various authors define the integral z-plane cut along the by Ei(z). For 2=z>O (e.g., in [5.10], [5.25]), then used to designate Correspondingly, El(z) _“, (et/t)& in the s positive real axis and denote it also additional notations such as %%(a$ E*(z) (in [5.2]), Ei*(z) (in [5.6]) are the principal value of the integral. is often denoted by - Ei( -z). FIGURE 5.2. y=&(x) n=O, 1, 2, 3, 5, 10 EXPONENTIAL INTEGRAL AND RELATED Y 4 5 229 FUNCTIONS Symmetry n=O n=l n.2 w3 n-4 n.5 n=6 5.1.13 am Relation .=Ez (2) 4 Recurrence 3 Relations 5.1.14 E.+,(z)=~[~-‘--zE.(z)] 2 5.1.15 1-s. 0 .5 I.0 I.5 2.0 2.5 FIGURE 5.3. 3.0 3.5 y=an(z) x (n=1,2,3,. aY,(z)=4-z+mYn-1(2) (n=1,2,3,. . .) (n=1,2,3,. . .) 5.1.16 zSn(z)=(-l)ne’-e-“+n~,-,(z) n=0(1)6 . .) Inequalities [5.8], [5.4] 5.1.17 n+En(x)<E.+l(x)<E,(x) (x>O;n=l, 2,3, . . .) 5.1.18 (x>O;n=1,2,3,. J%(4<J%1wL+1(4 . .) 5.1.19 I& (x>O;n=1,2,3,. <e%(x)<h $n<ezE,(x) (I+:) . .) 5.1.20 n:l \’ -151 FIGURE 5.4. Series Ei(z)=r+lnz+FI 5.1.10 y=&,(x) n=O, 1, 2, 5, 10, 15 5.1.21 &[&]>O (x>O;n=1,2,3,...) Continued Expansions a n$ . 5.1.12 E,(z)=e-* 5.1.23 (larg 4<4 $0)=--r, W=-r+~2~ 4 56649 . . *. is Euler’s Fraction 5.1.22 (XX) ( --$$-n$-&. . .) Special r=.57721 (x>O) am Values =A (n>l) E,(z)=$ 5.1.24 (n>l> constant. 5.1.25 d4 =$, @,,(,z)=~sinh z (b-g 4-G) 230 EXPONENTIAL INTEGRAL AND -dE”‘z)---E (n==1,2,3,. 12- 1 (z) dz FUNCTIONS 5.1.36 Derivatives 5 . 126 . RELATED le --Otsin bt . .) t S0 dt=arctan k +Y&(a+ib) (a>O, b real) 5.1.27 5.1.37 ~n[e”El(z)l=&l [e”E,(z)l 1 e”‘(l-cos +(-W-l)! 23 , , ,-** (+l 2” Definite and S t bt) Indefinite ) +WE,(-a+ib) 5.1.38 Integrals (a>O,b real) bt) t +%S(a+ib) 2-e-’ l S (a>O, b real) 1 e-“‘(l--OS (For more [5.3], [5.6], involving extensive [5.11], tables [5.12], of integrals see [5.13]. For integrals S 0 Z,(x) see [5.9].) 5.1.39 - e-“l o b+t dt=eubEl(ab) S 5.1.28 5.1.40 5.1.29 S op & +Ei(a) 0 dt=$ In (1+$)-&(a) 0 -t z e’-1 t dt=&(z)+ln dt=Ei S0 (x)-In z+y x--y (XX) 5.1.41 dt=eciQbE,(--iab) (a>% b>O) S G2 dx=$ [e-“E,(-a--ix)-ee”E,(u--ix)] 5.1.30 +const. S om s2 eta~dt=eabEl(ub) (a>% b>O) 5.1.42 a$ti - t+ib S 5.1.43 (a>% b>O) m S S e-“‘-emb’ t 0 5.1.33 [e-“I$(-a--ix)+e”E,(a--ix)] +const. o t2+b2 eI”ldt=e-““(-Ei(ub)+ir) 5.1.32 dx-; S 5.1.31 dt=h S b a a& 9(ef”EI(-x+ia))+const. (a>O) 5.1.44 S s2 mE:(t)dt=2 dx=-; ln 2 0 Relation dx=-L2(ef”EI(-x+ia))+const. e+E,(t)dt= Function E,(z)=P-'r(l-n, 2) 5.1.46 m S Gamma 5.1.45 5.1.34 to Incomplete (a>O) (see 6.5) q&)=2-*--lIyTL+1, 2) 0 q[ln 5.1.47 (1 +a) +g: q] (a>-1) 5.1.35 S ’ “’ SF bt dt=?r--arctan Relation 5.1.48 i +A?Z,(-a+ib) 0 (a>% b>O) j3,(z)=z-“-‘[r(n+l, to Spherical d4= -z)-Iyn+1, Bessel Functions z)] (see 10.2) EXPONENTIAL Number-Theoretic Significance INTEGRAL AND RELATED li (z) of ao= - .57721 566 al= .999!39 193 az= - .249!Jl 055 (Assuming Riemann’s hypothesis that all nonreal zeros of t(z) have a real part of 3) 5.1.50 li (x)-7r(x)=O(@ r(x) is the number to 2. In 2) of primes 5.1.54 (x+=> 231 PUNG!CIONS a3= .05519 968 a4= - .00976 004 as= .00107 857 l<z<=J less than or equal x2+a:x+T+6 x+b xe=J%c4=x2+b (2) I&> 1<5x 10-6 Y a,=2.334733 a2= 250621 5.1.55 b1=3.330657 bz= 1.681534 lOSx<w l&)l<lO-’ a1=4.03640 %=1.15198 5.1.56 I 0Y I 400 I 600 FIGUR.E 5.5. y=li(x) 200 Asymptotic I 000 I 1000 lix<w ~e(x>1<2Xlo-* L-X al= 8.57332 87401 as= 18.05901 69730 a3= 8.63476 08925 a(= .26777 37343 and y=~(x) Expansion 5.1.51 En(z)m e+il-n+n(n+l) 2 2 7- b,=5.03637 b2=4.19160 n(n+l)(n+2)+ z3 bI= 9.57332 b2=25.63295 b,=2 1.09965 b4= 3.95849 23454 61486 30827 69228 5.2. Sine and Cosine Integrals j *-* Definitions (larg Representation of E,(z) for Large 4 <%d 5.2.1 n 5.1.52 5.2.2 ta Ci(z)=r+ln +n(6x2-8nx+d) (x+nY +Nn, -.36nm4<R(n,x)< -’ x+n-1 Polynomial Approximations 5.1.53 &(x)+ln and Rational > 41 nT4 (x>O) 6 Ie(x)~<2Xlo-’ 6 The approximation 5.1.53 is from E. E. Allen, Note 169, MTAC 8, 240 (1954); approximations 5.1.54 and 5.1.56 are from C. Hastings, Jr., Approximations for digital computers, Princeton Univ. Press, Princeton, N.J., 1955; approximation 5.1.55 is from C. Hastings, Jr., Note 143, MTAC 7, 68 (1953) (with permission). (larg 4<4 s 0 ‘Ydt Shi(z)= 5.2.3 ‘I ‘sinh S 0 - t t dt 5.2.4 7 Chi(z)=r+ln O<s<l -- x=a0+aIx+azx2+a3i+a4x4+ap6+e(x) zf 6 Some authors z+ s 0 z coshtt-l [5.14], dt (lax 4<4 [5.16] use the entire function *(l- cos t)dt/t as the basic function and denote it by s Coin(a). We have Gin(z)= -Ci(z)+ln 2+-r. 7 The Cinh(z) = notations s Sib(z)= O’(cosh t- l)dt/t s0 ‘ sinh t dt/t, have also been proposed (5.14.1 EXPONENTIAL 232 Auxiliary Functions f(z)=Ci(z) 5.2.6 g(z)= 5.2.7 and Cosine in Shi(z)=~o co9 2 cos z-si(z) Integrals Functions - (-l)%!~” W) =r+ln z+zl 5.2.17 sin z--i(z) -Ci(z) AND RELATED FUNCTIONS 5.2.16 si(z.)=Si(z)-5 5.2.5 Sine INTEGRAL sin 2 Terms 5.2.8 Si(z)=i--f(z) 5.2.9 Ci(z)=f(z) sin z-g(z) Integral of co9 z-g(z) Auxiliary sin 2 (2n+$L+1)! Chi(z)=r+ln 5.2.18 dgl Symmetry Representations &, Relations Si(--z)=-Si(z), 5.2.19 2n(2n)r Si(Z)=si(z) 5.2.20 co9 2 Ci(-z)=Ci(z)--.i7r (O<w z<d C;(Z)=-) I Relation 5.2.10 si(z)=- e-2 cont cos (2 sin t)dt Si(z)=& 5.2.11 5.2.12 5.2.13 Ci(z) +&(z)=l’ to Exponential Integral 5.2.21 [E,(k)-EI(--iz)l+~ (kg zl<t$ e-’ coB’ sin (z sin t)dt 5.2.22 j(z)=S,- dt=l-g dt (.%'z>o) '$$ -cos (&>O)5.2.23 g(z)=t+z(jt= so t Si(iz)=E Ci(z)=-k 5.2.24 [Ei(z) +ITI(z)] (x>O) [E,(iz)+IG(--iz)] Ci(iz)=k (larg 4-C:) [Ei(z)--E,(x)]+i: Value b>O) at Infinity lim Si (2) =i z-P- 5.2.25 Integrals (For more extensive tables of integrals 15.31, [5.6], [5.11], [5.12], 15.131.) 5.2.26 sI OD‘!E$ * &=-si 5.2.27 FIGURE 5.6. Series y=Si(z) and y=Ci(z) *See page If. &=-Ci 5.2.28 a e-Wi s Cl (t)dt= Expansions 5.2.!29 5.2.14 5.2.15 my sE (2) S O SD m e-=‘si (2) -& (t)dt=--1 0 Si(z)=a gO JZ+t (g) 5.2.30 cos t Ci (t)dt= 0 (taris (!urg In (l+u*) arctan a a m S 4<4 zl<d (.@a>O)* (9?a>o) sin t si (t)dl=-5 0 see EXPONENTIAL 5.2.31 s0 5.2.32* OD (t)dt= Ci2 INTEGRAL AND RELATED 5.2.37 S m si2 (t)dt=E 233 l_<s<m 0 s0 .a Ci (t) si (t)dt= In 2 I4~>l<lo-* 5.2.33 ‘OSln +Ci bt &=i (b) s’(1-e-a’)(I+$) t 0 FUNCTIONS a,=7.547478 a~= 1.564072 5.2.38 b1=12.723684 * bz= 15.723606 * l<z<m + %?I$ (a+ ib) (a real, b>O) Asymptotic Expansions 5.2.34 ,(z)+(l-$+$-$+. . .) le(z)1<5Xlo-’ 38.027264 b1= 40.021433 &-265.187033 bz=322.624911 a3=335A77320 b3=570.236280 al= (larg zl<r) 5.2.35 a,= 5.2.39 Rational 5.2.36 Approximations 38.102495 b,=157.105423 l<s<=J * 152<a (&):<3Xlo-’ al = 42.242855 a,=302.757865 &=2.463936 Numerical 5.3. Use and Extension of the Tables Example 1. Compute Ci (.25) to 5D. From Tables 5.1 and 4.2 we have Ci (.25) - M.25) (.25)* -Y= _ .24g350 Ci (.25)=(.25)2(-.249350)+(-l.38629) +.577216= I Ei (8) =440.38. page II. 8 From C. Hastings, Jr., Approximations computers, Princeton Univ. Press, Princeton, (with permission). br= 449.690326 48.196927 Methods Example 3. Compute Si (20) to 5D. Since l/20=.05 from Table 5.2 we find j(20) = .049757, g(20) = .002464. From Table 4.8, sin 20 = .912945, cos 20 = .408082. Using 5.2.8 Si(20) =;-j(20) co9 20-g(20) sin 20 =1.570796-.022555=1.54824. -.82466. Example 2. Compute Ei (8) to 5s. From Table 5.1 we have ze-‘Ei (z) =1.18185 for s=8. From Table 4.4, e8=2.98096X103. Thus *see b3= 1114.978885 a*= 21.821899 bz=7.157433 bz= 482.485984 a,=352.018498 1~(z)1<2xlo-’ a1=7.241163 b1=9.068580 bl= for digital N.J., 1955 Example 4. Compute E,(z), n=l(l)N, to 5S for z= 1.275, N= 10. If z is less than about five, the recurrence relation 5.1.14 can be used in increasing order of n without serious loss of accuracy. By quadratic interpolation in Table 5.1 we get & (1.275) = .1408099, and from Table 4.4, e-l.*” =. 2794310. The recurrence formula 5.1.14 then yields 234 EXPONENTIAL n 1 2 3 4 5 EJ1.275) a&(1.275) .1408099 .0998984 .0760303 .0608307 .0504679 INTEGRAL 6 7 8 9 10 .0430168 .0374307 .0331009 .0296534 .0268469 Interpolating directly in Table 5.4 for n=lO we get E,,(1.275)= .0268470 as a check. Example 5. Compute E,(z), n= l(l)N; to 5s for 2= 10, N= 10. If, as in this example, z is appreciably larger than five and N<z, then the recurrence relation 5.1.14 may be safely used in decreasing order of n([5.5]). From Table 5.5 for z-*=.1 we get (s-t 10)eZElo(z)=1.02436 so that E,,(lO) =2.32529 x10-6. Using this as the initial value we obtain column (2). n 1 2 3 4 5 6 7 8 9 10 YfO) .41570 .38300 .355oz .33ozi .31oG .28= .27667 .25333 .25084 .22573 n 12 11 10 9 8 7 6 5 4 3 2 1 FUNCTIONS 106E,(12.3) .191038 .199213 208098 : 217793 .228406 240073 : 252951 .267234 283155 : 300998 .321117 .343953 106E,,(12.3) . 191038 . 183498 . 176516 170042 : 164015 158397 : 153144 . 148226 . 143608 n 12 13 14 15 16 17 18 19 20 From Tables 5.2 and 5.5 we find E1(12.3) = .343953 X lo-*, E,,(12.3)=.143609X lOmeas a check. Example 7. Compute (~~(2) to 6s for n= 1(1)5. The recurrence formula 5.1.15 can be used for all x>O in increasing order of n without loss of From 5.1.25 we have a0(2)=f esa =.0676676, so we get i 1 2 3 4 5 + .02032- .00043- .OOOOl) = 1.91038X lo-‘. Using the recurrence relation 5.1.14, we get RELATED accuracy. lw%Yo) .41570 .38302 .35488 .33041 .30898 .29005 .27325 .25822 .24472 .23253 From Table 5.2 we get zeZEl(z)= .915633 SO that E,(10)=4.15697X10-E as a check. Forward recurrence starting with E1(10)=4.1570X10-e yields the values in column (1). The underlined figures are in error. Example 6. Compute E,(z), n=l(l)N, to 5s for x=12.3, N=20. If N is appreciably larger than z, and x appreciably larger than five, then the recurrence relation 5.1.14 should be used in the backward direction to generate E,(x) for n<nO, and in the forward direction to generate E,(x) for n>nO, where n,=(x). From 5.1.52, with n0=12, x=12.3, we have En,(x) =e;;(l AND as indicated, ffnca .0676676 .101501 .169169 .321421 .710510 1.84394 Independent calculation with 5.1.8 yields the same result for (~~(2). The functions W,(X) and q(z) can be obtained from Table 10.8 using 5.1.48, 5.1.49. Example 8. Compute p,(x), n=O(l)N to 6s for x=1, N=5. Use the recurrence relation 5.1.16 in increasing order of n if x>.368N+.184 In N+.821 and in decreasing order of n otherwise [5.5]. From 5.1.9 with n=5 we get /3,(l)= -.324297 correctly rounded to 6D. Using the recurrence formula 5.1.16 in decreasing order of n and carrying 9D we get the values in column (2). n 0 1 2 3 4 5 w 2.35040 2 -.73575 9269 .87888 3849 -.44950 9722 .55236 3499 -.32434 3774 B# 2.35040 - .73575 .87888 - .44950 .55237 -.32429 2382 888_0 4629 7385 28g 7- Using forward recurrence instead, starting with EXPONENTIAL INTEGRAL AND p,(l)=2 sinh 1=2.350402 and again carrying 9D, we obtain column (1). The underlined figures are in error. The above shows that three significant figures are lost in forward recurrence, whereas about three significant figures are gained in backward recurrence! RELATED 0.lw) 10 n 5 k 0 1 8 7 6 The functions ,&(s) and &(x) can be obtained from Table 10.8 using 5.1.48, 5.1.49. Example +6.8943i. From Table 9. 2 3 . 059898 . 008174 -. 001859 . 000088 Compute E,(z) for z=3.2578 .095598i, e”oE,(zo)=.059898-.107895i. =f(zo+Az> =f(zo> +‘T -. 1078952’ f. 0024352’ +. 000110i -. -. -. 000212i 000003 000004i -.105354i -.022075i -.004716i .711093 -3.784225+-12.7i .278518 .010389 -!- -1.90572+12.7i+2.0900+12.7i - .0184106- .0736698i E,(z)z--1.87133-4.7054Oi. From Taylor’s formula with f(z) =ezEI (z) we have f(z) . 059898 . 003460 -. 000094 Example 10. Compute E,(z) for z=-4.2 + 12.7i. Using the formula at the bottom of Table 5.6 5.6 we have for z,,=z0+iy,=3+7i (zo) = .934958+ -. 10789% +. 012795i +. 000155i Repeating the calculation with zo=3+6i and Az=.2578+.8943,i we get the same result. An alternative ,procedure is to perform bivariate interpolation in t.he real and imaginary parts of ze“El (2). e”E,(z) = z&~El 5.1.27 (AZ) ff’k)(zo)/k! f(z)==.063261 e-‘==.031510 El(z)= --.000332 552373 - .449507 .878885 - .735759 2.350402 Thus with f’k)(&/k! - .324297 a”(l) 4 3 2 1 0 .280560 - .2oE?z .319908 -.253812 .40465 9 235 with Az=z-zo=.2578-.1057i. we get An alternative procedure is to start with an arbitrary value for n sufficiently large (see also 15.11). To illustrate, starting with the value zero at n= 11 we get 1: FUN(!TIONS AZ +jq (A,@+. .. References Texts [5.1] F. J. Corbat6, On the computation of auxiliary functions for two-center integrals by means of a high-speed computer, J. Chem. Phys. 24,452-453 (1956). [5.2] A. Erdelyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1953). [5.3] A. Erdelyi et al., Tables of integral transforms, ~01s. 1, 2 (McGraw-Hill Book Co., Inc., New York, N.Y., 1954). [5.6] W. [5.7] [5.8] [5.9] [5.4] W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. Phys. 38, 77-81 (1959). [5.10] [5.5] W. Gautschi, Recursive computation of certain integrals, J. Assoc. Comput. Mach. 8, 21-40 (1961). [5.11] Grijbner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). C. Hastings, *Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). E. Hopf, Mathematical problems of radiative equilibrium, Cambridge Tracts in Mathematics and Mathematical Physics, No. 31 (Cambridge Univ. Press, Cambridge, England, 1934). V. Kourganofl’, Basic methods in transfer problems (Oxford Un:v. Press, London, England, 1952). F. Lijsch and F. Schoblik, Die Fakultiit und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). N. Nielsen, Theorie des Integrallogarithmus (B. G. Teubner, Leipzig, Germany, 1906). EXPONENTIAL 236 INTEGRAL [5.12] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [5.13] I. M. Ryshik and I. S. Gradstein, Tables of series, products and integrals (VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1957). [ 5.141 S. A. Schelkunoff, Proposed symbols for the modified cosine and exponential integral, Quart. Appl. Math. 2, 90 (1944). (5.151 J. Todd, Evaluation of the exponential integral for large complex arguments, J. Research NBS 52, 313-317 (1954) RP 2508. [ 5.161 F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). Tables [5.17] British Association for the Advancement of Science, Mathematical Tables, vol. I. Circular and hyperbolic functions, exponential, sine and cosine integrals, etc., 3d ed. (Cambridge Univ. Press, Cambridge, England, 1951). Ei(z)-ln z, -El(z) --In z, Ci(z)-In 2, Si(z), z=O(.1)5, 11D; Ei(z), 2=5(.1)15, 10-115; &(z),z=5(.1)15, 13-14D; Si(x), Ci(z), x=5(.1)20(.2)40, IOD. [5.18] L. Fox, Tables of Weber parabolic cylinder functions and other functions for large arguments, Mathematical Tables, vol. 4, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1960). e-ZEi(z), e=&(z), x-1=0(.001).1, 10D; f(z), Q(Z), z-‘=O(.OOl).l, 10D. [5.19] J. W. L. Glaisher, Tables of the numerical values of the sine-integral, cosine-integral and expoTrans. Roy. Sot. nential integral, Philos. London 160, 367-388 (1870). Si(z), Ci(z), Ei(z), 18D, 2=1(.1)5(1)15, 11D. -l&(x), z=O(.Ol)l, [5.20] B. S. Gourary and M. E. Lynam, Tables of the auxiliary molecular integrals A,(s) and the auxiliary functions C,,(z), The Johns Hopkins Univ. Applied Physics Laboratory, CM Report 905, Baltimore, Md. (1957). a,,(z), n!e,,(z), z= .05(.05) 15, n=O(l) 18, 9s. [5.21] F. E. Harris, Tables of the exponential integral Ei(z), Math. Tables Aids Comp. 11, 9-16 (1957). El(z), eZEl(z), Ei(z), e-*Ei(z), z=1(1)4(.4)8(1)50, 18-19s. [5.22] Harvard University, The Annals of the Computation Laboratory, ~01s. 18, 19; Tables of the generalized sine- and cosine-integral functions, parts I, II (Harvard Univ. Press, Cambridge, Mass., =1-cosu 1949). 8&z)= cZ~dz,C(o,z)= c Tax, S’ s 6D; Ss(a, z)= s= s+ sin xdx, &(a, z) = s -zsin u co8 xdx, 6D; Cs(a, 2) = c” 7 sin z&r, s-0 u s Cc(a,x)=Jyy (l-cos 2)&r, 6D;u=&q$ 0 <a<25,0 Ix 125. [5.23] Harvard University, The Annals of the Computation Laboratory, vol. 21; Tables of the generalized exponential-integral functions (Harvard Univ. Press, Cambridge, Mass., 1949). E(a, z) = AND RELATED FUNCTIONS 2 l--e* u a?, Es&, 2) = ,” F dz, Ec(a, z) = s * l-e~cosu clx, 6D; ~=&+a~, O<a<lO, s- 0 s0 0 Ix< 10. 15.241 A. V. Hershey, Computing programs for the complex exponential integral, U.S. Naval Proving Ground, Dahlgren, Va., NPG Report No. 1646 (1959). -E,(-z),. z=x+iy, z=-20(1)20, y=O(1)20, 13s. t5.251E. Jahnke and F. Emde, Tables of functions, 4th ed. (Dover Publications, Inc., New York, N.Y., 1945). --El(x), Ei(s), 2=0(.01)1(.1)5(1)15, 46s; Si(s), Ci(z), 2=0(.01)1(.1)5(1)15(5)100(10) 200(100) 10s(1)7, generally 4-55; maxima and minima of Ci(z) and si(z), O<s<16, 55. [5.26] K. A. Karpov and S. N. Razumovskili, Tablitsy integral’ nogo logarifma (Izdat. Akad. Nauk SSSR., MOSCOW, U.S.S.R., 1956). ii(z), .r= O(.OOOl) 2.5 (001) 20 (.01)200 (1) 500 (1) 10000 (10) 25000, 75; Ii(z)--In /l--2], z=.95(.0001)1.05, 6D. [5.27] M. Kotani, A. Amemiya, E. Ishiguro, T. Kimura, Table of molecular integrals (Maruzen Co., Ltd., Tokyo, Japan, 1955). a(z), x=.25(.25)9(.5)19(1) 25, n=0(1)15, 11s; ,8,,(z), 2=.25(.25)8(.5)19(l) 25, n=0(1)8, 11s. [5.28] M. Mashiko, Tables of generalized exponential-, sine- and cosine-integrals, Numerical Computation Bureau Report No. 7, Tokyo, Japan (1953). E&r)+ln]z(=C&)+ln .&i&(t), z=[ee”, E=0(.05) 5, a=O”(2°)600(10)900, 6D; ze~E,(z) =A.&) exp [i&(q)], z=-! eh, q=.O1(.01).2, ~=0”(2~)60~(1~) 900, 5-6D? [5.29] G. F. Miller, Tables of generalised exponential integrals, Mathematical Tables, vol. 3, National Physical Laboratory (Her Majesty’s Stationery Office, London, England, 1958). (z+n)e=E.(z), z=O(.Ol)l, n=1(1)8, 2=0(.1)20, n=1(1)24, z-1=0(.001).05, n=1(1)24; 8D. [5.30] J. Miller, J. M. Gerhauser, and F. A. Matsen, Quantum chemistry integrals and tables (Univ. of Texas Press, Austin, Tex., 1959). a,,(z), z=.125 (.125)25,n=0(1)16, 14S;&,(z),s=O(.125)24.875, n=0(1)16, 12-14s. [5.31] J. Miller and R. P. Hurst, Simplified calculation of the exponential integral, Math. Tables Aids Comp. 12, 187-193 (1958). e-“l%(z), Ei(z), etE,(x), El(x), z=.2(.05)5(.1)10(.2)20(.5)50(1)80; 165. [5.32] National Bureau of Standards, Tables of sine, cosine and exponential integrals, vol. I (U.S. Government Printing Office, Washington, D.C., 1940). Si(z), Ci(z), Ei(s), El(z), z=O(.OOO1)2, z=O(.l)lO; 9D. [5.33] National Bureau of Standards, Tables of sine, cosine and exponential integrals, vol. II (U.S. Government Printing Office, Washington, D.C., 1940). Si(z), a(z), Ei(z), El(z), z=O(.OO1)10, 9-10 D or S; Si(z), Ci(z), 2=10(.1)40, 1OD; Ei(z), E,(z), z= 10(.1)15, 7-11s. EXPONENTIAL INTEGRAL [5.34] National Bureau of Standards, Table of sine and cosine integrals for arguments from 10 to 100, Applied Math. Series 32 (U.S. Government Printing Office, Washington, D.C., 1954). Si(r), Ci(x), z= lO(.Ol)lOO, 10D. [5.35] National Bureau of Standards, Tables of functions and of zeros of functions, Collected short tables of the Computation Laboratory, Applied Math. Series 37 (U.S. Government Printing Office, Washington, D.C., 1954). E,(z), n=0(1)20, s=O(.O1)2(.1)10, 4-9s; E&)-z In 2, x=0(.01) .5, 75; Ea(z)+~2* In 2, z=O(.Ol).l, 75. [5.36] National Bureau of Standards, Tables of the exponential integral for complex arguments, Applied Math. Series 51 (U.S. Government Printing Office, Washington, D.C., 1958). Ei(z)+ln z, 6D, 2=0(.02)1, y=O(.O2)1, 2=-l(.l)O, y= O(.l)l; E,(z), 6D, x=0(.02)4, y=O(.O2)3(.05)10, 2=0(1)20, y=O(1)20, x=-3.1(.1)0, y=O(.1)3.1, z= -4.5(.5)0, y= 0(.1)4(.5)10, z= - 10(.5) -4.5, y=O(.5)10,2= -2O(l)O, y=O(1)20; efEi(z), 6D, 2=4(.1) 10, y=O(.5) 10. [5.37] S. Oberliinder, Tabellen von Exponentialfunktlonen und-integralen zur Anwendung auf Gebieten der Thermodynamik, Halbleitertheorie und Gaskinetik (Akademie-Verlag. Berlin, Germanv. AND RELATED 237 FUNCTIONS El (ff), 1-$exp($$) T= 1000, 25(25) exp ( --&:I, El (g); T= z exp (-z-l), 150(10)390, AE=.2(.2)2, 3-4s; z-1, El (z-l), e’exp (-t-l)dt, s z-1 exp (x-1) El (z-l), I.--z-l exp (z-i) El (z-1) ; z = .Ol (.OOOl) .l, 5-6s. [5.38] .V. I. Pagurova, Tables of the exponential integral E,(z)= s ;’ e-zuu-*du. Translated from the Rus- sian by D. G. Fry (Pergamon Press, New York, N.Y.; Oxfford, London, England; Paris, France, 1961). E,(z), n=0(1)20, 2=0(.01)2(.1)10, 49s; Es(z) --a In z, 2=0(.01)5, 75; E&r) +&* In z, s=O(.Ol).l, 75; eZEn(z), n=2(1)10, %=10(.1)20, 7D; e*E,(x), r=O(.l)l, z=.01(.01)7(.05)12(.1)20, 7 S or D. [5.39] Tablitsy integral’nogo sinusa i kosinusa (Izdat. Akad. Na.uk SSSR., MOSCOW, U.S.S.R., 1954). Si(z), Ci(;:), ~=0(.0001)2(.001)10(.005)100, 7D; Ci(r) --In :G,~=0(.0001).01, 7D. [5.40] Tablitsy integral’nol pokasatel’nol funktsii (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1954). Ei(s), E1(~),r=0(.0001)1.3(.001)3(.0005)10(.1)15, 7D. [5.41] D. K. Trubey, A table of three exponential integrals, Oak: Ridge National Laboratory Report 2750, Oak. Ridge, Tenn. (June 1959). E,(z), R(z), E&z), ~=0(.0005).1(.001)2(.01)10(.1)20, 6s. 238 EXPONENTIAL Table 5.1 INTEGRAL SINE, COSINE AND RELATED FUNCTIONS AND EXPONENTIAL INTEGRALS 0. 01 0.02 0.03 0.04 r-ISi 1.00000 00000 0.99999 44444 0.99997 77781 0.99995 00014 0.99991 11154 r-z[Ci(s)-ln 2--y] I -0.25000 00000 -0.24999 89583 -0.24999 58333 -0.24999 06250 -0.24998 33339 0.05 0.06 0.07 0. 08 0.09 0.99986 11215 OI99980 06216 0.99972 78178 OI99964 45127 0.99955 01094 -0.24997 -0.24996 -0.24994 -0.24993 -0.24991 39598 25030 89639 33429 56402 1.01264 1.01520 1.01777 1.02036 1.02295 0202 2272 5836 0958 7705 0.98763 0.98519 0.98276 0.98035 0.97794 75971 77714 86889 02898 25142 0.10 0.11 0.12 0.13 0.14 0.99944 0.99932 0.99920 0.99906 0.99891 46111 80218 03455 15870 17512 -0.24989 -0.24987 -0.24985 -0.24982 -0.24979 58564 39923 00480 40244 59223 1.02556 1.02818 1.03081 1.03346 1.03611 6141 6335 8352 2259 8125 0.97554 0.97315 0.97078 0.96841 0.96606 53033 85980 23399 64710 09336 0.15 0.16 0.17 0.18 0.19 0.99875 0.99857 0.99839 0.99820 0.99799 08435 88696 58357 17486 66151 -0.24976 -0.24973 -0.24969 -0.24966 -0.24962 57422 34850 91516 27429 42598 1.03878 1.04146 1.04415 1.04686 1.04957 6018 6006 8158 2544 9234 0.96371 0.96138 0.95905 0.95674 0.95443 56702 06240 57383 09569 62237 0.20 0.21 0.22 0.23 0.24 0.99778 0.99755 0.99731 0.99706 0.99680 04427 32390 50122 57709 55242 -0.24958 -0.24954 -0.24949 -0.24944 -0.24940 37035 10749 63752 96056 07674 1.05230 1.05504 1.05780 1.06057 1.06334 8298 9807 3833 0446 9719 0.95214 0.94985 0.94758 0.94531 0.94306 14833 66804 17603 66684 13506 0.25 0.26 0.27 0.28 0.29 0.99653 0.99625 0.99595 0.99565 0.99533 42813 20519 88464 46750 95489 -0.24934 -0.24929 -0.24924 -0.24918 -0.24912 98618 68902 18540 47546 55938 1.06614 1.06894 1.07176 1.07459 1.07743 1726 6539 4232 4879 8555 0.94081 0.93857 0.93635 0.93413 0.93192 57528 98221 35046 67481 94997 0.30 0.31 0.32 0.33 0.34 0.99501 0.99467 0.99432 0.99396 0.99360 34793 64779 85570 97288 00064 -0.24906 -0.24900 -0.24893 -0.24886 -0.24879 43727 10933 57573 83662 89219 1.08629 1.08316 1.08604 1.08894 1.09185 5334 5293 8507 5053 5008 0.92973 0.92754 0.92536 0.92319 0.92103 17075 33196 42845 45510 40684 0. 35 0.36 0.37 0.38 0.39 0.99321 94028 OI99282 79320 0.99242 56078 0.99201 24449 0.99158 84579 -0.24872 -0.24865 -0.24857 -0.24850 -0.24842 74263 38813 82887 06507 09693 1.09477 1.09771 1.10066 1.10363 1.10660 8451 5458 6108 0481 8656 0.91888 0.91674 0.91460 0.91248 0.91036 27858 06533 76209 36388 86582 0.40 0.41 0.42 0.43 0. 44 0.99115 0.99070 0.99025 0.98978 0.98930 36619 80728 17063 45790 67074 -0.24833 -0.24825 -0.24816 -0.24808 -0.24799 92466 54849 96860 18528 19870 1.10960 1.11260 1.11562 1.11866 1.12170 0714 6735 6800 0991 9391 0.90826 0.90616 0.90407 0.90199 0.89992 26297 55048 72350 77725 70693 0.45 0.46 0.47 0.48 0.49 0.98881 0.98831 0.98780 0.98728 0.98675 81089 88008 88010 81278 67998 -0.24790 -0.24780 -0.24771 -0.24761 -0.24751 00913 61685 02206 22500 22600 1.12477 1.12784 1.13094 1.13404 1.13716 2082 9147 0671 6738 7432 0.89786 0.89581 0.89376 0.89173 0.88970 50778 17511 70423 09048 32920 0.50 0.98621 48361 o.“oo See Examples [c-y1 l-2. crl[Ei(z)-In 1.00000 1.00250 1.00502 1.00755 1.01008 -0.24741 02526 [c-y1 y=0.57721 56649 s--y1 0000 5566 2306 0283 9560 1.14030 2841 C-612 4 [ 1 rl[E:l(z)+ln z+y] 1.00000 00000 0.99750 55452 0.99502 21392 0.99254 97201 0.99008 82265 0.88768 41584 c-y [ 1 EXPONENTIAL SINE, INTEGRAL COSINE AND AND RELATED EXPONENTIAL Si(X) 239 FUNCTIONS INTEGRALS Ci(x) Ei(x) Table 5.1 El(x) 0.51 0.52 0.53 0.54 0.49310 0.50268 0.51225 0.52179 0.53132 74180 77506 15212 84228 al492 -0.17778 -0.16045 -0.14355 -0.12707 -0.11099 40788 32390 37358 07938 04567 0.45421 9905 2167 0633 0445 5931 0.55977 0.54782 0.53621 0.52495 0.51400 3595 2352 9798 1510 3886 0.55 0.56 0.57 0.58 0.59 0.54084 0.55033 0.55981 0.56926 0.57870 03951 48563 12298 92137 85069 -0.09529 -0.07998 -0.06503 -0.05044 -0.03618 95274 55129 65744 14815 95707 0.61529 0. 64667 0.67781 0.70872 0.73940 0657 74190 8642 5720 9764 0.50336 0.49301 0.48296 0.47317 0.46364 4081 9959 0034 3433 9849 0.60 0. 61 0.63 0.64 0.58812 0.59752 0.60691 0.61627 0.62561 88096 98233 12503 27944 41603 -0.02227 -0.00867 +o.O0460 0.01758 0.03026 07070 52486 59849 17424 03686 0.76988 0.80015 0.83022 0.86011 0.88983 1290 0320 6417 8716 5949 0.45437 0.44535 0.43656 0.42799 0.41965 9503 3112 1854 7338 1581 0.65 0.66 0.67 0.68 0.69 0.63493 0.64423 0.65351 0.66277 0.67200 50541 51831 42557 19817 80721 0.04264 0.05475 0.06659 0.07815 0.08946 98293 77343 13594 76659 33195 0.91938 0.94877 0.97801 1.00711 1.03607 6468 a277 9042 6121 6576 0.41151 0.40358 0.39585 0.38830 0.38095 6976 6275 2563 9243 0010 0.70 0.71 0.72 0.73 0.74 0.68122 0.69041 0.69958 0.70873 0.71785 22391 41965 36590 03430 39660 0.10051 0.11131 0.12187 0.13220 0.14229 47070 79525 a9322 32879 64404 1.06490 1.09361 1.12220 1.15068 1.17905 7195 4501 4777 40159 8208 0.37376 0.36675 0.35991 0.35323 0.34671 8843 9981 7914 7364 3279 0.75 0.76 0.77 0.78 0.79 0.72695 0.73603 0.74508 0.75411 Oi76311 42472 09067 36664 22494 63804 0.15216 0.16180 0.17123 0.18045 0.18946 36010 97827 98110 a3335 98290 1.20733 1.23551 1.26360 1.29161 1.31954 28:16 33:19 4960 2805 17!j3 0.34034 0.33411 0.32803 0.32208 0.31627 0813 5321 2346 7610 7004 0.80 0.81 0.82 0.83 0.84 0.77209 0.78105 0.78997 0.79888 0.80776 57855 01921 93293 29277 07191 0.19827 0.20688 0.21530 0.22352 0.23156 a6160 88610 45659 96752 78824 1.34739 1.37518 1.40290 1.43056 1.45816 6548 1783 1245 3978 0.31059 0.30504 0.29961 0.29429 0.28910 6579 2539 1236 9155 2918 0. a5 0.86 0.87 0.88 0.89 0.81661 0.82543 0.83423 0.84300 0.85175 24372 78170 65953 a5102 33016 0.23942 0.24709 0.25459 0.26192 Oi26907 28368 80486 69153 27264 86687 1.48571 1.51321 1.54067 1.56808 1.59546 4176 574'1 2664 8534 7036 0.28401 0.27904 0.27417 0.26941 0.26474 9269 5070 7301 3046 9496 0.90 0.91 0.92 0.93 0.94 0.86047 0.86916 0.87782 0.88645 0.89506 07107 04808 23564 60839 14112 0.27606 0.28289 0.28955 0.29606 0.30241 78305 32065 77018 41358 52458 1.62281 1.65012 1.67741 1.70467 1.73191 1714 6019 3317 6891 994.6 0.26018 0.25571 0.25133 0.24704 0.24285 3939 3758 6425 9501 0627 0.95 0.96 0. 97 0.98 0.99 0.90363 0.91218 0.92070 0.92919 0.93765 80880 58656 44970 37370 33420 0.30861 0.31466 0.32056 0.32631 0.33193 36908 20547 28495 85183 14382 1.75914 1.78635 1.81355 1.84074 1.86793 5612 6947 6941 8519 4543 0.23873 0.23470 0.23075 0.22689 0.22309 7524 7988 9890 1167 9826 1.00 0.94608 30704 c-p4 o.;o 0.62 c1 0.337:03;229 II 6 1 0.48703 0.51953 0.55173 0.58364 19110 1.89511 7816 c-y4 [ 1 0.21938 3934 C-514 5 II 1 240 EXPONENTIAL Table 5.1 SINE, X INTEGRAL COSINE AND AND EXPONENTIAL Ci(x) Si(X) RELATED FUNCTIONS INTEGRALS Ei(X) El(x) 1.00 1.01 1.02 1.03 1.04 0.94608 Oi95448 0.96285 Oi97119 0.97949 30704 26820 19387 06039 a4431 0.33740 0.34273 0.34793 0.35300 0.35793 39229 82254 65405 10067 37091 1.89511 1.92230 1.94948 1.97667 2.00387 7816 1085 7042 8325 7525 0.21938 0.21574 0.21217 0.20867 0.20523 3934 1624 1083 0559 8352 1.05 1.06 1. 07 1.08 1.09 0.98777 0.99602 1.00423 liO1241 1.02056 52233 07135 46846 69091 71617 0.36273 0.36741 0.37196 0.37638 0.38069 66810 19060 13201 68132 02312 2.03108 2.05830 2.08554 2.11280 2.14007 7184 9800 7825 3672 9712 0.20187 0.19857 0.19533 0.19216 0.18904 2813 2347 5403 0479 6118 1.10 1.11 1.12 1.13 1.14 1.02868 1703677 1.04482 1.05284 1.06083 52187 08583 38608 40082 10845 0.38487 0.38893 0.39288 0.39671 0.40043 33774 80142 58645 86134 79090 2.16737 2:19470 2.22205 2.24943 2.27684 8280 1672 2152 1949 3260 0.18599 0.18299 0.18005 0.17716 0.17433 0905 3465 2467 6615 4651 1.15 1.16 1. 17 1.18 1.19 1.06878 1.07670 1.08459 1.09244 1.10026 48757 51696 17561 44270 29760 0.40404 0.40754 0.41093 0.41421 0.41738 53647 25593 10390 23185 78816 2.30428 2.33176 2.35928 2.38684 2.41444 8252 9062 7800 6549 7367 0.17155 0.16882 0.16615 0.16352 0.16094 5354 7535 0040 1748 1567 1.20 1.21 1.22 1.23 1.24 1.10804 1.11579 1.12351 1.13119 1.13883 71990 68937 18599 18994 68160 0.42045 0.42342 0.42629 0.42906 0.43172 91829 76482 46760 16379 98802 2.44209 2.46978 2.49752 2.52531 2.55315 2285 3315 2442 1634 2836 0.15840 0.15592 0.15347 0.15108 0.14872 8437 1324 9226 1164 6188 1.25 1.26 1.27 :'22: . 1.14644 1.15402 1.16155 1.16906 1.17652 64157 05063 88978 14023 78340 0.43430 0.43677 0.43915 0.44144 0.44363 07240 54665 53815 17205 57130 2.58104 2.60899 2.63700 2.66507 2.69320 7974 8956 7673 5997 5785 0.14641 0.14414 0.14191 0.13971 0.13756 3373 1815 0639 8989 6032 1.30 1.31 1.32 1.33 1.34 1.18395 1.19135 1.19870 1.20602 1.21331 80091 17459 88649 91886 25418 0.44573 0.44775 0.44967 0.45151 G.45326 85675 14723 55955 20863 20753 2.72139 2.74965 2.77798 2.80637 2.83484 8880 7110 2287 6214 0677 0.13545 0.13337 0.13133 0.12932 0.12735 0958 2975 1314 5224 3972 1.35 1.36 1.37 1.38 1.39 1.22055 1.22776 1.23493 1.24207 1.24916 87513 76460 90571 28180 87640 (1.45492 0.45650 0.45800 0.45941 0.46075 66752 69811 40711 90071 28349 2.86337 2.89198 2.92067 2.94943 2.97828 7453 8308 4997 9263 2844 0.12541 0.12351 0.12164 0.11980 0.11799 6844 3146 2198 3337 5919 1.40 1.41 1.42 1.43 1.44 1.25622 1.26324 1.27022 1.27717 1.28407 67328 65642 81004 11854 56658 0.46200 0.46318 0.46427 0.46529 0.46624 65851 12730 78995 74513 09014 3.00720 3.03621 3.06530 3.09448 3.12375 7464 4843 6691 4712 0601 0.11621 0.11447 0.11275 0.11106 0.10940 9313 2903 6090 8287 8923 1.45 1.46 1.47 1.48 1.49 1.29094 1.29776 1.30455 1.31130 1.31801 13902 82094 59767 45473 37788 0.46710 0.46790 0.46862 0.46927 0.46984 92094 33219 41732 26848 97667 3.15310 3.18255 3.21209 3.24172 3.27145 6049 2741 2355 6566 7042 0.10777 0.10617 0.10459 0.10304 0.10151 7440 3291 5946 4882 9593 1.50 1.32468 35312 C-j)5 [ 1 0.47035 63172 c-5)2 5 ll 1 3.30128 5449 C--5)1 5 I: 1 0.10001 9582 [(-;I9 1 EXPONENTIAL SINE, COSINE INTEGRAL AND EXPONENTIAL RELATED 241 FUNCTIONS Table INTEGRALS Ci(x) Si(X) l.GO 1.32468 1.51 1.33131 AND Ei(x’l 5.1 El(x) 35312 36664 1.33790 40489 1.34445 45453 1.35096 50245 0.47035 0.47079 0.47116 0.47146 0.47169 63172 32232 13608 15952 47815 3.30128 3.33121 3.36124 3.39137 3.42161 5449 3449 2701 4858 1576 0.10001 9582 0.09854 0.09709 0.09566 0.09426 4365 3466 6424 2786 1.55 1.56 1.57 1.58 1.59 1.35743 1.36386 1.37025 1.37660 1.38291 53577 54183 50823 42275 27345 0.47186 0.47196 0.47200 0.47197 0.47188 17642 33785 04495 37932 42164 3.45195 3.48240 3.51296 3.54363 3.57442 4503 5289 5580 7024 1266 0.09288 0.09152 0.09018 0.08887 0.08758 2108 3960 7917 3566 0504 1.60 1.61 1.62 1.63 1.64 1.38918 1.39540 1.40159 1.40773 1.41384 04859 73666 32640 80678 16698 0.47173 0.47151 0.47124 0.47091 0.47052 25169 94840 58984 25325 01507 3.60531 3.63633 3.66746 3.69871 3.73009 9949 4719 7221 9099 1999 0.08630 0.08505 0.08382 0.08261 0.08142 8334 6670 5133 3354 0970 1.65 1.66 1.67 1.68 1.69 1.41990 1.42592 1.43190 1.43784 1.44373 39644 48482 42202 19816 80361 0.47006 0.46956 0.46899 0.46837 0.46769 95096 13580 64372 54812 92169 3.76158 3.79320 3.82495 3.85682 3.88882 7:569 7456 3.310 0.08024 0.07909 0.07795 0.07683 0.07573 7627 2978 6684 8412 1.70 1.71 1.72 1.73 1.74 1.44959 1.45540 1.46117 1.46690 1.47258 22897 46507 50299 33404 94974 0.46696 0.46618 0.46534 0.46445 0.46351 83642 36359 57385 53716 32286 3.92096 3.95322 3.98562 4.01816 4.05084 3201 9462 9972 6395 0400 0.07465 0.07358 0.07253 0.07150 0.07048 4644 8518 9154 6255 9527 1.75 1.76 1.77 1.78 34189 50249 42379 09830 51872 0.46251 0.46147 0.46038 0.45924 99967 63568 29839 05471 1.79 1.47823 1.48383 1.48939 1.49491 1.50038 0.45804 97097 4.08365 4.11660 4.14970 4.18294 4.21633 3659 7647 4645 5736 2809 0.06948 0.06850 0.06753 0.06657 0.06563 8685 3447 3539 8691 8641 1.80 1.81 1.82 1.83 1.84 1.50581 1.51120 1.51655 1.52185 1.52711 67803 56942 18633 52243 57165 0.45681 0.45552 0.45419 0.45281 0.45139 11294 54585 33436 54262 23427 4.24986 4.28355 4.31738 4.35137 4.38551 7557 1681 6883 4872 7364 0.06471 0.06380 0.06290 0.06202 0.06115 3129 1903 4715 1320 1482 1.85 1.86 1.87 1.88 1.89 1.53233 32813 1.53750 78626 1.54263 94066 0.44992 0.44841 0.44685 0.44526 0.44362 47241 31966 83813 08948 13486 4.41981 4.45427 4.48888 4.52366 4.55860 6080 2746 9097 6872 7817 0.06029 0.05945 0.05862 0.05780 0.05699 4967 1545 0994 3091 7623 1.90 1.55777 53137 1.56273 42192 1.56764 98545 0.44194 0.44021 0.43845 0.43665 0.43481 03497 85005 63991 38088 4.59371 4.62898 4.66442 4.70003 4.73582 3687 62142 7249 8485 1734 0.05620 0.05542 0.05465 0.05389 0.05314 3149 3731 5927 9540 44941 72752 27288 14273 39391 4.77177 4.80791 4.84422 4.88071 4.91738 8785 1438 1501 0791 1131 0.05241 0.05169 0.05097 0.05027 0.04958 4380 0257 6988 4392 2291 4356 0.04890 1.52 1.53 1.54 1.91 1.92 1.93 1.94 1.54772 78621 1.55277 31800 1.57252 1.57735 21801 11591 1.95 1.96 1.97 1.98 1.99 1.58213 67567 1.58687 89407 1.59623 29502 1.60084 47231 0.43293 0.43101 0.42906 0.42707 0.42504 2.00 1.60541 29768 0.42298 1.59157 76810 [C-j)5 1 46388 cc-y1 08288 4.95423 6783 9528 c1 (592 7839 4378 0511 c1 (-!I3 242 EXPONENTIAL Table SINE, 5.1 INTEGRAL AND COSINE AND 1.60541 1.64869 1.68762 1.72220 1.75248 Si(x) 29768 86362 48272 74818 55008 0.42298 0.40051 0.37507 0.34717 0.31729 1.77852 1.80039 1.81821 1.83209 1.84219 01734 44505 20765 65891 01946 0.28587 0.25333 0.22008 0.18648 0.15289 1.84865 1.85165 1.85140 1.84808 1:84191 25280 93077 08970 07828 39833 1.83312 1.82194 1.80862 1.79339 1.77650 FUNCTIONS EXPONENTI.iL INTEGRALS se-"Ei(x) 1.34096 5420 1.37148 6802 1.39742 1992 1.41917 1534 1.43711 8315 sexI 0.72265 0.73079 0.73843 0.74562 0.75240 (x) 7234 1502 1132 2149 4829 11964 66161 48786 83896 53242 1.45162 1.46303 1.47166 1.47780 1.48174 5159 3397 2153 8187 6162 0.75881 0.76488 0.77063 0.77610 0.78130 4592 2722 6987 2123 0252 0.11962 0.08699 0.05525 +0.02467 -0.00451 97860 18312 74117 82846 80779 1.48372 1.48398 1.48274 1.48017 1.47646 9204 9691 0191 4491 8706 0.78625 0.79097 0.79548 0.79979 0.80391 1221 2900 1422 1408 6127 53987 81156 16809 03548 13604 -0.03212 -0.05797 -0.08190 -0.10377 -0.12349 85485 43519 10013 81504 93492 1.47178 1.46625 1.46003 1.45321 1.44590 2389 9659 0313 0902 5765 0.80786 0.81165 0.81529 0.81878 0.82214 7661 7037 4342 8821 8967 1.75820 1.73874 1.71836 1.69731 1.67583 31389 36265 85637 98507 39594 -0.14098 -0.15616 -0.16901 -0.17950 -0.18766 16979 53918 31568 95725 02868 1.43820 1.43020 1.42195 1.41354 1.40501 8032 0557 6813 1719 2424 0.82538 0.82849 0.83149 0.83439 0.83718 2600 6926 8602 3794 8207 1.65414 1.63246 1.61100 1.58997 1.56955 04144 03525 51718 52782 89381 -0.19349 -0.19704 -0.19839 -0.19760 -0.19477 11221 70797 12468 36133 98060 1.39641 1.38780 1.37920 1.37066 1.36219 9030 5263 9093 3313 6054 0.83988 0.84249 0.84501 0.84745 0.84982 7144 5539 7971 8721 1778 1.54993 1.53125 1.51367 1.49731 1.48230 12449 32047 09468 50636 00826 -0.19002 -0.18347 -0.17525 -0.16550 -0.15438 97497 62632 36023 59586 59262 1.35383 1.34558 1.33748 1.32953 1.32175 1278 9212 6755 7845 3788 0.85211 0.85432 0.85648 0.85856 0.86059 0880 9519 0958 8275 4348 1.46872 1.45666 1.44619 1.43735 1.43018 40727 83847 75285 91823 43341 -0.14205 -0.12867 -0.11441 -0.09944 -0.08393 29476 17494 07808 06647 26741 1.31414 1.30671 1.29947 1.29241 1.28555 3566 4107 0536 6395 3849 0.86256 0.86447 0.86633 0.86813 0.86989 1885 3436 1399 8040 5494 1.42468 1.42086 1.41870 1.41817 1.41922 75513 73734 68241 40348 29740 -0.06805 -0.05198 -0.03587 -0.01988 -0.00418 72439 25290 30193 82206 14110 1.27888 1.27240 1.26612 1.26002 1.25411 3860 6357 0373 4184 5417 0.87160 0.87327 0.87489 0.87647 0.87801 5775 0793 2347 2150 1816 1.42179 1.42581 1.43120 1.43786 1.44570 42744 61486 53853 84161 24427 +0.01110 0.02582 0.03985 0.05308 0.06539 15195 31381 54400 07167 23140 1.24839 1.24284 1.23748 1.23228 1.22726 1155 8032 2309 9952 6684 0.87951 0.88097 0.88240 0.88379 0.88515 2881 6797 4955 8662 9176 1.45459 66142 (-;)5 0.07669 52785 1.22240 8053 r( -$"l 0.88648 7675 c1 Ci(x) 08288 19878 45990 56175 16174 RELATED 11 C-i’4 L ( -I [ C-i’6 1 EXPONENTIAL SINE, INTEGRAL COSINE AND AND RELATED EXPONEN’TJAL 243 FUNCTIONS INTEGRAL!? Table .a-rEi(r) :fPlr:, 3.1 1.51068 1.52331 1.53610 1.54893 1.56167 15309 37914 92381 74581 10702 Ci (.r) 0.07669 52785 0.08690 68881 0.09595 70643 0.10378 86664 0.11035 76658 0.11563 32032 0.11959 75293 0.12224 58319 0.12358 59542 0.12363 80071 1.57418 1.58636 1.59809 1.60921 1.61980 68217 66225 85106 75419 65968 0.12243 0.12001 0.11644 0.11176 0.10607 38825 66733 00055 72931 09196 1.18184 1.17849 1.17524 1.17210 1.16906 7987 2509 6676 6376 7617 0.89823 0.89927 0.90029 0.90129 0.90227 7113 7888 7306 6033 4695 1.62959 1.63856 1.64665 1.65379 1.65993 70996 96454 45309 21861 35052 1.66504 1.66908 1.67204 1.67392 1.67412 1.67446 1.67315 1.67084 1.66756 1.66338 1.65834 00758 43056 94480 95283 91725 33423 69801 45697 96169 40566 75942 0.09943 0.09193 0.08367 0.07475 0.06528 0.05534 0.04506 0.03455 0.02391 0.01325 13586 62396 93696 97196 03850 75313 93325 49134 33045 24187 1.16612 1.16327 1.16052 1.15785 1.15526 1.15275 1.15032 1.14797 1.14568 1.14347 6526 9354 2476 2390 5719 9209 9724 4251 9889 3855 0.90323 0.90417 0.90509 0.90600 0.90688 0.90775 0.90861 0.90944 0.91027 0.91107 3900 4228 6235 0459 7415 7602 1483 9530 2177 9850 +O. 00267 -0.00770 -0.01780 -0.02751 80588 70361 40977 91811 -0.03616 39563 1.14132 1.13923 1.13720 1.13524 1.13332 3476 6185 9523 1130 8746 0.91187 0.91265 0.91341 0.91416 0; 91490 2958 1897 7043 8766 7418 Si (.T) 1.45459 66142 1.46443 32441 1.47508 90554 1.48643 64451 1.49834 47533 (2) 1.22240 1.21770 1.21316 1.20877 1.20452 1.20042 1.19645 1.19261 1.18890 1.18531 -0.04545 64330 8053 9472 6264 3699 7026 1500 2401 5063 4881 7334 0.88648 0.88778 0.88905 0.89029 0.89150 0.89268 0.89384 0.89497 0.89608 0.89717 7675 5294 3119 2173 3440 7854 6312 9666 8737 4302 1.13147 0205 (-55)2 [1 0.91563 3339 (-46)4 [1 Table SINE, COSINE AND EXPONENTIAL INTEGRALS FOR 0351 4427 9171 1776 9405 9188 8244 3695 2682 2385 0.94885 0.95323 0.95748 0.96160 0.96557 39 18 44 17 23 .~.v*Ei (.v) 1.13147 021 1.12249 671 1.11389 377 1.10564 739 1.09773 775 0.075 0.070 0.065 0; 060 0.055 0.98191 0.98353 0.98509 0.98660 0.98803 0.98940 0.99070 0.99193 0; 99308 0.99415 0.96938 0.97302 0.97649 0.97976 0.98283 56 98 35 47 17 1.09014 087 1; 08283 054 1.137578 038 1.06896 548 1.06236 365 0.050 0.045 0.040 0.035 0.030 0.99514 0.99604 0. 99685 0.99758 0.99821 0052 3013 8722 4771 a937 0.98568 0.98830 0.99068 0.99282 0.99469 0.025 0.020 0.015 0.010 0.005 0.99875 0.99920 0.99955 0.99980 0.99995 9204 3795 1207 0239 0015 0.99629 0.99761 0.99865 0.99940 0.99985 0.000 24 52 al 12 37 57 a9 60 12 01 00 1.00000 0000 1.00000 c (-5)l 5 c (-Z)4 3 3 Si (,I,)= ; -f (.v)cos .I.-q(.r) sin .I’ 1.05595 591 1.04972 640 1.04366 194 1.03775 135 I.03198 503 1.02635 451 1.02085 228 1.01547 157 1.01020 625 1.00505 077 1.00000 000 (65)5 .c-’ 0.100 0.095 0.090 0.085 0.080 .r”f (I) .J”g(.r) ; =1.57079 63268 See Exat~~ple 3. II 1 LARGE ARGtiMEhTS .NEl 0. 91563 0.91925 0.92293 0.92665 0.93044 0.93427 0.93817 n. 94213 i: 94614 0.95022 (.I.) 33394 68286 15844 90998 09399 87466 424% 924% 56670 55126 0.95437 0.95858 0.96286 0.96722 0.97165 0.97616 0.98075 0.98543 0.99019 0.99504 1.00000 09099 41038 74711 35311 49596 46031 54965 08813 42287 92646 00000 [c-y1 Ci (,c)=J (.Y)sin .~-y [I,) ~0s .L’ <*>=nearest 5.2 integer to ,!,. <s> 10 :: :23 13 14 15 :87 :; 22; 33 40 50 lob07 200 cm 244 EXPONENTIAL Table 5.3 INTEGRAL SINE AND COSINE AND RELATED FUNCTIONS INTEGRALS FOR ARGUMENTS TX Si(sx) 1.63396 48 1.63088 98 1.62211 92 1.60871 21 1.59212 99 1.57408 24 1.55635 75 1.54064 a2 1.52839 53 1.52065 96 Cin (*.r) 3.32742 23 3.36670 50 3.40335 al 3.43582 68 3.46297 82 3.48419 47 3.49941 45 3.50911 a9 3.51426 a9 3.51619 al 1.51803 39 1.52060 20 1.52794 77 1.53921 04 1.55318 17 1.56843 12 1.58344 97 1.59679 62 1.60723 30 1.61383 a5 3.51647 44 3.51674 38 3.51857 25 3.52330 06 3.53192 30 3.54500 55 3.56264 55 3.58447 72 3.60972 10 3.63727 15 1.61608 55 1.61388 08 1.60756 la 1.59785 21 1.58578 13 1.57257 88 1.55954 96 1.54794 a1 1.53885 a4 1.53309 50 3.66581 26 3.69395 05 3.72034 97 3.74385 98 3.76362 13 3.77914 01 3.79032 64 3.79749 22 3.80131 21 3.80274 91 2.80993 76 2.87498 49 2.93491 77 2.98737 63 3.03074 73 3.06427 25 3.08807 51 3.10310 38 3.11100 53 3.11393 95 1.53113 13 1.53306 26 1.53860 67 1.54713 99 1.55776 52 1.56940 54 1.58091 06 1.59117 06 1.59922 11 1.60433 29 3.80295 56 3.80315 a3 3.80453 aa 3.80812 16 3.81467 97 3.82466 68 3.83818 15 3.85496 61 3.87444 05 3.89576 52 1.49216 12 1.49599 24 1.50687 40 1.52343 40 1.54382 74 1.56593 04 1.58755 15 1.60664 04 1.62147 45 1.63080 69 3.11435 65 3.11475 a2 3.11746 60 3.12441 61 3.13699 91 3.15595 79 3.18134 a4 3.21256 74 3.24843 a5 3.28734 92 1.60607 69 1.60435 a5 1.59942 00 1.59180 91 1.58232 00 1.57191 16 1.56161 12 1.55241 46 1.54519 00 1.54059 74 3.91792 84 3.93984 77 3.96047 61 3.97890 22 3.99443 58 4.00666 94 4.01551 22 4.02119 22 4.02422 80 4.02537 29 1.63396 48 (-s3)5 3.32742 23 (-s3)6 Si(Rx) 0.00000 00 0.31244 la 0.61470 01 0.89718 92 1.15147 74 1.37076 22 1.55023 35 1.68729 94 1.78166 12 1.83523 65 Cin(ux) 0.00000 00 0.02457 28 0.09708 67 0.21400 75 0.36970 10 0.55679 77 0.76666 63 0.98995 93 1.21719 42 1.43932 68 1.85193 70 1.83732 28 1.79815 90 1.74191 10 1.67621 68 1.60837 27 1.54487 36 1.49103 51 1.45072 37 1.42621 05 1.64827 75 1.83737 48 2.00168 51 2.13821 22 2.24595 41 2.32581 a2 2.38040 96 2.41370 98 2.43067 75 2.43680 30 6. 0 1.41815 16 1.42569 13 1.44667 38 1.47794 03 1.51568 40 1.55583 lo 1.59441 60 1.62792 16 1.65355 62 1.66945 05 2.43765 34 2.43844 23 2.44365 73 2.45676 95 2.48004 47 2.51446 40 2.55975 53 2.61452 59 2.67647 93 2.74269 41 7. 0 1.67476 la 1.66968 11 1.65535 02 1.63369 a2 1.60721 aa 1.57870 92 1.55099 62 1.52667 49 1.50788 19 1.49612 20 Ci are K )I NT 21” 2:: 55:: E 5: a 5. 9 z-: 6: 3 2: 6: 6 2: 6: 9 ;: 7: 3 ::5" ::7" ::9" 1.53902 91 4.02553 78 c-y (-;)7 [ I [ I Ci(m)=r+ln *+ln x-Cin(rxi) r+ln r=1.72194 55508 maximum values of si(x) if 7~>0is odd, and minimum values if n>O is even. [1 Si(nr) a: 2 ?r ,are odd. We have maximum 10.0 [1 values of ci(.z) if n>O is even, and minimum values if n>Ois si(7z7r)mi-@$[1-n+2+&-. . .] (n+~0) EXPONENTIAL INTEGRAL EXPONENTIAL RELATED INTEGRALS 245 FUNCTIONS En(z) Table 5.4 0.01 0. 02 0.03 0. 04 &(x)-x 1.00000 0.99572 0.99134 0.98686 0.98229 0.05 0.06 0.07 0.08 0.09 0.97762 11 0.97285 08 0.96798 34 0.96301 94 0.95795 93 0.45491 0.44676 0.43883 0.43111 0.42360 88 09 27 97 96 0.30949 0.30498 0.30055 0.29620 0.29193 45 63 85 89 54 0.10503 0.10386 0.10270 0.10155 0.10042 63 24 18 44 00 0.04992 0.04940 0.04888 0.04837 0.04786 60 19 33 02 24 0. 10 0.11 0. 12 0.13 0. 14 0.95280 0.94755 0.94220 0.93676 0.93123 35 26 71 72 36 0.41629 0.40915 0.40219 0.39539 0.38876 15 57 37 77 07 0.28773 0.28360 0.27955 0.27556 0.27164 61 90 24 46 39 0.09929 0.09818 0.09709 0.09600 0. 09,493 84 96 34 95 80 0.04736 0.04686 0.04637 0.04588 0.04540 00 29 10 43 27 0.15 0. 16 0.17 0.18 0.19 0.92560 67 0.91988 70 0.91407 48 0.90817 06 0.90217 50 0.38227 0.37593 0.36974 0.36367 0.35774 61 80 08 95 91 0.26778 0.26399 0.26026 0.25660 0.25299 89 79 96 26 56 0.09387 0.09,283 0.09179 0.09077 0.08975 86 12 56 18 95 0.04492 0.04445 0.04398 0.04352 0.04306 62 47 82 66 98 0.20 0.21 0. 22 0.23 0. 24 0.89608 82 0.88991 09 0.88364 33 0.87728 60 0.87083 93 0.35194 0.34626 0.34070 0.33525 0.32991 53 38 05 18 42 0.24944 72 0.24595 63 0.24252 16 0.23914 19 0.23581 62 0.081375 0.08'776 0.08679 0. 08!582 0.08486 87 93 10 38 75 0.04261 0.04217 0.04172 0.04129 0.04085 79 07 82 03 71 0.25 0.26 0.27 0.28 0.29 0.86430 37 0.85767 97 0.85096 76 0.84416 78 0.83728 08 0.32468 0.31955 0.31453 0.30960 0.30477 41 85 43 86 87 0.23254 32 0.22932 21 0.22615 17 0.22303 11 0.21995 93 0.08392 20 0.08298 72 0.08206 30 0.08U4 92 0.08024 57 0.04042 0.04000 0.03958 0.03916 0.03875 85 43 46 93 84 0.30 0.83030 71 0.82324 69 0.81610 07 0.80886 90 0.80155 21 0.30004 0.29539 0.29083 0.28636 0.28197 18 56 74 52 65 0.21693 0.21395 0.21102 0.20814 0.20529 52 81 70 11 94 0.07935 24 0.07846 93 0.07759 60 0.07673 27 0.07587 90 0.03835 0.03794 0.03755 0.03715 0.03676 18 95 15 76 78 8% p; . 0.79415 0.78666 0.77909 0.77144 0.76370 04 44 43 07 39 0.27766 0.27344 0.26929 0.26521 0.26121 93 16 13 65 55 0.20250 0.19974 0.19703 0.19435 0.19172 13 58 22 97 76 0.07503 0.07420 0.07337 0 072!55 0:071!75 50 06 55 97 31 0.03638 0.03600 0.03562 0.03524 0.03487 22 06 31 95 98 0.40 0.41 0.42 0.43 0.44 0.75588 0.74798 0.73999 0.73193 0.72378 43 23 82 24 54 0.25728 0.25342 0.24963 0.24591 0.24225 64 76 73 41 63 0.18913 0.18658 0.18406 0.18158 0.17914 52 16 64 87 79 0.07095 0.07016 0.06938 0.06061 0. 06785 57 71 75 67 45 0.03451 40 0.03415 21 0.03379 39 0.03343 96 0.03308 89 0.45 0. 46 0.47 0.48 0. 49 0.71555 0.70724 0.69886 0.69039 0.68184 75 91 05 21 43 0.23866 0.23513 0.23166 0.22825 0.22489 25 13 12 08 90 0.17674 33 0.17437 44 0.17204 05 0.16974 10 0.16747 53 0.06710 0.06635 0.06561 0.06489 0.064117 09 58 91 07 04 0.03274 0.03239 0.03205 0.03172 0.03139 20 87 90 29 03 0.50 0.67321 0.16524 0. 06?145 83 0.03106 (-;I7 12 2 0. 00 i-33: 0: 33 0.34 0.35 &X Examples la x 00 22 50 87 39 AND 75 c-y1 1. 4-6. E3(4 E4 (x) &o(x) 1~1oCx) 0.49027 66 0.48096 83 0.47199 77 0.46332 39 0.33333 0.32838 0.32352 0.31876 0.31408 33 24 64 19 55 0.11111 0.10986 0.10863 0.10742 0.10622 11 82 95 46 36 0.05263 0.05207 0.05153 0.05099 0.05045 16 90 21 11 58 0.50000 00 0.22160 44 C-i)5 II 1 28 (-;)I II 1 c(-92 1 c1 246 EXPONENTIAL Table EXPONENTIAL 5.4 Edx) Ed4 o.“so 0.51 0.52 0. 53 0. 54 0.55 0. 56 0.51 0.58 0. 59 0.32664 39 0.31568 63 0.32110 62 0.31038 07 0.30518 62 0.30009 96 0.29511 79 0.29023 0.28545 0.28077 82 78 39 0.60 0.61 0. 62 0. 63 0. 64 0.27618 39 0.26295 35 0. 65 0.66 0. 61 0. 68 0.69 0.25455 0.25048 0.24648 0.24256 0.23872 97 44 74 67 06 0. 70 0. 71 0.23494 71 0.23124 46 0. 72 0. 73 0. 74 0.22761 0.22404 0.22054 0.75 0. 76 0. 77 0. 78 0. 79 0.21711 09 0.21373 88 0.27168 55 0.26727 61 0.25871 54 14 57 61 0.21042 0.20717 0.20398 82 77 60 0.80 0.81 0. 82 0. 83 0. 84 0.20085 0.19777 17 36 0.85 0.86 0. 87 0.88 0.89 0.18599 86 0.18318 33 0.18041 73 0.90 0.91 0.92 0.93 0.94 INTEGRAL 0.19475 04 10 0.18886 41 0.19178 0.17769 0.17502 94 87 0.17240 0.16982 0.16728 41 47 95 0.16479 77 0.16234 82 0.95 0.96 0.97 0.98 0.99 0.15757 0.15524 0.15295 0.15070 0.15994 04 1.00 0.14849 32 59 78 79 cc-y1 55 RELATED INTEGR.4LS 44 57 18 16 0.20897 39 0.20594 0.20297 0.20004 75 15 48 0.19716 64 0.19433 53 0.16524 FUNCTIONS E,(r) EIO(X) E4b3 0.22160 0.21836 0.21518 0.21205 0.19155 0.18881 0.18611 0.18346 0.18085 AND 28 0.16304 30 0.16087 53 0.15873 0.15663 92 41 0.15455 0.15251 96 50 0.15050 00 0.14851 39 0.14655 65 83 42 80 96 89 0.06001 0.05935 0.05869 59 05 25 0.05804 19 0.05489 0.05428 69 89 50 12 77 33 55 0.02652 0.02624 0.02596 0.02569 0.02542 04 25 75 54 62 0.02515 0.02489 0.02463 0.02437 0.02412 98 62 53 72 19 0. 02386 92 0.16606 12 0.12678 08 0.05078 0.12513 0.12350 0.12190 0.12032 19 61 31 24 0.04966 0.04911 0.11876 38 0.15476 0.15261 0.15049 0.14840 0.14634 67 25 17 37 79 0.14432 0.14233 0.14036 0.13843 0.13653 0.11722 70 0.05192 43 0.05134 97 15 0.05021 96 0.04803 0.04750 0.04697 0.04645 0.04594 44 33 81 88 53 0.04543 0.04493 0.04443 0.04394 0.04346 15 70 33 38 07 81 55 24 0.11129 0.10985 00 67 0.10704 0.10567 93 44 0.13465 0.13281 0.13099 0.12920 0.12744 81 22 43 37 0.10431 0.10298 85 12 0.04298 0.04250 0.04203 01 0.09907 80 0.12570 0.12399 0.12230 0.12064 0.11901 30 19 63 59 02 0.04066 0.09656 0.09533 0.09411 0.09291 39 24 77 94 0.11739 0.11581 0.11424 0.11270 88 13 72 63 0.11118 80 0.09173 0.09057 0.08942 0.08828 0.08716 74 13 11 63 69 0.10969 20 0.08606 25 0.03639 [ c-y1 0.10166 22 0.10036 12 0.09781 23 40 47 0.04857 15 0.11571 0.11421 0.11274 0.10844 33 25 0.02855 01 48 18 58 94 03 60 78 49 0.02885 0.02795 0.02766 0.02737 0.02708 0.02680 0.05613 36 0.05551 18 0.17083 0.16842 0.16373 0.16143 0.15917 0.15695 70 0.02915 81 26 0.05368 0.05309 0.05250 0.17328 10 0.02946 0.05676 55 53 01 95 33 58 46 91 08 0.13538 0.13361 0.13187 0.13014 0.12845 0.17576 0.03041 34 0.03009 0.02977 0.02825 0.17829 10 0.13718 13 12 56 86 0.14462 71 53 07 28 0.03106 0.03073 0.05739 06 14 66 56 73 0.14272 0.14085 0.13900 &o(x) 0.06345 0.06275 0.06205 0.06136 0.06068 19 0.02361 91 0.02337 17 0.02288 46 76 56 91 82 28 0.02264 0.02240 49 78 29 82 89 0.04157 49 0.02148 0.02125 0.02103 0.02081 37 87 61 58 0.04111 60 0.02059 78 22 0.02038 0.02016 0.01995 0.01974 21 87 75 86 0.04021 35 0.03976 98 0.03933 11 0.02312 69 0.02217 31 0.02194 08 0.02171 11 0.03889 73 0.01954 18 0.03846 83 0.03762 0.03720 0.03679 46 98 96 0.01933 0.01913 0.01893 0.01873 0.01854 0.03804 41 c(-;I1 1 40 72 47 44 62 01 0.01834 60 [I(-P4 1 EXPONENTIAL INTEGRAL AND RELATED 1.00 1. 01 1.02 1. 03 1.04 82 (.r) 0.14849 55 0.14631 99 0.14418 04 0.14207 63 0.14000 68 EXPONENTIAL &::I(J) 0.10969 20 0.10821 79 0.10676 54 0.10533 42 0.10392 38 INTEGRALS 1.05 1. 06 1. 07 1. 08 1.09 0.13797 0.13596 0.13399 0.13206 0.13015 13 91 96 22 62 0.10253 0.10116 0.09981 0.09848 0.09717 39 43 45 42 31 0.08075 0.07974 0.07873 0.07774 0.07676 1.10 1. 11 1.12 1.13 1. 14 0.12828 0.12643 0.12462 0.12283 0.12107 11 62 10 50 75 0.09588 0.09460 0.09335 0.09211 0.09089 09 74 21 49 53 1.15 1. 16 1.17 1. 18 1.19 0.11934 0.11764 0.11597 0.11432 0.11270 81 62 14 31 08 1.20 1. 21 1.22 1.23 1.24 0.11110 0.10953 0.10798 0.10646 0.10496 1.25 1.26 1.27 1.28 1.29 Table 5.4, 0.01834 0.01815 0.01796 0.01777 0.01758 (.I.) 60 39 39 59 98 90 06 57 42 59 0.03443 0.03405 0.03367 0.03330 0.03294 28 35 85 77 10 0.01740 0.01722 0.01704 0.01686 0.01668 57 35 33 49 84 0.07580 0.07484 0.07390 0.07298 0.07206 07 83 85 12 61 0.03257 0.03221 0.03186 0.03151 0.03116 84 98 52 45 78 0.01651 0.01634 0.01616 0.01600 0.01583 37 09 99 07 33 0.08969 32 0.08850 83 0.08734 02 0.08618 88 0.08505 37 0.07116 0.07027 0.06939 0.06852 0.06766 32 22 30 53 91 0.03082 0.03048 0.03015 0.02981 0.02949 49 58 05 89 10 0.01566 0.01550 0.01534 0.01518 0.01502 76 37 14 09 21 41 25 55 27 37 0.08393 0.08283 0.08174 0.08067 0.07961 47 15 39 17 46 0.06682 0.06599 0.06516 0.06435 0.06355 42 04 75 55 40 0.02916 0.02884 0.02852 0.02821 0.02790 68 61 90 55 54 0.01486 0.01470 0.01455 0.01440 0.01425 49 94 55 32 26 0.10348 0.10203 0.10060 0.09919 0.09781 81 53 51 70 06 0.07857 0.07754 0.07653 0.07553 0.07454 23 47 16 26 76 0.06276 0.06198 0.06121 0.06045 0.05970 31 25 22 19 15 0.02759 88 0.02729 55 0.02699 57 0.02669 91 0.02640 59 0.01410 0.01395 0.01381 0.01366 0.01352 35 59 00 55 26 1.30 1. 31 1.32 1.33 1.34 0.09644 0.09510 0.09377 0.09247 0.09119 55 15 80 47 13 0.07357 0.07261 0.07167 0.07074 0.06982 63 86 42 29 46 0.05896 0.05822 0.05750 0.05679 0.05609 09 99 85 64 36 0.02611 0.02582 0.02554 0.02526 0.02498 59 91 55 51 78 0.01338 0.01324 0.01310 0.01296 0.01283 11 12 27 57 01 1.35 1. 36 1. 37 1. 38 1. 39 0.08992 0.08868 0.08745 0.08624 0.08506 75 29 71 99 10 0.06891 91 0.06802 60 0.06714 53 0.06627 68 0.06542 03 0.05539 0.05471 0.05403 0.05337 0.05271 98 51 93 22 37 0.02471 0.02444 0.02417 0.02390 0.02364 35 23 41 88 65 0.01269 0.01256 0.01243 0.01230 0.01217 59 31 17 17 31 1. 40 1.41 1.42 1.43 1.44 0.08388 0.08273 0.08160 0.08048 0.07937 99 65 04 13 89 0.06457 0.06374 0.06292 0.06211 0.06131 55 24 07 04 11 0.05206 0.05142 0.05078 0.05016 0.04954 37 22 89 37 66 0.02338 0.02313 0.02287 0.02262 0.02237 72 06 70 61 80 0.01204 0.01191 0.01179 0.01167 0.01154 58 98 52 19 99 1.45 1.46 1.47 1. 48 1.49 0.07829 0.07722 0.07616 0.07513 0.07410 30 33 94 13 85 0.06052 0.05974 0.05897 0.05822 0.05747 27 52 82 17 55 0.04893 0.04833 0.04774 0.04715 0.04657 74 61 25 65 80 0.02213 0.02189 0.02165 0.02141 0.02117 27 01 01 28 82 0.01142 0.01130 0.01119 0.01107 0.01095 91 96 14 44 86 1.50 1.51 1.52 1.53 1.54 0.07310 0.07210 0.07112 0.07016 0.06921 08 80 98 60 64 0.05673 0.05601 0.05529 0.05459 0.05389 95 35 73 08 39 0.04600 0.04544 0.04488 0.04433 0.04379 70 32 67 72 48 0.02094 0.02071 0.02048 0.02026 0.02004 61 67 97 53 33 0.01084 0;01073 0.01061 0.01050 0.01039 40 07 85 75 77 1.55 1.56 1. 57 1.58 1.59 0.06828 0.06735 0.06645 0.06555 0.06467 07 87 02 49 26 0.05320 0.05252 0.05185 0.05119 0.05054 64 83 92 92 81 0.04325 0.04273 0.04220 0.04169 0.04118 93 07 87 35 47 0.01982 0.01960 0.01939 0.01917 0.01896 38 67 21 98 98 0.01028 0.01018 0.01007 0.00996 0.00986 90 15 50 97 56 1.60 0.06380 32 C-3615 0.04990 57 C-i)3 0.04068 25 0.01876 22 0.00976 24 C-l)3 [1[1 (A) 25 30 81 76 13 E,(z) 247 ~,~(.r) 0.03639 40 0.03599 29 0.03559 63 0.03520 41 0.03481 63 .I: 0.08606 0.08497 0.08389 0.08283 0.08179 FUNCTIONS JY4 c-y [ (-;I6 I [ I -f320 [1 248 EXPONENTIAL Table INTEGRAL EXPONENTIAL 5.4 l1.iO . 61 0.06380 0.06294 32 64 0.06044 97 0.06210 20 0.06126 98 1. 65 1.66 1. 61 0.05964 13 1. 70 1.71 1. 72 1.73 1.74 0.05884 0.05805 0.05728 0.05652 46 94 54 26 0.05577 0.05502 0.05429 0.05357 0.05286 06 94 88 86 86 1.75 1.76 1.77 1.78 1.79 0.05216 87 0.05147 88 86 0.05012 81 1.80 1.81 1. 82 1. 83 1. 84 0.04881 53 0.04817 27 1. 85 1. 86 1.87 1. 88 0.05079 0.04946 70 0.04753 92 0.04629 87 0.04569 0.04509 0.04450 0.04392 0.04334 15 28 24 03 63 1.90 1.91 1.92 1.93 1.94 1.95 1. 96 1.97 1.98 1. 99 0.04278 0.04222 0.04059 2. 00 0.03753 25 66 70 0.03873 0.04682 0.04622 0.04564 0.04506 0.04449 64 09 84 39 72 82 0.03826 0.03779 0.03734 0.03688 0.03643 52 99 06 70 92 0.04393 0.04338 67 27 0.03599 0.03556 70 04 0.04176 45 0.03428 34 0.03386 0.03345 0.03305 0.03265 0.03225 84 86 39 44 98 0.04283 61 0.04229 67 0.04123 97 51 71 0.03871 0.03823 0.03775 0.03728 0.03681 0.04691 46 57 08 22 00 39 0.03635 40 0.03545 21 0.04072 11 0.03329 0.03288 0.03247 60 55 22 59 67 0.03207 27 43 5.5 0.03013 34 0.03167 46 0.03128 17 0.03089 39 0.03051 12 T-1 (.r+2)@&(“.) (.c+3)eZE&) 1.10937 1.11329 0. 07 0.06 0.05 0.04 0. 03 0.02 0.01 0.00 1.04770 1.03522 1.02325 1.01240 1.01045 1.00861 1.00688 1.00528 1.00384 1.00258 1.00152 1.00071 1.00019 0.02563 31 0.02532 61 0.02502 28 1.10285 1.09185 1.08026 1.06808 1.05536 1.04222 1.02895 1.01617 1.01377 1.01147 1.00927 1.00721 1.00531 1.00361 1.00217 1.00103 1.00027 INTEGRALS E,,(x) (r+4)e”E,(s) 1.10937 1.10071 1.09136 1.08125 1.07031 1.05850 1.04584 1.03247 1.01889 1.01624 1.01366 1.01116 1.00878 1.00654 1.00451 1.00275 1.00133 1.00036 FOR 46 90 51 29 24 35 63 07 67 43 34 41 64 01 54 21 (z+lO)e=E,o(z) 1.07219 1.06926 1.06586 1.06187 1.05712 1.05138 1.04432 1.03550 1.02436 1.02182 1.01917 1.01642 :* %!8 1:00790 1.00516 1.00271 1.00081 <.r>=nearest integer to .I’. 72 63 94 22 0.00799 92 54 34 34 53 90 45 18 10 19 LARGE 03 56 0.00833 0.00825 79 42 24 27 50 63 0.00888 18 0.00878 90 0.00851 64 0.00842 74 0.00816 60 0.00808 07 0.00791 28 0.00783 02 0.00774 0.00766 0.00758 84 74 74 0.00750 81 97 0.00735 21 0.00742 0.00727 53 0.00712 0.00704 0.00697 0.00690 0.00683 42 98 62 33 12 0.00675 0.00668 99 93 0.00648 20 0.00719 93 [1 EXPONENTIAL 1.09750 1.08533 1.07292 1.06034 75 87 40 29 0.00869 0.00860 16 37 (-l)3 0. 50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.09 0.08 23 30 79 70 02 0.02657 0.02625 0.02594 86 46 59 11 Table 0.02823 0.02789 0.02755 0.02722 0.02690 0.00926 0.00916 61 0.01349 0.01334 0.01320 0.01305 0.01291 0.01277 0.01263 0.01249 0.01236 0.01222 0.01209 38 24 04 95 96 07 0.00907 0.00897 0.01425 0.01409 0.01394 0.01379 0.01364 0.02857 59 30 68 38 30 43 79 37 16 0.01590 0.01573 0.01556 0.01539 0.01522 0.01505 0.01489 0.01473 0.01457 0.01441 0.02999 41 0.02963 28 61 0.03371 80 0.00976 0.00966 0.00955 0.00945 0.00936 0.01680 0.01662 0.01644 0.01626 0.01608 04 99 0.02927 0.02892 0.03501 00 0.03457 37 0.04167 18 0.04112 91 c-y 0.03073 0.03035 22 0.01718 0.01699 0.03187 02 0.03148 55 0.03110 56 0.03590 01 0.03414 0.01855 0.01835 0.01815 0.01795 0.01775 0.01756 0.01737 0.03512 93 0.03470 37 93 0.04020 0.03970 0.03920 0.01876 0.03921 36 13 Go (J) J%o(.r) 0.04068 0.04018 0.03969 38 0.04006 0.03954 0.03903 0.03852 0.03802 57 20 67 99 FUNCTIONS E,,(z) E14(T) 0.04990 0.04927 0.04864 0.04802 0.04742 03 22 1. 89 INTEGRALS Id:< (.r) I& (.r) 1. 62 1.63 1. 64 1.68 1. 69 AND RELATED 0.00661 95 0.00655 04 0.00641 43 [(-p1 ARGUMENTS (a+20)e~E~o(.c) 1.04270 1.04179 1.04067 t 00:;:: 1:03543 1.03249 1.02837 1.02222 1.02060 1.01883 1.01688 1.01472 1.01234 :* ~0096;; 1:00401 1.00137 EXPONENTIAL EXPONENTIAL St? INTEGRAL INTEGRAL .f R -19 Y\X 9 -18 AND RELATED FOR COMPLEX zeZEl(2) $3 .Y- -17 249 FUNCTIONS Table 5.6 ARGUMENTS 4 ?z -16 & Jf -15 1.063087 1.062827 1.062061 1.060829 1.059190 0.000001 0.004010 0.007918 0.011633 0.015079 1.067394 1.067073 1.066135 1.064636 1.062657 0.000002 0.004584 0.009032 0.013226 0.017075 1.072345 1.071942 1.070774 1.068925 1.066508 0.000006 0.005296 0.010403 0.015172 0.019486 1.078103 1.077584 1.076102 1.073783 1.070793 0.000014 0.006195 0.012118 0.017579 0.022432 1.057215 1.054981 1.052565 1.050037 1.047458 0.018202 0.020969 0.023364 0.025391 0.027066 1.060297 1.057655 1.054829 1.051905 1.048958 0.020512 0.023505 0.026044 0.028141 0.029824 1.063659 1.060510 1.057187 1.053795 1.050421 0.023272 0.026499 0.029167 0.031306 0.032960 1.067318 1.063538 1.059610 1.055664 1.051797 0.026598 0.030055 0.032823 0.034957 0.036527 1.044880 1.042345 1.039882 1.037515 1.035259 0.028412 0.029461 0.030245 0.030796 0.031148 1.046045 1.043212 1.040490 1.037901 1.035456 0.031130 0.032102 0.032781 0.033211 0.033431 1.047129 1.043967 1.040965 1.038140 1.035501 0.034183 0.035034 0.035567 0.035836 0.035888 1.048081 1.044559 1.041259 1.038192 1.035359 0.037609 0.038282 0.038616 0.038677 0.038520 0.029445 0.029511 0.029296 1.033123 1.031110 1.029222 1.027456 1.025809 0.031330 0.031368 0.031288 0.031110 0.030854 1.033162 1.031019 1.029025 1.027174 1.025459 0.033476 0.033377 0.033162 0.032855 0.032474 1.033049 1.030780 1.028685 1.026756 1.024981 0.035765 0.035502 0.035129 0.034672 0.034150 1.032754 1.030365 1.028180 1.026183 1.024360 0.038193 0.037735 0.037179 0.036552 0.035873 0.029080 1.024275 0.030534 1.023872 0.032037 1.023349 0.033582 1.022695 0.035160 1.084892 1.084200 1.082276 1.079313 1.075560 0.000037 0.007359 0.014306 0.020604 0.026075 1.093027 1.092067 1.089498 1.085635 1.080853 0.000092 0.008913 0.017161 0.024471 0.030637 1.102975 1.101566 1.098025 1.092873 1.086686 0.000232 0.011063 0.020981 0.029507 0.036422 1.115431 1.113230 1.108170 1.101137 1.093013 0.000577 0.014169 0.026241 0.036189 0.043843 1.131470 1.127796 1.120286 1.110462 1.099666 0.001426 0.018879 0.033700 0.045218 0.053451 1.071279 1.066708 1.062046 1.057448 1.053021 0.030642 0.034303 0.037117 0.039174 0.040580 1.075522 1.069960 1.064412 1.059054 1.053997 0.035599 0.039405 0.042169 0.044041 0.045176 1.079985 1.073185 1.066578 1.060352 1.054606 0.041724 0.045552 0.048115 0.049644 0.050359 1.084526 1.076197 1.068350 1.061159 1.054687 0.049336 0.052967 0.055093 0.056057 0.056158 1.088877 1.078701 1.069450 1.061235 1.054046 0.058817 0.061886 0.063225 0.063322 0.062566 1.048834 1.044928 1.041320 1.038010 1.034989 0.041444 0.041867 0.041938 0.041734 0.041321 1.049303 1.044997 1.041080 1.037537 1.034344 0.045719 0.045801 0.045531 0.044999 0.044277 1.049380 1.044674 1.040464 1.036713 1.033378 0.050452 0.050084 0.049384 0.048452 0.047365 1.048933 1.043853 1.039389 1.035473 1.032040 0.055640 0.054695 0.053465 0.052056 0.050547 1.047807 1.042417 1.037766 1.033752 1.030282 0.061249 0.059584 0.057719 0.055758 0.053773 1.032241 1.029747 1.027486 1.025437 1.023580 0.040751 0.040066 0.039301 0.038481 0.037629 1.031474 1.028895 1.026579 1.024499 1.022628 0.043422 0.042477 0.041475 0.040444 0.039401 1.030414 1.027781 1.025438 1.023352 1.021489 0.046180 0.044941 0.043679 0.042417 0.041170 1.029026 1.026377 1.024043 1.021981 1.020155 0.048991 0.047428 0.045883 0.044374 0.042912 1.027274 1.024658 1.022375 1.020375 1.018617 0.051808 0.049894 0.048049 0.046282 0.044599 1.021896 0.036759 1.020942 0.038361 1.019824 0.039950 1.018533 0.041505 1.017066 0.043001 1.152759 1.146232 1.134679 1.120694 1.106249 0.003489 0.026376 0.044579 0.057595 0.065948 1.181848 1.169677 1.151385 1.131255 1.111968 0.008431 0.038841 0.060814 0.074701 0.082156 1.222408 1.199049 1.169639 1.140733 1.115404 0.020053 0.060219 0.085335 0.098259 0.102861 1.278884 1.233798 1.186778 1.146266 1.114273 0.046723 0.097331 0.122162 0.130005 0.128440 1.353831 1.268723 1.196351 1.142853 1.105376 0.105839 0.160826 0.175646 0.170672 0.158134 1.092564 1.080246 1.069494 1.060276 1.052450 0.070592 0.072520 0.072580 0.071425 0.069523 1.094818 1.080188 1.067987 1.057920 1.049645 0.085055 0.084987 0.083120 0.080250 0.076885 1.094475 1.077672 1.064339 1.053778 1.045382 0.102411 0.099188 0.094618 0.089537 0.084405 1.089952 1.071684 1.057935 1.047493 1.039464 0.122397 0.114638 0.106568 0.098840 0.091717 1.079407 1.061236 1.048279 1.038838 1.031806 0.143879 0.130280 0.118116 0.107508 0.098337 1.045832 1.040241 1.035508 1.031490 1.028065 0.067197 0.064664 0.062063 0.059482 0.056975 1.042834 1.037210 1.032539 1.028638 1.025359 0.073340 0.069803 0.066381 0.063128 0.060070 1.038659 1.033231 1.028808 1.025171 1.022152 0.079462 0.074821 0.070524 0.066576 0.062962 1.033205 1.028260 1.024300 1.021090 1.018458 0.085271 0.079488 0.074315 0.069688 0.065542 1.026459 1.022317 1.019052 1.016439 1.014319 0.090413 0.083544 0.077561 0.072320 0.067702 :; 19 1.025132 1.022608 1.020426 1.018530 1.016874 0.054573 0.052291 0.050135 0.048106 0.046201 1.022583 1.020219 1.018192 1.016444 1.014929 0.057215 0.054559 0.052094 0.049806 0.047684 1.019626 1.017494 1.015681 1.014129 1.012790 0.059658 0.056638 0.053874 0.051341 0.049015 1.016277 1.014452 1.012912 1.011600 1.010476 0.061817 0.058460 0.055424 0.052670 0.050161 1.012577 1.011130 1.009915 1.008887 1.008009 0.063610 0.059962 0.056694 0.053752 0.051092 20 1.015422 0.044413 1.013607 0.045714 1.011629 0.046875 1.009505 0.047870 1.007254 0.048675 f 1.059090 1.059305 0.003539 0.000000 : 4 1.057431 1.058456 1.056058 0.010310 0.007000 0.013410 5 6 ; 1.054391 1.052490 1.050413 1.048217 0.016252 0.018806 0.021055 0.022996 9 1.045956 0.024637 10 1.043672 0.025993 :: 1.039177 1.041402 0.027940 0.027086 :i 1.034942 1.037018 0.028581 0.029034 :z 1.032959 1.031076 0.029477 0.029326 :'8 19 1.027620 1.029296 1.026046 20 1.024570 -13 -14 Y\X :: 14 :'b For [21>4, linear interpolation yield about six decimals. See Example39 -10. -10 -6 -7 -8 :i -11 -12 -5 will yield about four decimals, eight-point interpolation will 250 EXPONENTIAL Table EXPONENTIAL 5.6 .?A INTEGRAL P .f 9f -4 Y\X 0 INTEGRAL AND RELATED FOR COMPLEX .zeZEl (2) .% ./ -2 -3 FUNCTIONS ARGUMENTS 9 92 & -1 9 0 1.438208 1.287244 1.185758 1.123282 1.085153 0.230161 0.263705 0.247356 0.217835 0.189003 1.483729 1.251069 1.136171 1.080316 1.051401 0.469232 0.410413 0.328439 0.262814 0.215118 1.340965 1.098808 1.032990 1.013205 1.006122 0.850337 0.561916 0.388428 0.289366 0.228399 0.697175 0.813486 0.896419 0.936283 0.957446 1.155727 0.570697 0.378838 0.280906 0.222612 0.577216 0.621450 0.798042 0.875873 0.916770 0.000000 0.343378 0.289091 0.237665 0.198713 1.061263 1.045719 1.035205 1.027834 1.022501 0.164466 0.144391 0.128073 0.114732 0.103711 1.035185 1.025396 1.019109 1.014861 1.011869 0.180487 0.154746 0.135079 0.119660 0.107294 1.003172 1.001788 1.001077 1.000684 1.000454 0.187857 0.159189 0.137939 0.121599 0.108665 0.969809 0.977582 0.982756 0.986356 0.988955 0.183963 0.156511 0.136042 0.120218 0.107634 0.940714 0.955833 0.965937 0.972994 0.978103 0.169481 0.147129 0.129646 0.115678 0.104303 1.018534 1.015513 1.013163 1.011303 1.009806 0.094502 0.086718 0.080069 0.074333 0.069340 1.009688 1.008052 1.006795 1.005809 1.005022 0.097181 0.088770 0.081673 0.075609 0.070371 1.000312 1.000221 1.000161 1.000119 1.000090 0.098184 0.089525 0.082255 0.076067 0.070738 0.990887 0.992361 0.993508 0.994418 0.995151 0.097396 0.088911 0.081769 0.075676 0.070419 0.981910 0.984819 0.987088 0.988891 0.990345 0.094885 0.086975 0.080245 0.074457 0.069429 :8' 19 1.008585 1.007577 1.006735 1.006025 1.005420 0.064959 0.061086 0.057640 0.054555 0.051779 1.004384 1.003859 1.003423 1.003057 1.002747 0.065803 0.061786 0.058227 0.055052 0.052202 1.000070 1.000055 1.000043 1.000035 1.000028 0.066102 0.062032 0.058432 0.055224 0.052349 0.995751 0.996246 0.996661 0;997011 0.997309 0.065838 0.061812 0.058246 0.055066 0.052214 0.991534 0.992518 0.993342 0.994038 0.994631 0.065024 0.061135 0.057677 0.054583 0.051801 20 1.004902 0.049267 1.002481 0.049631 1.000023 0.049757 0.997565 0.049640 0.995140 0.049284 0.596347 0.777514 0.673321 0.000000 0.147864 0.1865.70 0.722657 0.747012 0.796965 0.000000 0.045686 0.078753 0.096659 0.103403 0.825383 0.831126 0.846097 0.865521 0.885308 0.000000 0.030619 0.055494 0.072180 0.081408 0.852111 0.855544 0.864880 0.877860 0.892143 0.000000 0.021985 0.040999 0.055341 0.064825 : 3 4 2 7 9" :; :: 14 :2 1 Y\X 2 4 3 0.847468 0.891460 0.165207 0.181226 0.844361 0.881036 0.131686 0.132252 0.919826 0.938827 0.952032 0.961512 0.968512 0.148271 0.132986 0.119807 0.108589 0.099045 0.907873 0.927384 0.941722 0.952435 0.960582 0.125136 0.116656 0.107990 0.099830 0.092408 0.903152 0.921.006 0.934958 0.945868 0.954457 0.103577 0.100357 0.095598 0.090303 0.084986 0.903231 0.918527 0.931209 0.941594 0.950072 0.085187 0.085460 0.083666 0.080755 0.077313 0.906058 0.918708 0.929765 0.939221 0.947219 0.070209 0.072544 0.072792 0.071700 0.069799 :: 13 14 0.973810 0.977904 0.981127 0.983706 0.985799 0.090888 0.083871 0.077790 0.072484 0.067822 0.966885 0.971842 0.975799 0.979000 0.981621 0.085758 0.079836 0.074567 0.069873 0.065679 0.961283 0.966766 0.971216 0.974865 0.977888 0.079898 0.075147 0.070769 0.066762 0.063104 0.957007 0.962708 0.967423 0.971351 0.974646 0.073688 0.070080 0.066599 0.063300 0.060206 0.953955 0.959626 0.964412 0.968464 0.971911 0.067447 0.064878 0.062242 0.059630 0.057096 15 0.987519 0.063698 :; 18 19 0.990149 0.988949 0.991167 0.992036 0.056745 0.060029 0.053792 0.051122 0.983791 0.985606 0.987138 0.988442 0.989561 0.061921 0.058539 0.055485 0.052717 0.050199 0.980414 0.982544 0.984353 0.985902 0.987237 0.059767 0.056723 0.053941 0.051394 0.049057 0.977430 0.979799 0.981827 0.983574 0.985089 0.057322 0.054644 0.052162 0.049861 0.047728 0.974858 0.977391 0.979579 0.981478 0.983135 0.054671 0.052371 0.050200 0.048160 0.046245 20 0.992784 0.048699 0.990527 0.047900 0.988395 0.046909 0.986410 0.045749 0.984587 0.044449 0.871606 0.873827 0.880023 0.889029 0.899484 0.000000 0.016570 0.031454 0.043517 0.052380 0.886488 0.888009 0.892327 0.898793 0.906591 0.000000 0.012947 0.024866 0.034995 0.042967 0.898237 0.899327 0.902453 0.907236 0.913167 0.000000 0.010401 0.020140 0.028693 0.035755 0.907758 0.908565 0.910901 0.914531 0.919127 0.000000 0.008543 0.016639 0.023921 0.030145 0.915633 0.916249 0.918040 0.920856 0.924479 0.000000 0.007143 0.013975 0.020230 0.025717 0.910242 0.920534 0.929945 0.938313 0.945629 0.058259 0.061676 0.063220 0.063425 0.062714 0.914952 0.923283 0.931193 0.938469 0.945023 0.048780 0.052667 0.054971 0.056047 0.056211 0.919729 0.926481 0.933096 0.939359 0.945154 0.041242 0.045242 0.047942 0.049570 0.050349 0.924336 0.929836 0.935365 0.940731 0.945812 0.035208 0.039123 0.041986 0.043936 0.045128 0.928664 0.933175 0.937807 0.942398 0.946833 0.030334 0.034063 0.036944 0.039060 0.040514 0.951965 0.957427 0.962128 0.966178 0.969673 0.061408 0.059735 0.057855 0.055877 0.053874 0.950850 0.955987 0.960495 0.964444 0.967903 0.055725 0.054790 0.053560 0.052146 0.050627 0.950427 0.955176 0.959421 0.963201 0.966559 0.050481 0.050135 0.049444 0.048514 0.047425 0.950535 0.954870 0.958814 0.962379 0.965591 0.045711 0.045818 0.045563 0.045038 0.044319 0.951035 0.954959 0.958586 0.961913 0.964949 0.041413 0.041861 0.041948 0.041755 0.041347 :9" 0.972699 0.975326 0.977617 0.979622 0.981384 0.051894 0.049966 0.048109 0.046332 0.044641 0.970935 0.973551 0.975940 0.978009 0.979839 0.049062 0.047489 0.045935 0.044419 0.042951 0.969539 0.972185 0.974538 0.976632 0.978500 0.046236 0.044992 0.043724 0.042456 0.041205 0.968477 0.971067 0.973393 0.975481 0.977357 0.043463 0.042516 0.041512 0.040477 0.039431 0.967710 0.970214 0.972484 0.974540 0.976402 0.040780 0.040095 0.039329 0.038508 0.037653 20 0.982938 0.043036 0.981465 0.041538 0.980169 0.039980 0.979047 0.038388 0.978090 0.036781 : : 2 i 9 10 6 Y\X 0 : 3 4 5 ; i 10 :: 13 14 15 :7" * 0.000000 0.075661 0.118228 5 0.786251 0.797036 0.823055 0.853176 0.880584 0 7 8 10 9 If ~~10 or y>lO then (see [5.15]) eZEl(z)x 0.711093 +------+--------+e,161<3~10-6. 0.278518 0.010389 2+0.415775 z+2.29428 z+6.2900 El(iy)=-Ci(y)+i si(y) (y real) l S8e page Ix. EXPONENTIAL INTEGRAL EXPONENTIAL 9 .f ‘J .P 11 y\x INTEGRAL 12 AND RELATED 251 FUNCTIONS FOR COMPLEX zezE1 (2) 9 Y 13 Table ARGUMENTS x 9 .A4 14 5.6 4 15 0.922260 0.922740 0.924143 0.926370 0.929270 0.000000 0.927914 0.928295 0.929416 0.931205 0.933560 0.000000 0.003972 0.007847 0.011540 0.014974 0.940804 0.941014 0.941636 0.942643 0.943994 0.000000 0.004528 0.008932 0.013098 0.016934 0.937055 0.937308 0.938055 0.939261 0.940870 0.000000 0.005212 0.010258 0.014991 0.019295 0.932796 0.933105 0.934013 0.935473 0.937408 0.000000 0.006063 0.011902 0.017321 0.022171 0.932672 0.936400 0.940297 0.944229 0.948093 0.026361 0.029857 0.032670 0.034847 0.036453 0.936356 0.939462 0.942757 0.946132 0.949500 0.023091 0.026339 0.029036 0.031205 0.032887 0.939729 0.942338 0.945140 0.948047 0.950985 0.020373 0.023378 0.025934 0.028052 0.029756 0.942816 0.945024 0.947419 0.949933 0.952502 0.018095 0.020867 0.023273 0.025315 0.027004 0.945640 0.947522 0.949582 0.951765 0.954018 0.016169 0.018725 0.020980 0.022931 0.024582 10 11 12 13 14 0.951816 0.955347 0.958659 0.961739 0.964583 0.037566 0.038261 0.038612 0.038684 0.038534 0.952792 0.955958 0.958968 0.961800 0.964447 0.034134 0.035004 0.035552 0.035833 0.035893 0.953895 0.956729 0.959454 0.962049 0.964499 0.031081 0.032068 0.032761 0.033201 0.033428 0.955075 0.957610 0.960073 0.962443 0.964702 0.028365 0.029426 0.030221 0.030781 0.031140 0.956296 0.958563 0.960787 0.962947 0.965026 0.025949 0.027052 0.027915 0.028564 0.029024 El 17 18 19 0.967199 0.969597 0.971789 0.973792 0.975621 0.038211 0.037756 0.037200 0.036572 0.035893 0.966907 0.969184 0.971285 0.973220 0.974999 0.035775 0.035515 0.035144 0.034687 0.034166 0.966799 0.968947 0.970946 0.972802 0.974521 0.033479 0.033384 0.033172 0.032865 0.032485 0.966843 0.968860 0.970752 0.031327 0.031370 0.031293 0.972521 0.974172 0.031117 0.030862 0.967011 0.968897 0.970680 0.972359 0.973936 0.029320 0.029476 0.029512 0.029448 0.029301 20 0.977290 0.035179 0.976634 0.033597 0.976112 0.032049 0.975709 0.030542 0.975414 0.029086 0.000000 0.000000 0.002290 0.004549 0.006745 0.008853 0.954371 0.954467 0.954752 0.955219 0.955856 0.000000 0.002527 0.005016 0.007430 0.009735 0.952181 0.952291 0.952619 0.953156 0.953887 0.000000 0.002804 0.005560 0.008223 0.010754 0.949769 0.949897 0.950277 0.950898 0.951741 0 : i y\x 0 17 16 19 18 0.003512 0.006949 0.010242 0.013331 20 0.944130 0.944306 0.944829 0.945678 0.946824 0.000000 0.003128 0.006196 0.009150 0.011940 0.947100 0.947250 0.947693 0.948416 0.949395 0.948226 0.949842 0.951624 0.953527 0.955509 0.014529 0.016886 0.018994 0.020847 0.022445 0.950600 0.951995 0.953545 0.955212 0.956960 0.013121 0.015296 0.017265 0.019019 0.020555 0.952782 0.953995 0.955349 0.956815 0.958363 0.011904 0.013916 0.015753 0.017409 0.018878 0.954793 0.955853 0.957043 0.958337 0.959712 0.010847 0.012709 0.014425 0.015986 0.017387 0.956650 0.957581 0.958631 0.959779 0.961004 0.009922 0.011649 0.013253 0.014723 0.016056 0.957530 0.959559 0.961568 0.963534 0.965443 0.023797 0.024917 0.025823 0.026534 0.027070 0.958758 0.960576 0.962391 0.964181 0.965931 0.021878 0.022998 0.023927 0.024679 0.025271 0.959966 0.961598 0.963238 0.964868 0.966472 0.020163 0.021270 0.022207 0.022984 0.023616 0.961144 0.962612 0.964097 0.965582 0.967052 0.018628 0.019712 0.020645 0.021436 0.022094 0.962288 0.963611 0.964956 0.966310 0.967658 0.017250 0.018305 0.019227 0.020021 0.020694 19 0.967280 0.969038 0.970712 0.972300 0.973800 0.027453 0.027700 0.027831 0.027862 0.027809 0.967628 0.969264 0.970832 0.972328 0.973751 0.025720 0.026041 0.026249 0.526361 0.026388 0.968039 i;9695i8 0.971023 0.972430 01973775 0.024114 0.024493 0.024765 0.024943 0;0i503s 0.968496 0.969906 0.971273 0.972594 0.973863 0.022629 0.023052 0.023375 0.023607 0.023760 0.968990 0.970297 0.971571 0.972808 0.974004 0.021255 0.021712 0.022075 0.022352 0.022552 20 0.975215 0.027685 0.975099 0.026343 0.975057 0.025062 0.975079 0.023842 0.975155 : 4' 2 7 9" :1" :: 14 15 16 :i EXPONENTIAL I .9? -4.0 y\r 2: -0.359552 -0.347179 -0.333373 -0.318556 -0.303109 -0.287369 Y\" -0.057540 -0.078283 -0.096648 -0.112633 -0.126301 -0.137768 0.636779 0.000000 2 0:8 -3.890531 -4.094686 -3.611783 -3.265262 1.260867 1.859922 2.422284 2.937296 0.2 0.4 0.6 0.8 1.0 -0.133374 -0.126168 -0.104687 -0.069328 -0.020743 +0.040177 0.5 0.000000 0.157081 0.312331 0.463961 0.610264 0.749655 -2.895820 -2.867070 COMPLEX .B? .f -0.094868 -0.119927 -0.141221 -0.158890 -0.173169 -0.184355 -0.494576 -0.156411 -0.462493 -0.429554 -0.396730 -0.364785 -0.334280 -0.185573 -0.208800 -0.226575 -0.239500 -0.248231 El(z)+ln -1.0 0.342700 0.679691 1.005410 1.314586 1.602372 -0.811327 0.000000 0.462804 -1.875155 0.917127 1.354712 1.767748 -2.210344 2.149077 -1.418052 0.000000 0.126210 0.251143 0.373547 0.492229 0.606074 -0.928842 0.505485 0.509410 0.521123 0.540441 0.567061 0.600568 4 -0.425168 -0.451225 -0.463193 -0.464163 -0.457088 -0.444528 0.000000 0.103432 0.205962 0.306707 0.404823 0.499516 0 0.000000 0.258840 0.513806 0.761122 0.997200 1.218731 -0.577216 -0.567232 -0.537482 -0.488555 -0.421423 -0.337404 2.0 1.5 1.0 0.219384 0.224661 0.240402 0.266336 0.302022 0.346856 -0.670483 -0.587558 -0.510543 -0.441128 -0.380013 -0.327140 -0.5 -1.147367 -1.133341 -1.091560 -1.022911 -1.895118 -2.781497 -2.641121 -2.449241 -0.257878 -0.289009 -0.310884 -0.324774 -0.332047 -0.334043 .@ -2.0 5.7 z 0.000000 -1.815717 -1.718135 -1.584591 -0.580650 -0.528987 -0.478303 -0.429978 -0.384941 -0.343719 0.022684 Table ARGUMENTS -2.5 -1.5 -4.219228 -4.261087 Y\X 0.0 -0.420509 -0.400596 -0.379278 -0.357202 -0.334923 -0.312894 FOR SMALL .f .?f -2.5 -.- -2.0 FE 1.0 INTEGRAL .~~__ 0.002085 0.004144 0.006151 0.008084 0.742048 0.745014 0.753871 0.768490 0.788664 0.814107 0.000000 0.086359 0.172075 0.256515 0.339075 0.419185 0.000000 0.199556 0.396461 0.588128 0.772095 0.946083 2.5 0.941206 0.943484 0.950289 0.961532 0.977068 0.996699 0.000000 0.073355 0.146246 0.218215 0.288822 0.357653 6. Gamma Function and Related Functions PHILIP J. DAVIS 1 Contents Page Mathematical Properties. ................... . . . . . . 255 258 258 260 260 263 ...................... 263 6.7. Use and Extension of the Tables. . . . . . . . . . . . . 6.8. Summation of Rational Series by Means of Polygamma Functions. . . . . . . . . . . . . . . . . . . . . . . . . 263 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. Gamma Function. . . . . . Beta Function . . . . . . . Psi (Digamma) Function. . . Polygamma Functions. . . . Incomplete Gamma Function. Incomplete Beta Function. . Numerical Methods 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 .......................... 265 Gamma, Digamma and Trigamma Functions (1 Is_< 2) . . 267 References. Table . . . . . . 255 r(x), In I’(z), #(z), +‘(x)), x=1(.005)2, 10D ... 271 6.3. Gamma and Digamma Functions for Integer and HalfInteger Values (1 In< 101) . . . . . . . . . . . . . . . . . . 272 Table 6.2. Tetragamma and Pentagamma Functions (15 ~12) $“(s), $b3’(x), x=1(.01)2, 10D Table r(n), 11s uw, 9s r(n+3), ddn), 1011 n!/[(27r)%P++]e?, In n+(n), 8s 8D 8D n=l(l.>lOl Table 6.4. Logarithms of the Gamma Function (1 In < 101). . . . . . loglo r(n), 8s log,tl r(n+i), loglo r(n+*), 8s klo In r(n) - 274 8s (n- +) In nfn, 8D 8s r(n+$>, n=l(l)lOl 1 National Bureau of Standards. 253 254 GAMMA FUNCTION AND RELATED FUNCTIONS Page 6.5. Auxiliary Functions for Gamma and Digamma Functions (66I:ZIm) ....................... Table 2!/[(2~)%P+k-~], z-l=.015 Table (-.OOl)O, 6.6. Factorials for Large Arguments (lOO<nI 1000) . . . . . 276 6.7. Gamma Function for Complex Arguments . . . . . . . . 277 In I’(z+iy), Table In X+X, In z++(2) 8D n!, ??I=100(100) 1000, Table In r(z)-((2-i) 276 x=1(.1)2, 20s y=O(.l)lO, 12D 6.8. Digamma Function for Complex Arguments . . . . . . . #(r+iy), x=1(.1)2, %‘W+iy), S?$(l+iy)-ln y=O(.l)lO, 288 5D 10D y, y-l=.11 (-.Ol)O, The author acknowledges the assistance the tables; and the assistance of Patricia 8D of Mary 01-r in the preparation Farrant in checking the formulas. and checking of 6. Gamma Function and Related Mathematical 6.2. Gamma (Factorial) Euler’s 6.1.1 r(z)= s0 =k Properties Function Integral - t’-‘e-’ Functions 0% s omt’-‘em”’ Euler’s W’z>O) dt (9?2>0, 9’k>O) Formula 6.1.2 r(z)=lim n!n' . . . (z+n) n+os z&+1) Euler’s Infinite (z#O,-1,-2, . . .) Product -3 l+k+j!j+a+. r= lim m+- =.57721 . . +;-ln 56649. . . Y is known as Euler’s constant and is given to 25 decimal places in chapter 1. I’(Z) is single valued and analytic over the entire complex plane, save for the points z=-n(n=O, 1, 2, . . . ) where it possesses simple poles with residue (- l)“/n!. Its reciprocal i/r (2) is an entire function possessing simple zeros at the points z=-n(n=O, 1, 2, . . .). Hankel’s 6.1.4 3’ i,’ PA ,‘\ ” n /r -5 7)X-j &=&Jo Contour FIGURE 6.1. Gamma junction. -, y=r(z), r(3/2)=#=.88622 6.1.10 Integral 6.1.9 r(,t+)=1’5’9’13 l-(+)=3.62560 The path of integration C starts at + a~ on the real axis, circles the origin in the counterclockwise direction and returns to the starting point. Factorial and Integer 6.1.6 r(n+i)=i.2.3 6.1.11 r(n+#)=1a4e7’10 r(+)=2.67893 II Notations II(Z) =2!= r(2+ 1) 6.1.5 y=l/l-(z) 69254. . . c(3)! k;l’ (4n-3) r(t) (-t)-‘es’& n ‘L” \-’ - - - -, * 6.1.12 r(n+$) 6.1.13 r(n+#)=2’5’8’11 99082. , . ‘3’“’ (3n-2) r(+) 85347. . . =1*3*5*7 * ;; (2n-1) r(3) Values . . . (7+i)n=d ‘3; (3n-1) r(g) 6.1.7 lb ----L-E 2-ln u-4 o= l (-n-l)! Fractional w =2J 0 e-,‘&=&=1.77245 r(#)=i.3541179394.. (n=O, 1, 2) . . .) Values 6.1.14 r(n+f) =3’7’11’15 . h;’ (4n-1) r(j) 38509 . . . =(-$)! r&)=1.22541 -*see page II. 67024 . . . 255 256 GAMMA Recurrence FUNCTION AND Formulas RELATED FUNCTIONS i.1.29 r(i~)r(-iy)=lr(iy)[2=ysi~ny il.30 r(d-+iy)r(t--iy)=lr(~+i21)12=~y il.31 r(l+iy)r(l-iy)=lr(l+iy)l2=~ r(z+1)=zr(z)=z!=z(z-l)! 6.1.15 6.1.16 r(n+z)=(n-l+z)(n-2+2) =(n-l+z)! . . . (l+z)r(l+z) =(n-l+z)(n-2+2) . . . (l+z)z! Reflection 6.1.17 Formula r(2)r(i--2)=-2r(--2)rQ)=t csc TZ Power Series 5.1.33 III r(l+2)=-h(l+z)+z(i--y) Duplication 6.1.18 r(22)=(2?r)-*22z-* r(2) r(2++) Triplication c(m) is the Riemann 23). Formula Series 6.1.19 r(32)=(2d-138*-* Gauss’ 6.1.20 Multiplication Binomial a uz+l) r(w+i)r+-w+i) Pochhammer’s Symbol 6.1.22 (z>c= 1, Gamma 6.1.23 6.1.24 . . . @+?I-l)=W Function Expansion * for (see chapter 1 /r(z) 6.1.34 Formula Coefficient ?V--z 0’ sw!(z--w)! (z).=z(z+l)(z+2) Zeta Function r(z)r(2++)r(2++) r(nz)=(2rr)t"-"'~~'-l~~r(~+~) 6.1.21 ew> +-g2 (-w-(+ll~“/~ Formula in the Complex Plane r(z)=r(2); In r(B)=ln rl.2) arg r(z+i)=arg r(z)+arhnf 6.1.26 6.1.27 mgr b+id =8m +n$o (--&-arctan&-) (z+iy#0,-1,-2, where ~W=rWr(2) 6.1.28 r(i+iy)=iy r(iy) .. .) k '1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 l.oooooc~oooo 0.57721 56649 -0.65587 80715 -0.04200 26350 0.16653 86113 -0.04219 77345 -0.00962 19715 0.00721 89432 -0.00116 51675 -0.00021 52416 0.00012 80502 -0.00002 01348 -0.00000 12504 0.00000 11330 -0.00000 02056 0.00000 00061 0.00000 00050 -0.00000 00011 0.00000 00001 0.00000 00000 -0.00000 00000 0.00000 00000 -0.00000 00000 -0.00000 00000 0.00000 00000 0.00000 00000 000000 015329 I- *-, 202538 \ I 340952' 822915'5 555443Lit 278770 ' 466630 918591 741149 823882 547807 0 934831 1 272320 2, 338417 3 160950 '-f 020075 3 812746 b 0434271 0778234 036968 A 005100 I' 000206k, 000054b 000014 i; 000001 (5 2 The coeffhxents ck are from H. T. Davis, Tables of higher mathematical functions, 2 vols., Principia Press, Bloomington, Ind., 1933, 1935 (with permission); with corrections due to H. E. Salzer. GAMMA Polynomial FUNCTION AND Approximationsa 6.1.35 FUNCTIONS Error r(z+1)=2!=1+a,z+u~Z+u,23+a,~~+a,zbf~(~) for Asymptotic Expansion If ~e(z))115XlW~ R,(z)= 57486 46 .95123 63 aa=-. 69985 88 r (z)-(z-+) In In Z+z--4 al= .42455 49 a5=-. 10106 78 al=-. aa= 6.1.36 Term 257 6.1.42 Olxll . RELATED -2 In (2~) 2n(iL)z an-1 then O<x<l r(x+1)=2!=1+~,z+b,22+ . . . +b,xa+,(x) where (&)(~3XlO-7 bl=-. 57719 1652 bz= .98820 5891 ba=-. 89705 6937 b4= . 91820 6857 Stirling’s bs= -. 75670 4078 b,,= .48219 9394 b,=-. 19352 7818 bs= .03586 8343 K(z) =uppe;,~oundlzz/(3+24) 1 For z real and positive, R, is less in absolute value than the first term neglected and has the same sign. Formula 6.1.43 6.1.37 S?ln r(iy)=Sln 4 r(--iy) ln (%I -+7y-*ln v, cy--++ a> 6.1.44 x1=&rxz+&p(-x+&J Asymptotic (x>O, O<O<l) 3% r(iy)=arg =-An r(-@) r(iy)=-mg q--iy) Formulas 6.1.39 r(az+b) $iiie-ae(uz)‘u+b-~ (lw 4<*, C-0) 6.1.40 6.1.45 r(2) ~(z-+) In Z--Z++ In (274 +& In (y++ (2-m 2m(2mB>)z”-1 -1 ~~21r)-“lr(z+iy)le~~1u11y11-“=1 m in Jarg z/<T) For B, see chapter 23 6.1.41 In r(2) ‘v(2-*) In z-z+& Zt,-a r(z+a)-,+(a-b)(a+b-1) w+b) In (2~)+&&: (z+w in larg zl<~) * Prom C. Hastings, Jr., Approximations computers, Princeton Univ. Press, Princeton, (with permission). for digital N.J., 1955 +&pi”) 22 (3(a+b--1)2-a+b-l);+ ... as Z+W along any curve joining z=O and z= m, providingzf --a, --a-l, . . . ; z# -b, -b-l, .... 258 GAMMA Continued FUNCTION AND RELATED FUNCTIONS Fraction 6.1.48 In r(z)+z-(z-*) In z-+ln a0 =- _-al z+ z+ \ i &:/’ 1 a2 ---a3 z+ z+ (27r) a4 z+ a5 z+ (92 ... > 0) 1 53 695 1 &=---, al=-9 a2=-) a3=-, 12 30 210 371 I/ 22999 10953534lOO9 48264275462 29944523 a4=ii%%’ “=19733142’ Wallis’ “= Formula 4 6.1.49 -4 -6 FIGURE 6.2. Psi junction. y=$+)=dln Some Definite Integrals 6.1.50 In r(z)= o- (z-l) e-Qy-$J S[ =(z-&) In z-z+& In 2s f (g z > 0) 6.3.2 r : nteger Values I n-1 p-l 7, It(n)=--+ lw)= x_. rkr)/dx . ..’ (n22> 2+ Fractional Values (s?~>O) 6.3.3 6.2. Beta Function 9G) =-y-2ln 6.2.1 1 B(z,w) = S S p-1 (l--t)“-’ dt= 0 Psi (Digamma) +(z) =d[ln 6.3.4 at #(n, +$)=--r--2ln2+2 r (2)1/h= ’ ( l+i+... Function r’(2)p 6Some authors write $(z) =-$ In I’(z+ Recurrence 5 functions. Formulas 6.3.5 (2) 1) and similarly +A) (nil) 2) 4 Some authors employ the special double factorial tion as follows: (27~) ! ! =2*4-f? . . . (2n)=2”n i (2n-1) ! ! =1*3*5. . (27&--1)=*-f 2” r(n++) the polygamma 00260 21423 . . . (92~>0,~‘>00) f&Y ,w)2(2)r(wLB(w r(z+w) 6.3. 6.3.1 om(l;;;z+w 4 o (sin t)2E-1 (cos t)2w-1 dt =2 6.2.2 s 2=-1.96351 nota- 6.3.6 Hn+ for + &+&+w+4 GAMMA Reflection FUNCTION AND RELATED FUNCTIONS 259 6.3.19 Formula Td 6.3.7 ~(1--2)=~(2)+7r cot ?Tz Duplication b.3.8 Formula ~@~)=W(z)++h Psi Function in 6.3.9 (zS9) the bm Complex =ln y+&+i+l+. 2 Sin .. 12oy4 252y6 Plane (y+w) =1Lo Extremae of r(z) - Zeros of +(r) 6.3.10 - s?~(iy)=sz~(-iy)=%?‘1L(l+iy)=5z’9(1--iy) - n 6.3.11 Y-$(iy)=+y-‘+; 6.3.12 Y+(i+&) 6.3.13 ?r coth ny > j$(l+iy)=-+y++ =Yn~l(n’+Y2~ Series _ +o.sss -3.545 +2.302 -0.888 $0.245 -XL053 +o. 009 -0.001 $1.462 =+T tanh 7~ rcoth r (x?J 2% __ -0.504 - 1.573 my -2.611 -3.635 -4.653 -5.667 -6.678 -1 Expansions - 6.3.14 #(l+~)=--r+~~(-l)~{(n)~-~ - (lzl<l) . -- x0=1.46163 I’(x,)= .88560 6.3.15 ~(l+z)=+z-‘-+ cot ?fz-(l--22)-1+1--y -~~Ik(2n+~wn 6.3.20 xd=-n+(ln Ma Definite 21449 68362 gi 31944 10889 n>-‘+o[(ln Integrals I+& \3 n>-‘1 - 6.3.21 6.3.16 @f--1,-2,-3,. $(l+~)=-r+~$~& . .> +j-y[;-‘-g&q W’z>O) dt . .m 1 6.3.17 = ~~(l+iy)=l--Y-& =ln S tat z--I[ 0 e-’ -- l222 +gl (-l)“+‘[r(2n+l)--lly2” n$l n-‘(n2Sy2) -l (--~<Y<~) Asymptotic $ 0 (t”+z”)(P-1) (I&E 4< ;) (lYl<2) =-r+y2 &)z 6.3.22 Formulas 6.3.18 #(z> --In 2 -L&-~$~ =ln -&-&+&4-&o+. &” .. (z+ m in larg zI<7r) 6 From W. Sibagaki, Theory and applications of the gamma function, Iwanami Syoten, Tokyo, Japan, 1952 (with permission). 0 260 GAMMA 6.4. Polygamma FUNCTION AND Functions’ RELATED FUNCTIONS Series 6.4.9 6.4.1 0 $‘“‘(l+z)=(-l)“+’ (n=1,2,3, Expansions [n![(n+i) (n+l)! -Tl(n+2)[email protected],*{(n+3)9-... . ...) 1 Tifl<l) 6.4.10 )L(“)(2),(?%=0,1, . . .), is a single valued analytic function over the cnt,irc complex plane save at, the points z=-m(m=0,1,2, . . . ) where it possesses poles of order (n + 1). +‘“‘(z)=(-l)n+Ln!~O (z+k)-“-1 (z#O,-1,-2,. Asymptotic . .) Formulas 6.4.11 0 Integer Values 6.4.2' #‘“‘(l)=(- l)“+‘n!{(n+ (n=l,2,3, 1) . .) 6.4.3 (2 + 02 in 1arg 2 1<T) 6.4.12 $(m)(n+l)=(-lpm! 1 i+ +2m+’ - - * +nm+l 1 . Fractional 6.4.4 ' '-' p”‘(*) = (- l)“+‘n!(2”+‘- #‘(n++) =*nJ-4 Recurrence l)l(n+ 1) (z--,-in 21-&j I arg 2 I<*) 6.4.14 PI (2k- 1) -* (z+- Formuh IL(“)(2+i)=~(n)(zj+(-i)112!2--.-1 6.4.6 in 1arg Values (12=1,2, . . 6.4.5 (z+6.4.13 6.5. Incomplete Gamma in I arg 2 I<T) Function (see also 26.4) Reflection Formula 6.5.1 6.4.7 ~(“‘(l-z)+(-l).+lll’.‘(z)=(-l)n~~ cot *z P(u, s) = & s0 * e-W1 dt 6.5.2 6.4.8 * Multiplication P(mz)=bln m+ & Formula gv) r(a, x) =P(a, (z+&) n=O b=O, n>O . s oze-fte-l dt 6.5.3 (s&z>Ol 0 r (a, T) = r(u) +a, b=l, 7 $’ is known as the trigamma function. $‘I, $(a), $tr) nre the t&a-, penta-, and hesagamma functions respectively. Some authors write $(z) =d[ln I’(z+ l)]/dz, and similnrly for the polygamma functions. *See page II. 2) r(u) = T) = S I e-W1 dt 6.5.4 y*(u, r) =x-=qz, r) = &$ (a, 4 y* is a single valued analytic function z possessing no finite singularities. of a and GAMMA FUNCTION AND RELATED 261 FUNCTIONS a t -3 -4 FIGURE *see page Il. 2 3 6.3. Incomplete gamma function. y*(a += From F. G. Tricomi, I 0 -I -2 * e-V-W r(a) s 0 Sulla funeione gamma incompleta, Annali di Matematica, IV, 33, 1950 (with permission). 4 GAMMA FUNCTION 262 AND RELATED FUNCTIONS 6.5.5 6.5.16 Integral of the y (3, x2)=2 6.5.17 Probability r(+,z2)=2 +Distribution 6.5.18 6.5.6 G S m S oze-t2dt=J;; erfx e-f2dt=J;; erfc x 2 S x7*(+,-x2)= ‘ef2dt 0 (Pearson’s Form of the Incomplete Gamma Function) 6.5.19 164 r(pl+l) e-v P> sum & = =Np+l, S(X,U)=~P-~ 6.5.8 w] [C(z,a)--iiS( dp+l) Recurrence Wu<l) Sta-1 co5t dt z C(x,u)= [El(x)-e-<g r(a,ix)=e*rf” 6.5.20 0 6.5.7 r(--n,x)=T P(a+l, 6.5.21 sin t dt W’a<l) Formulas x>=P(u, x)-& 6.5.22 6.5.23 6.5.9 ~(u+l,x)=uy(u,x)-xsae-z r’(a-l,x)=xr*(u,x)+~ E, (x)=~-~-~~~-“df=x”-Lr(l--n,x) 6.5.10 Derivatives a(dsm ,= and Differential Equations 6.5.24 e-=‘t”dt=x-“-lr(1+7,x) (?j$)mmo 1 ar (w) -=ax 6.5.11 6.5.25 Incomplete Gamma Hypergeometric 6.5.12 Function Function Special 13) ar(a2) ax ------1-=y-le-z f$ [2-ar(a,x)]=(-l)n2-a-“r(a+n,x) (n=O, 1,2, . . .) 1+a,-x) 6.5.27 g [e%.?y* (u,x>]=e”x-r*(u-n, 2) Values 6.5.13 P(n,X)=l- 2 6.5.26 l+a,x) M(a, x=-&(x)--In as a Confluent (see chapter y(u,x)=u-lxae-zM(l, =u-lx= =-lmeT-ln (T&=0,1,2,. ( 1+X+$+. . . +& .> emz 6.5.28 . .> x~+(u+l+x)~+u-y*=O =l-e,-l(x)e-Z For relation to the Poisson distribution, 26.4. 6.5.14 6.5.15 Series see Developments 6.5.29 r*(--n, r (0, x>= x)=x” (2) Sme-fl-ldt=El 2 ~*(u,z)=e+ 2’ n-o r(a+n+l) _ 12 C-2)" r(d n-~ b+dn! (IzI<=> GAMMA FUNCTION AND RELATED 263 FUNCTIONS Definite 6.5.30 r(a, x+Y)-r(a, cm n=o ,e-zxa-l m S Integrals 6.5.36 5) (a-1)(a-2). _____ . . (4[l-e-Ye ?I Xn (y)J e 0 -“‘r(b,ct) ha l- y &] (c%%+c)>O,9b>-1) (lYl<l4) Continued 6.5.37 m trt=!%!.k!i S p-lr(b,t) Fraction 6.5.31 1 1-a ( &1+icp r(u,x)=e-‘.P a 0 1 2-u 2 1+ s+“’ (9 (a+ b) >O, > (x>O,bl< m> Asymptotic 6.6. Incomplete Expansions B,(a,b)= 6.6.1 Beta Bu>O> Function S = ta-‘(l--2)b-‘dt 0 6.5.32 r(a, Z)-p-le-r [ ,+!!$+(a-y-2)+. 6.6.2 . .] For statistical applications, see 26.5. ( 2-m in jarg zI<F Symmetry > !nn(w)! and sign I?,(a,z)=sign 6.5.34 I&b)= 6.6.3 R,(u,z)=u,+,(u,z)+ . . . is t,lie remainder after R terms in this scrics. Then if a,2 arc real, WC have for n>u-2 SUppOW 6 .5 .33 I,(a,b)=B,(u,b)lB(u,b) Relation 6.6.4 _<lu,+,b,z)l Expansion I,(a,n-a+l)=I$ Recurrence r@ 2)--&!?2a+n __n=o (u+n)n! (a-+ ()$(1--p)‘see 26.1. Formulas a) 6.6.5 6.6.6 (a+ b-m)l&,b> =u(l-x)l,(u+l,b-l)+bl,(u,b+l) 6.6.7 0 for a>1 lim !l?LkTEJz 4 for a=1 p ,1--f1 for O<a<l r,(a,b)=~:l,(a--l,b)+(l--s)r,(cl;,b-1) (a+b>Iz(a,b) =alz(a+ Relation (z-03 in 1arg zl<+f) Numerical Use and Extension , to Hypergeometric B,(a,b)=a-‘x”F(u,l-bb; 6.6.8 * l,b)+bIz(a,b+ 1) Function a+l; x) Methods of the Tables Compute r(6.88) to 8s. Using the rccurrcIicc relation 6.1.16 nntl Table 6.1 WC 11nvr, r(6.R8)=[(5.38)(4.:~8)(:~.R8)(2.38)(1..78)]r(1.38) Example to Binomial For binomial distribution, zr,z+,(u,z). 6.5.35 6.7. 1--I,-,(b,a) 1. =232.43671. Example 2. Compute ln r(56.38), using Table 6.4 and linear illtcrpolntion ill.f,. \\-c have In I’(56.38) = (56.38---a) In (56.38) - (56.38) +J2(56.38) The error of linear interpolation in the table of the function.fi is smaller tlian lo-’ in this region. Hence, f,(56.38)=.92041 67 and In r(56.38)= 169.85497 42. Direct interpolation in Table 6.4 of log,, I’(n) eliminates tlir necessity of employing logarithms. However, the error of linear interpolation is .002 so that log,0 r(n) is obtained wit11 a relative error of lo-“. Wee page 11. 264 GAMMA Example FUNCTION AND 3. Compute recurrence relation lt(6.38) to 8s. Using the 6.3.6 and Table 6.1. $(6.38)=1+1+1+1+1+$(1.38) 5.38 4.38 3.38 =1.77275 2.38 3.38 59. 4. Compute #(56.38). Using Table 6.3 we have J/(56.38) =ln 56.38-ja(56.38). The error of linear interpolation in the table of the function f3 is smaller than 8X10-’ in this region. Hence&56.38) = .00889 53 and#(56.38) = 4.023219. Example Example 5. Compute In I’(l-i). reflection principle 6.1.23 and In I’(l--i)=lnI’(l+i)=---.6509+.3016i. Compute In I’(+++$. of the recurrence relation Example the 6.7, FUNCTIONS 6.8. Summation of Rational Series of Polygamma Functions Taking by Means An infinite series whose general term is a rational function of the index may always be reduced to a finite series of psi and polygamma functions. The method will be illustrated by writing the explicit formula when the denominator contains a triple root. Let the general term of an infinite series have the form 2-W U”=&(n)d2(nMn) where dl(n)=(n+arl)(n+az) 6. . . . (n + 4 6.1.15 we r (++*i) =ln r (g+g;)-In (4+&Q -.23419+.03467i =- (+ In *+i arctan 1) =.11239-.75073i The logarithms from 4.1.2. of complex numbers Un)=(n+W(n+W . . . (n+i%Y ~dn)=(n+~d3(n+rd3 the logarithm have, In From Table RELATED . . . (n+-d3 where p(n) is a polynomial of degree m + 2r + 3s - 2 at most and where the constants ai, pi, and yc are distinct. Expand u, in partial fractions as follows are found Example 7. Compute In I’(3+7i) using the duplication formula 6.1.18. Taking the logarithm of 6.1.18, we have -+ In 27r= - .91894 (;+7i) In 2= 1.73287+ 4.85203i In r(++Ji)=-3.31598+ In r(Z+$i) = -2.66047+ 2.32553i 293869i In I’(3+7i)=-5.16252+10.11625i Example 8. the asymptotic Then, we may express 2 u, in terms of the 75-l constants appearing sion as follows in this partial fraction expan- Compute In I’(3+7i) to 5D using formula 6.1.41. We have ln (3+7i)=2.03022 15+1.16590 45i. Then, (2.5$7i) In (3+7i) =-3. 0857779+ 17.1263119i -(3+7i)=-3. ooooooo7. oooooooi + In (2~r)= . 9189385 [12(3+7i)]-‘= .0043103.0100575i -[360(3+7i)3]-1= .0000059.0000022i ---------_-__ ln I’(3+7i)=-5. 16252 +lO. 11625i Higher order repetitions in the denominator are If the denominator contains handled similarly. GAMMA FUNCTION AND RELATED only simple or double roots, omit the corresponding lines. 265 FUNCTIONS Therefore s=16~(1)-16~(1~)+~‘(1)+~‘(1~)=.013499. Example 9. Find Example 11. 1 S=%(%+1)(2n+l)(4n+l) nsl i we have 72 i --i Hence, al=-t a2=--, 6 6 1 1 ( nii-n--2i >* i --i as=-) ad=--’ 12 12 cY1=i, az=---i, (Y3=2i, (yq=-2i, and therefore Example 10. F-y Find s= 2 ’ .)&=I n2p3n+ 1)“’ Since 1 n2(8n+1)2=-x 16 16 1 +m+++;li+m [$(l+i)-P(l-i)l+; [#(142i)--$(l-2i)]. By 6.3.9, this reduces to 1 sf &(l+i)-; 99(1f2i). we have, From Table 6.8, s= .13876. References Tables Texts [6.1] E. Artin, Einfiihrung in die Theorie der Gammafunktion (Leipzig, Germany, 1931). [6.2] P. E. Bohmer, Differenzengleichungen und bestimmte Integrale, chs. 3, 4, 5 (K. F. Koehler, Leipzig, Germany, 1939). [6.3] G. Doetsch, tion, vol. Switzerland, Handbuoh der II, pp. 52-61 1955). Laplace-Transforma(Birkhauser, Basel, [6.4] A. Erdelyi et al., Higher transcendental functions, vol. 1, ch. 1, ch. 2, sec. 5; vol. 2, ch. 9 (McGrawHill Book Co., Inc., New York, N.Y., 1953). [6.5] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). [6.61-F. L6soh and F. Schoblik, Die. Fakultiit und verwandte Funktionen (B. G. Teubner, Leipzig, Germany, 1951). [6.7] W. Sibagaki, Theory and applications of the gamma function (Iwanami Syoten, Tokyo, Japan, 1952). 16.81 E. T. Whittaker and G. N. Watson, A course of modern analysis, ch. 12, 4th ed. (Cambridge Univ. Press, Cambridge, England, 1952). [6.9] A. Abramov, Tables of In r(z) for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1960). In r(z+iy), 2=0(.01)10, y=O(.O1)4, 6D. [6.10] Ballistic Research Laboratory, A table of the factorial numbers and their reciprocals from l! through lOOO! to 20 significant digits. Technical Note No. 351, Aberdeen Proving Ground, Md., 1951. [6.1 I] British Association for the Advancement of Science, Mathematical tables, vol. 1, 3d ed., pp. 40-59 (Cambridge Univ. Press, Cambridge, England, 1951). The gamma and polygamma functions. Also lf = loglo (t)!dt, z=o(.Ol)l, 10D. s0 [6.12] H. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press, Bloomington, Extensive, many place tables Ind., 1933, 1935). of the gamma and polygamma functions up to @o(x) and of their logarithms. [6.13] F. J. Duarte, Nouvelles tables de log10 n! 2133 d&imales depuis n= 1 jusqu’il n=3000 (Kundig, Geneva, Switzerland; Index Generalis, Paris, France, 1927). 266 GAMMA FUNCTION AND RELATED [6.14] National Bureau of Standards, Tables of nl and P(n++) for the first thousand values of n, Applied Math. Series 16 (U.S. Government Printing Office, Washington, D.C., 1951). n!, lGS;P(n+&), 85. [6.15] National Bureau of Standards, Table of Coulomb wave functions, vol. I, pp. 114-135, Applied Math. Series 17 (U.S. Government Printing Office, Washington, D.C., 1952). &‘[F’(l+i~)/F(l +iq],rl=O(.OO5)2 (.01)6 (.02)10(.1) 2O(.2)60(.5)110,10D;a~gF(l+i~),~=0(.01)1(.02) 3 (.05)10(.2)20(.4)30(.5)85, 8D. [6.16] National Bureau of Standards, Table of the gamma function for complex arguments, Applied Math. Series 34 (U.S. Government Printing Office, Washington, D.C., 1954). In F(z+iy), z=9(.1)10, y=O(.l)lO, 12D. Contains an extensive bibliography. (6.171 National Physical Laboratory, Tables of Weber parabolic cylinder functions, pp. 226-233 (Her Majesty’s Stationery Office, London, England, 1955). RealandimaginarypartsoflnF($c+$a),k=0(1)3, a=O(.1)5(.2)20, 8D; (Ir(4+aia)/r(t+~ia)I)-“1 a=0(.02)1(.1)5(.2)20, 8D. FUNCTIONS (6.181 E. S. Pearson, Table of the logarithms of the complete P-function, arguments 2 to 1200, Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922). Loglo P(p), p=2(.1) 5(.2)70(1)1200, 10D. [6.19] J. Peters, Ten-place logarithm tables, vol. I, Appendix, pp. 58-68 (Frederick Ungar Publ. Co., New York, N.Y., 1957). nl, n=1(1)60, exact; (n!)-I, n=1(1)43, 54D; Logto( n=1(1)1200, 18D. [6.20] J. P. Stanley and M. V. Wilkes, Table of the reciprocal of the gamma function for complex argument (Univ. of Toronto Press, Toronto, Canada, 1950). x=-.5(.01).5, y=O(.Ol)l, 6D. [6.21] M. Zyczkowski, Tablice funkcyj eulera i pokrewnych (Panstwowe Wydawnictwo Naukowe, Warsaw, Poland, 1954). Extensive tables of integrals involving gamma and beta functions. For references to tabular material on the incomplete gamma and incomplete beta functions, see the references in chapter 26. GAMMA FUNCTION GAMMA, DIGAMMA 267 AND RELATED FUNCTIONS AND TRIGAMMA Table FUNCTIONS 6.1 ti’ (x) 1.00000 0.99713 0.99432 0.99156 0.98884 00000 85354 58512 12888 42033 In r(x) 0.00000 00000 -0.00286 55666 -0.00569 03079 -0.00847 45187 -0.01121 84893 3 (.r) 1.000 1.005 1.010 1.015 1.020 -0.57721 56649 -0.56902 09113 -0.56088 54579 -0.55280 85156 -0.54478 93105 1.6449340668 1.6329941567 1.6212135283 1.6095891824 1.5981181919 0.000 1.025 1.030 1.035 1.040 1.045 0.98617 0.98354 0.98097 0.97843 0.97594 39633 99506 15606 82009 92919 -0.01392 -0.01658 -0.01921 -0.02179 -0.02434 25067 68539 18101 76511 46490 -0.53682 70828 -0.52892 10873 -0.52107 05921 -0.51327 48789 -0.50553 32428 1.5867976993 1.5756249154 1.5645971163 1.55371 16426 1.5429658968 0.025 0.030 0.035 0.040 0.045 1.050 1.055 1.060 1.065 1.070 0.97350 0.97110 0.96874 0.96642 0.96415 42656 25663 36495 69823 20425 -0.02685 -0.02932 -0.03175 -0.03414 -0.03650 30725 31868 52537 95318 62763 -0.49784 49913 -0.49020 94448 -0.48262 59358 -0.47509 38088 -0.46761 24199 1.5323573421 1.5218835001 1.51154 19500 1.5013303259 1.4912463164 0.050 0.055 0.060 0.065 0.070 1.075 1.080 1.085 1.090 1.095 0.96191 0.95972 0.95757 0.95545 0.95338 83189 53107 25273 94882 57227 -0.03882 -0.04110 -0.04335 -0.04556 -0.04773 57395 81702 38143 29148 57114 -0.46018 11367 -0.45279 93380 -0.44546 64135 -0.43818 17635 -0.43094 47988 1.4812876622 1.4714521556 1.4617376377 1.4521419988 1.4426631755 0.075 0.080 0.085 0.090 0.095 1.100 1.105 1.110 1.115 1.120 0.95135 0.94935 0.94739 0.94547 0.94359 07699 41778 55040 43149 01856 -0.04987 -0.05197 -0.05403 -0.05606 -0.05806 24413 33384 86341 85568 33325 -0.42375 49404 -0.41661 16193 -0.40951 42761 -0.40246 23611 -0.39545 53339 1.4332991508 1.4240479514 1.4149076482 1.4058763535 1.3969522213 0.100 0.105 0.110 0.115 0.120 1.125 1.130 1.135 1.140 1.145 0.94174 0.93993 0.93815 0.93641 0.93471 26997 14497 60356 60657 11562 -0.06002 -0.06194 -0.06383 -0.06569 -0.06751 31841 83322 89946 53867 77212 -0.38849 26633 -0.38157 38268 -0.37469 83110 -0.36786 56106 -0.36107 52291 1.3881334449 1.3794182573 1.3708049288 1.3622917670 1.3538771152 0.125 0.130 0.135 0.140 0.145 1.150 1.155 1.160 1.165 1.170 0.93304 0,93140 0.92980 0.92823 0.92669 09311 50217 30666 47120 96106 -0.06930 -0.07106 -0.07278 -0.07447 -0.07612 62087 10569 24716 06558 58106 -0.35432 66780 -0.34761 94768 -0.34095 31528 -0.33432 72413 -0.32774 12847 1.3455593520 1.3373368900 1.3292081752 1.3211716859 1.3132259322 0.150 0.155 0.160 0.165 0.170 1.175 1.180 1.185 1.190 1.195 0.92519 0.92372 0.92229 0.92088 0.91951 74225 78143 04591 50371 12341 -0.07774 -0.07933 -0.08089 -0.08242 -0.08391 81345 78240 50733 00745 30174 -0.32119 48332 -0.31468 74438 -0.30821 86809 -0.30178 81156 -0.29539 53259 1.3053694548 1.2976008248 1.2899186421 1.2823215358 1.2748081622 0.175 0.180 0.185 0.190 0.195 1.200 1.205 1.210 1.215 1.220 0.91816 0.91685 0.91557 0.91432 0.91310 87424 72606 64930 61500 59475 -0.08537 -0.08680 -0.08820 -0.08956 -0.09090 40900 34780 13651 79331 33619 -0.28903 98966 -0.28272 14187 -0.27643 94897 -0.27019 37135 -0.26398 37000 1.2673772054 1.2600273755 1.2527574090 1.2455660671 1.2384521360 0.200 0.205 0.210 0.215 0.220 1.225 1.230 1.235 1.240 1.245 0.91191 0.91075 0.90962 0.90852 0.90744 56071 48564 34274 10583 74922 -0.09220 -0.09348 -0.09472 -0.09593 -0.09711 78291 15108 45811 72122 95744 -0.25780 90652 -0.25166 94307 -0.24556 44243 -0.23949 36791 -0.23345 68341 1.2314144258 1.2244517702 1.2175630254 1.2107470707 1.2040028063 0.225 0.230 0.235 0.240 0.245 1.250 0.90640 24771 -0.09827 18364 -0.22745 35334 1.1973291545 0.250 f lny! f$ In y! Y r(3’) lny! Y! [C-5616 1 For a>2 see Examples l-4. [i-56)51 log,, e=0.43429 [(-56)71 0.005 0.010 0.015 0.020 [C-55)2 1 44819 Compiled from H. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press,Bloomington,Ind., 1933,1935)(with permission). Known error hasbeencorrected. 208 GAMMA Table 6.1 $50 1.255 FUNCTION GAMMA, r(x) AND DIGAMMA In r(x) RELATED AND FUNCTIONS TRIGAMMA e> FUNCTIONS +’(4 1.260 1.265 1.270 0.90640 0.90538 0.90439 0.90343 0.90250 24771 57663 71178 62946 30645 -0.09827 -0.09939 -0.10048 -0.10154 -0.10258 18364 41651 67254 96809 31932 -0.22745 -0.22148 -0.21554 -0.20964 -0.20376 35334 34266 61686 14193 88437 1.19732 1.19072 1.18418 1.17772 1.17131 91545 50579 94799 14030 98301 0.250 0.255 0.260 0.265 0.270 1.275 1.280 1.285 1.290 1.295 0.90159 0.90071 0.89986 0.89904 0.89824 71994 84765 66769 15863 29947 -0.10358 -0.10456 -0.10550 -0.10642 -0.10731 74224 25269 86634 59872 46519 -0.19792 -0.19211 -0.18634 -0.18059 -0.17487 81118 88983 08828 37494 71870 1.16498 1.15871 1.15250 1.14635 1.14027 37821 22990 44385 92764 59053 0.275 0.280 0.285 0.290 0.295 1.300 1.305 1.310 1.315 1.320 0.89747 0.89672 0.89600 0.89530 0.89464 06963 44895 41767 95644 04630 -0.10817 -0.10900 -0.10981 -0.11058 -0.11133 48095 66107 02045 57384 33587 -0.16919 -0.16353 -0.15790 -Oil5231 -0.14674 08889 45526 78803 05782 23568 1.13425 1.12829 1.12238 1.11654 1.11075 34350 09915 77175 27706 53246 0.300 0.305 0.310 0.315 0.320 1.325 1.330 1.335 1.340 1.345 0.89399 0.89337 0.89278 0.89221 0.89167 66866 80535 43850 55072 12485 -0.11205 -0.11274 -0.11341 -0.11404 -0.11465 32100 54356 01772 75756 77697 -0.14120 -0.13569 -0.13020 -0.12475 -0.11932 29305 20180 93416 46279 76069 1.10502 1.09934 1.09372 1.08816 1.08265 45678 97037 99497 45379 27136 0.325 :*;:: 1:360 1.365 1.370 0.89115 0.89065 0.89018 0.88973 0.88931 14420 59235 45324 71116 35074 -0.11524 -0.11579 -0.11632 -0.11682 -0.11730 08974 70951 64980 92401 54539 -0.11392 -0.10855 -0.10321 -0.09789 -0.09259 80127 55827 00582 11840 87082 1.07719 1.07178 1.06643 1.06112 1.05587 37361 68773 14226 66696 19286 0.350 0.355 1.375 1.380 1.385 1.390 1.395 0.88891 0.88853 0.88818 0.88785 0.88754 35692 71494 41041 42918 75748 -0.11775 52707 -0.11817 88209 -0.11857 62331 -0.11894 '76353 -0.11929 31538 -0.08733 -0.08209 -0.07687 -0.07168 -0.06652 23825 19619 72046 78723 37297 1.05066 1.04550 1.04040 1.03533 1.03032 65216 97829 10578 97036 50881 0.375 0.380 0.385 1.400 1.405 1.410 1.415 1.420 0.88726 0.88700 0.88676 0.88654 0.88635 38175 28884 46576 89993 57896 -0.11961 -0.11990 -0.12017 -0.12041 -0.12063 29142 70405 56559 88823 68406 -0.06138 45446 -0.05627 00879 -0.05118 01337 -0.04611 44589 -0.04107 28433 1.02535 1.02043 1.01555 1.01072 1.00593 65905 36002 55173 17518 17241 0.400 0.405 0.410 0.415 0.420 1.425 1.430 1.435 1.440 1.445 0.88618 0.88603 0.88590 0188580 0.88572 49081 62361 96587 50635 23397 -0.12082 -0.12099 -0.12114 -0.12125 -0.12135 96505 74307 02987 83713 17638 -0.03605 -0.03106 -0.02609 -0.02114 -0.01621 50697 09237 01935 26703 81479 1.00118 0.99648 0.99181 0.98719 0.98261 48640 06113 84147 77326 80318 0.425 0.430 0.435 0.440 0.445 1.450 1.455 1.460 1.465 1.470 0.88566 0.88562 0.88560 0.88560 0.88563 13803 20800 43364 80495 31217 -0.12142 -0.12146 -0.12148 -0.12148 -0.12145 05907 49657 50010 08083 24980 -0.01131 -0.00643 -0.00158 +0.00325 0.00806 64226 72934 05620 39677 64890 0.97807 0.97357 0.96911 0.96469 0.96031 87886 94874 96215 86921 62091 0.450 0.455 0.460 0.465 0.470 1.475 1.480 1.485 1.490 1.495 0.88567 0.88574 0.88583 0.88594 0.88607 94575 69646 55520 51316 56174 -0.12140 -0.12132 -0.12122 -0.12110 -0.12095 01797 39621 39528 02585 29852 0.01285 0.01762 0.02237 0.02710 0.03180 71930 62684 39013 02758 55736 0.95597 0.95166 0.94739 0.94316 0.93896 16896 46592 46509 12052 38700 0.475 0.480 0.485 0.490 0.495 1.500 0.88622 69255 -0.12078 22376 0.93480 22005 0.500 2/! [C-56)4I *See page II. lng! 0.03648 99740 * dy lny! d [c-y 1 (-i)4 [ I log,, e=O.43429 44819 * $$ lny! ” (-;)9 [1 "0% 0:340 0.345 Ez 0:370 K%5" . !/ GAMMA FUNCTION GAMMA, t DIGAMMA r(z) AND AND RELATED TRIGAMMA FUNCTIONS 269 FUNCTIONS +w In r(r) Table 6.1 ti’ (z) 1.500 1.505 1.510 1.515 1.520 0.88622 0.88639 0.88659 0.88680 0.88703 69255 89744 16850 49797 87833 -0.12078 -0.12058 -0.12037 -0i12Oi3 -0.11986 22376 81200 07353 01860 65735 0.03648 0.04115 0.04579 0.05041 0.05502 99740 36543 67896 95527 21146 0.93480 0.93067 0.92658 0.92252 0.91850 22005 57588 41142 68425 35265 0.500 0.505 0.510 0.515 0.520 1.525 1.530 1.535 1.540 1.545 0.88729 0.88756 0.88786 0.88817 0.88851 30231 76278 25287 76586 29527 -0.11957 -0.11927 -0.11893 -0.11858 -0.11820 99983 05601 83580 34900 60534 0.05960 0.06416 0.06871 0.07323 0.07773 46439 73074 02697 36936 77400 0.91451 0.91055 0.90663 0.90274 0.89888 37552 71245 32361 16984 21253 0.525 0.530 0.535 0.540 0.545 1.550 1.555 1.560 1.565 1.570 0.88886 0.88924 0.88963 0.89005 0.89048 83478 37830 91990 45387 97463 -0.11780 -0.11738 -0.11693 -0.11647 -0.11598 61446 38595 92928 25388 36908 0.08222 0.08668 0.09113 0.09556 0.09997 25675 83334 51925 32984 28024 0.89505 0.89125 0.88749 0.88375 0.88005 41371 73596 14249 59699 06378 0.550 0.555 0.560 0.565 0.570 1.575 1.580 1.585 1.590 1.595 0.89094 0.89141 0.89191 0.89242 0.89296 47686 95537 40515 82141 19949 -0.11547 -0.11494 -0.11438 -0.11380 -0.11321 28415 00828 55058 92009 12579 0.10436 0.10873 0.11309 0.11742 0.12174 38544 66023 11923 77690 64754 0.87637 0.87272 0.86911 0.86552 0.86196 50766 89402 18871 35815 36921 0.575 0.580 0.585 0.590 0.595 1.600 1.605 1.610 1.615 1.620 0.89351 0.89408 0.89468 0.89529 0.89592 53493 82342 06085 24327 36685 -0.11259 -0.11195 -0.11128 -0.11060 -0.10990 17657 08127 84864 48737 00610 0.12604 0.13033 0.13459 0.13884 0.14307 74528 08407 67772 53988 68404 0.85843 0.85492 0.85145 0.84800 0.84457 18931 78630 12856 18488 92455 0.600 0.605 0.610 0.615 0.620 1.625 1.630 1.635 1.640 1.645 0.89657 0.89724 0.89793 0.89864 0.89936 42800 42326 34930 20302 98138 -0.10917 -0.10842 -0.10765 -0.10687 -0.10606 41338 71769 92746 05105 09676 0.14729 0.15148 0.15566 0.15983 0.16398 12354 87158 94120 34529 09660 0.84118 0.83781 0.83446 0.83115 0.82785 31730 33330 94315 11790 82897 0.625 0.630 0.635 0.640 0.645 1.650 1.655 1.660 1.665 1.670 0.90011 0.90088 0.90166 0.90247 0.90329 68163 30104 83712 28748 64995 -0.10523 -0.10437 -0.10350 -0.10261 -0.10170 07282 98739 84860 66447 44301 0.16811 0.17222 Oil7632 0.18040 0.18447 20776 69122 55933 82427 49813 0.82459 0.82134 Oi818i2 0.81493 0.81176 04826 74802 90092 48001 45875 0.650 0.655 0.660 0.665 0.670 1.675 1.680 1.685 1.690 1.695 0.90413 0.90500 0.90588 0.90678 0.90770 92243 10302 18996 18160 07650 -0.10077 -0.09981 -0.09884 -0.09785 -0.09684 19212 91969 63351 34135 05088 0.18852 0.19256 0.19658 0.20058 0.20457 59282 12015 09180 51931 41410 0.80861 0.80549 0.80239 0.79931 0.79626 81094 51079 53282 85198 44350 0.675 0.680 0.685 0.690 0.695 1.700 1.705 1.710 1.715 1.720 0.90863 0.90959 0.91057 0.91156 0.91258 87329 57079 16796 66390 05779 -0.09580 -0.09475 -0.09368 -0iO9259 -0.09147 76974 50552 26573 05785 88929 0.20854 0.21250 0.21645 0.22037 0.22429 78749 65064 01462 89037 28871 0.79323 0.79022 0.78723 0:78427 0.78132 28302 34645 61012 05060 64486 0.700 0.705 0.710 0.715 0.720 1.725 1.730 1.735 1.740 1.745 0.91361 0.91466 0.91573 0.91682 0.91793 34904 53712 62171 60252 47950 -0.09034 -0.08919 -0.08802 -0.08683 -0.08562 76741 69951 69286 75466 89203 0.22819 0.23207 0.23594 0.23980 0.24364 22037 69593 72589 32061 49038 0.77840 0.77550 0.77262 0.76976 0.76692 37011 20396 12424 10915 13714 0.725 0.730 0.735 0.740 0.745 1.750 0.91906 25268 -0.08440 11210 0.24747 24535 0.76410 18699 0.750 Y! [1 (El)3 lny! f [1 (-46)3 log,, e=0.43429 lny! [1 (-f)3 44819 d$2 lny! [1 (-56)4 Y 270 GAMMA Table 6.1 GAMMA, 2 1.750 1.755 1.760 1.765 1.770 0.91906 0.92020 0.92137 0192255 0.92376 1.775 1.780 1.785 1.790 1.795 r(x) FUNCTION AND DIGAMMA AND RELATED FUNCTIONS TRIGAMMA In r(r) FUNCTIONS +’(2) d+) 25268 92224 48846 95178 31277 -0.08440 -0.08315 -0.08188 -0.08060 -0.07929 11210 42192 82847 33871 95955 0.24747 0.25128 0.25508 0.25887 0.26264 24535 59559 55103 12154 31686 0.76410 0.76130 0.75852 0.75576 0.75302 18699 23773 26870 25950 19003 0.750 0.755 0.760 0.765 0.770 0.92498 0.92622 O-92748 0:92876 0.93006 57211 73062 78926 74904 61123 -0.07797 -0.07663 -0.07527 -0.07389 -0.07249 69782 56034 55386 68509 96070 0.26640 0.27014 0.27387 0.27759 0.28129 14664 62043 74769 53776 99992 0.75030 0.74759 0.74491 0.74224 0.73960 04040 79107 42268 91617 25271 0.775 0.780 0.785 0.790 0.795 1.800 1.805 1.810 1.815 1.820 0.93138 0.93272 0.93407 0.93545 0.93684 37710 04811 62585 11198 50832 -0.07108 -0.06964 -0.06819 -0.06672 -0.06523 38729 97145 71969 63850 73431 0.28499 0.28866 0.29233 0.29598 0.29962 14333 97707 51012 75138 70966 0.73697 0.73436 0.73177 0.72919 0.72663 41375 38093 13620 66166 93972 0.800 0.805 0.810 0.815 0.820 1.825 1.830 1.835 1.840 1.845 0.93825 0.93969 0.94114 0.94261 0.94410 81682 03951 17859 23634 21519 -0.06373 -0.06220 -0.06066 -0.05910 -0.05752 01353 48248 14750 01483 09071 0.30325 0.30686 0.31046 0.31405 0.31763 39367 81205 97335 88602 55846 0.72409 0.72157 0.71907 0.71658 0.71411 95297 68426 11662 23333 01788 0.825 0.830 0.835 0.840 0.845 1.850 1.855 1.860 1.865 1.870 0.94561 0.94713 0.94868 0.95025 0.95184 11764 94637 70417 39389 01855 -0.05592 -0.05430 -0.05267 -0.05102 -0.04935 38130 89276 63117 60260 81307 0.32119 0.32475 0.32829 0.33182 0.33533 99895 21572 21691 01056 60467 0.71165 0.70921 0.70679 0.70438 0.70199 45396 52546 21650 51138 39461 0.850 0.855 0.860 0.865 0.870 1.875 1.880 1.885 1.890 1.895 0.95344 0.95507 0.95671 0.95837 0.96006 58127 08530 53398 93077 27927 -0.04767 -0.04596 -0.04424 -0.04251 -0.04075 26854 97497 93824 16423 65875 0.33884 0.34233 0.34581 0.34928 0.35273 00713 22577 26835 14255 85596 0.69961 0.69725 0.69491 0.69258 0.69027 85089 86512 42236 50790 10717 0.875 0.880 0.885 0.890 0.895 1.900 1.905 1.910 1.915 1.920 0.96176 0.96348 0.96523 0.96699 0.96877 58319 84632 07261 26608 43090 -0.03898 -0.03719 -0.03538 -0.03356 -0.03172 42759 47650 81118 43732 36054 0.35618 0.35961 0.36304 0.36645 0.36985 41612 83049 10646 25136 27244 0.68797 0.68568 0.68341 0.68116 0.67892 20582 78965 84465 35696 31293 0.900 0.905 0.910 0.915 0.920 1.925 1.930 1.935 1.940 1.945 0.97057 0.97239 0.97423 0.97609 0.97797 57134 69178 79672 89075 97861 -0.02986 -0.02799 -0.02609 -0.02419 -0.02226 58646 12062 96858 13581 62778 0.37324 0.37661 Oij7998 0.38334 0.38668 17688 97179 66424 26119 76959 0.67669 0.67448 0.67228 0.67010 0.66793 69903 50194 70846 30559 28044 0.925 0.930 0.935 0.940 0.945 1.950 1.955 1.960 1.965 1.970 0.97988 0.98180 0.98374 0.98570 0.98768 06513 15524 25404 36664 49838 -0.02032 -0.01836 -0.01639 -0.01439 -0.01239 44991 60761 10621 95106 14744 0.39002 0.39334 0.39665 0.39996 0.40325 19627 54805 83163 05371 22088 0.66577 0.66363 0.66150 0.65938 0.65728 62034 31270 34514 70538 38134 0.950 0.955 0.960 0.965 0.970 1.975 1.980 1.985 1.990 1.995 0.98968 0.99170 0.99375 0.99581 0.99789 65462 84087 06274 32598 63643 -0.01036 -0.00832 -0.00626 -0.00419 -0.00210 70060 61578 89816 55291 58516 0.40653 0.40980 0.41306 0.41631 0.41955 33970 41664 45816 47060 46030 0.65519 0.65311 0.65105 0.64900 0.64696 36104 63266 18450 00505 08286 0.975 0.980 0.985 0.990 0.995 2.000 1.00000 00000 0.64493 40668 1.000 Y! II 3 c-y 0.00000 00000 0.42278 43351 lny! $ In y! [1 (512 log,o e=0.43429 [ c-y 44819 Y I GAMMA FUNCTION TETRAGAMMA ., #” (.I,) AND RELATED AND PENTAGAMMA +m) (.r) .I’ 271 FUNCTIONS FUNCTIONS Tal,le 6.2 V(T) $(:I) (x) 1.00 1.01 1.02 1.03 1.04 38063 86771 42052 85963 63855 6.49393 6.25106 6.01969 5.79918 5.58891 94023 18729 49890 38573 68399 0. 00 0. 01 0.02 0.03 0.04 1.50 1.51 1.52 1.53 1.54 -0.82879 -0.81487 -0.80129 -0.78803 -0.77509 66442 76121 51399 87419 83287 1.40909 1.37489 1.34177 1.30967 1.27856 10340 70527 21104 56244 88154 0.50 0.51 0.52 0.53 0.54 1.05 1.06 1.07 1.08 1.09 -2.10815 -2.05523 -2.00419 -1.95493 -1.90737 80219 94833 19194 13213 82154 5.38832 5.19686 5.01404 4.83939 4.67247 23132 56970 67303 69702 74947 0. 05 0.06 0.07 0.08 0.09 1.55 1.56 1.57 1.58 1.59 -0.76246 -0.75012 -0.73807 -0.72630 -0.71480 41904 69793 76946 76669 85441 1.24841 1.21917 1.19082 1.16332 1.13663 46160 75841 38216 08979 77770 0.55 0.56 0.57 0.58 0.59 1.10 1.11 1.12 1.13 1.14 -1.86145 -1.81709 -1.77423 -1.73279 -1.69272 73783 75731 13035 45852 67342 4.51287 4.36020 4.21411 4.07424 3.94028 67903 88083 11755 35447 60717 0.10 0. 11 0.12 0.13 0.14 1.60 1.61 1.62 1.63 1.64 -0.70357 -0.69259 -0.68185 -0.67136 -0.66110 22779 11105 75627 44220 47316 1.11074 1.08561 1.06121 1.03752 1.01452 47490 33658 63792 76835 22608 0.60 0.61 0.62 0.63 0.64 1.15 1.16 1.17 1.18 1.19 -1.65397 -1.61647 -1.58017 -1.54503 -1.51100 01677 02206 49731 50903 36723 3.81193 3.68891 3.57095 3.45780 3.34922 80220 64540 50416 29554 38402 0.15 0. 16 0. 17 0.18 0.19 1.65 1.66 1.67 1.68 1.69 -0.65107 -0.64125 -0.63166 -0.62226 -0.61308 17793 90881 04061 96973 11332 0.99217 0.97046 0.94937 0.92886 0.90893 61290 62927 06973 81843 84502 0.65 0.66 0.67 0.68 0.69 1.20 1.21 1.22 1.23 1.24 -1.47803 -1.44608 -1.41512 -1.38510 -1.35598 61144 99765 48602 22950 56308 3.24499 3.14490 3.04875 2.95636 2.86754 48647 58422 84139 52925 95589 0.20 0. 21 0.22 0.23 0.24 1.70 1.71 1.72 1.73 1.74 -0.60408 -0.59528 -0.58667 -0.57823 -0.56997 90841 81112 29593 85490 99702 0.88956 0.87072 0.85239 0.83456 0.81722 20066 01433 48922 89940 58660 0.70 0.71 0.72 0.73 0.74 1.25 1.26 1.27 1.28 1.29 -1.32773 -1.30033 -1.27372 -1.24790 -1.22282 99375 19112 97857 32496 33691 2.78214 2.69999 2.62093 2.54484 2.47158 40092 05478 96227 97000 67746 0.25 0. 26 0.27 0.28 0.29 1.75 1.76 1.77 1.78 1.79 -0.56189 -0.55397 -0.54621 -0.53861 -0.53116 24756 14738 25238 13291 37320 0.80034 0.78392 0.76793 0.75237 0.73721 95719 47929 68005 14300 50564 0.75 0.76 0.77 0.78 0.79 1.30 1.31 1432 1.33 1.34 -1.19846 -1.17479 -1.15179 -1.12943 -1.10769 25147 42923 34794 59642 86881 2.40102 2.33304 2.26752 2.20436 2.14345 39143 08348 35032 37678 90132 0. 30 0. 31 0.32 0.33 0.34 1.80 1.81 1.82 1.83 1.84 -0.52386 -0.51671 -0.50970 -0.50283 -0.49609 57084 33630 29242 07396 32712 0.72245 0.70807 0.69407 0.68042 0.66712 45705 73565 12710 46226 61527 0.80 0.81 0.82 0.83 0.84 1.35 1.36 1.37 1.38 1.39 -1.08655 -1.06599 -1.04599 -1.02652 -1.00757 95925 75682 24073 47586 60850 2.08471 2.02802 1;97332 1.92051 1.86951 18367 97472 48830 37473 69616 0.35 E 0:38 0.39 1.85 1.86 1.87 1.88 1.89 -0.48948 -0.48300 -0.47665 -0.47042 -0.46431 70921 88813 54207 35909 03677 0.65416 0.64153 0.62921 0.61720 0.60549 50169 07680 33389 30270 04793 0.85 0.86 0.87 0.88 0.89 1.40 1.41 1.42 1.43 1.44 -0.98912 -0.97116 -0.95366 -0.93662 -0.92002 86236 53479 99322 67177 06808 1.82025 1.77266 1.72667 1.68221 1.63923 90339 81419 59295 73161 03178 0.40 0. 41 0. 42 0.43 0.44 1.90 1.91 1.92 1.93 1.94 -0.45831 -0.45242 -0.44665 -0.44098 -0.43542 28188 81007 34549 62055 37563 0.59406 0.58292 0.57205 0.56144 0.55108 66772 29238 08299 23020 95304 0.90 0.91 0.92 0.93 0.94 1.45 1.46 1.47 1.48 1.49 * -2.40411 -2.34039 -2.27905 -2.21996 -2.16303 -0.90383 -0.88806 -0.87268 -0.85768 -0.84306 74031 30426 43070 84281 31376 1.59765 1.55743 1.51852 1.48085 1.44439 58792 77157 21649 80478 65370 0.45 0. 46 0.47 0.48 0.49 1.95 1.96 1.97 1.98 1.99 -0.42996 -0.42460 -0.41934 -0.41417 -0.40909 35876 32537 03805 26631 78630 0.54098 0.53112 0.52149 0.51208 0.50290 49774 13668 16733 91127 71324 0.95 0.96 0.97 0.98 0.99 1.50 -0.82879 66442 1.40909 10340 0.50 2.00 -0.40411 38063 0.49393 94023 1.00 Lf- In 2/! dy* cln W y! Y $ln g! 5 In 2/! Y -65)4 6J [(-74)31 T. Davis, Tables of the higher mathematical[(functions, 2 ~01s.c(Principia Press, 1 rL -4)11 Compiled from H. Bloomington, Ind., 1933, 1935) (with permission). *‘See page II. * 272 GAMMA ‘l-al ,Ic 6.3 FUNCTION GAMMA AND DICAMMA r(u) AND Fl’NC’lM)NS RELATED FOR FUNCTIONS lh’l’IS(;l3R r(u+i) l/r00 /III) II.\I,F-I;\‘I’FXER ,I’1 (1,) +Oc) Vi\I,c’E:s ,1’.;01) ( 0)1.00000 00000 0)1.00000 000 -1)8.86226 93 -0.57721 56649 1.08443 755 0.57721 566 i 0i 2.00000 00000 1.00000 6.00000 ( 1)2.40000 00000 I - 111.00000 000 0) 1.66666 667 5.00000 (- 2)4.16666 667 I 013.32335 10 +0.42278 43351 1 1.16317 28 0.92278 76684 0)1.32934 04 1.25611 ( 1)5.23427 78 1.50611 76684 1.02100 830 0.13017 669 1.04220 712 0.27036 285 1.02806 452 0.17582 795 1.01678 399 0.10332 024 (- 3)8.33333 333 ( 2)2.87885 28 3)1.87125 43 4)1.40344 07 (- 6)2.75573 192 I 1.70611 76684 1.87278 43351 2.01564 14780 2.14064 14780 2.25175 25891 1.01397 285 1.01196 776 1.01046 565 1.00929 843 1.00836 536 0.08564 180 0.07312 581 0.06380 006 0.05658 310 0.05083 250 (- 7)2.75573 192 (-11)1.14707 456 ( 7)1.18994 23 I 8)1.36843 37 9 1.71054 21 10 2.30923 18 11)3.34838 61 2.35175 25891 2.44266 16800 2.52599 50133 2.60291 80902 2.67434 66617 1.00760 243 1.00696 700 1.00642 958 1.00596 911 1.00557 019 0.04614 268 0.04224 497 0.03895 434 0.03613 924 0.03370 354 I -13 17.64716 733 -14 4.77947 373 (12)5.18999 74 13 8.56349 85 2.80351 33283 2.74101 1.00491 343 0.02970 539 1.00522 124 0.03157 002 -15)2.81145 725 I 15I 1.49861 21 2.91789 24133 2.86233 68577 2.97052 39922 1.00439 519 0.02654 657 1.00463 988 0.02803 490 1.00417 501 0.02520 828 (-19)4.11031 762 (19)1.10827 98 3.02052 39922 3.06814 30399 3.11359 75853 3.15707 58462 3.19874 25129 1.00397 584 1.00379 480 1.00362 953 1.00347 806 1.00333 872 0.02399 845 0.02289 941 0.02189 663 0.02097 798 0.02013 331 (-26)6.44695 029 (-27)2.47959 626 (-29 9.18368 986 i-30 3.27988 924 1-31I 1.13099 629 (25)7.87126 49 2712.08588 52 28 5.73618 43 3.23874 25129 3.27720 40513 3.31424 10884 3.34995 53741 3.38443 81327 1.00321 011 1.00309 105 1.00298 050 1.00287 758 1.00278 154 0.01935 403 0.01863 281 0.01796 342 0.01734 046 0.01675 925 (3212.65252 85981 86542 83693 76188 79904 -33)3.76998 763 -34)1.21612 504 (33)1.47092 26 (39)1.74039 42 3.41777 14660 3.45002 95305 3.48127 95305 3.51158 25608 3.54099 43255 1.00269 170 1.00260 748 1.00252 837 1.00245 392 1.00238 372 0.01621 574 0.01570 637 0.01522 803 0.01477 796 0.01435 374 47966 32679 53091 61747 82081 (-4:)9.67759 296 (-47)4.90246 976 40)6.17839 94 42)2.25511 58 43)8.45668 42 45)3.25582 34 47)1.28605 02 3.56956 57541 3.59734 35319 3.62437 05589 3.65068 63484 3.67632 73740 1.00231 744 1.00225 474 1.00219 534 1.00213 899 1.00208 546 0.01395 318 0.01357 438 0.01321 560 0.01287 530 0.01255 208 (47)8.15915 28325 i 49i 6.04152 26613 52 3.34525 63063 51 1.40500 61178 (-48)1.22561 744 (-53)1.65521 673 I -50 12.98931 087 -52 7.11740 083 48 5.20850 35 51 3.99612 67 50 9.18649 81 5312.16152 90 3.70132 73740 3.77278 71417 3.74952 76179 3.72571 29557 1.00203 455 0.01224 469 1.00193 606 0.01195 200 1.00189 983 0.01140 668 1.00198 570 0.01167 297 (54)2.65827 15748 (+5)3.76184 288 55)1.77827 64 3.79551 02284 1.00185 354 0.01115 226 I -60 I 3.86662 471 -63 18.05547 607 -58 1.64397 540 -57 1.81731 851 -62 8.35965 084 (63 I 4.29046 74 58 1.78713 18 61 3.76238 44 60 8.66760 82 56 8.09115 29 3.83947 15811 3.81773 24506 3.90198 96734 3.88158 15102 3.86074 81768 1.00166 210 0.01023 895 1.00170 460 0.01003 879 1.00181 321 0.01090 333 1.00173 759 0.01045 283 1.00177 803 0.01067 602 (-65)3.28794 942 (65)2.16668 38 3.92198 96734 1.00163 530 0.00983 596 11 :3 :54 :76 18 19 20 21 22 23 24 25 26 El 29 30 42 43 3.99168 00000 I13 2.09227 89888 12)1.30767 43680 I 14i 3.55687 42810 15 6.40237 37057 (17)1.21645 10041 5.1OYO942172 (25)1.55112 10043 3.04888 34461 46 47 48 ;; 51 (64)3.04140 93202 (11-l)! p,!= (2,)1,,4,4 l/(,1-1)! I I ()I -a)! r(tr)=(2r)~,r"-~,,-)~f, (.)I) 2 Zn(tc-l)! * (2#=2.50662 82746 31001 +(u)=ln ~/-f:~(~~) $0)) compiled from I-I. T. Davis, Tables of the higher mathematical functions, 2 ~01s. (Principia Press, Bloomington, Ind., 1933, 1935) (with permission). fl(II) GAMMA FUNCTION AND RELATED FUNCTIONS GAMMA AND DICAMMA IWiXc'I'IOIvS FOR IKTE(;ER AND IIALF-Ih.rE(;ER VALl!& tl r(tt+3) l/rot) rot) JOI) ./I 00 273 'I‘;,],!,. (,.:s .f:;(l,) I - 70)2.33924 942 ( 65)2.16668 38 65)3.28794 964 68 1.23979 515 67 6.44695 993 16 12 21 1.00160 355 0 00928 363 100154 383 0.00946 620 1.00157 438 0.00964 784 1.00163 530 0.00983 596 (1 72)4.33193 547 63 3:99821 47288 1:OOlSl 628 ( 73)9.47993 5.35616 3.07979 1.80167 ( 81)1.07199 44 29 37 93 92 4.01639 4.03425 4.05179 4.06903 4.08598 65470 1.00148 36899 1.00146 75495 1.00143 89288 1.00141 80814 1.00138 919 0.00895 514 304 0.00879 758 780 0.00864 546 341 0.00849 852 984 0.00835 648 ( 82)6.48559 ( 84)3.98864 2.49290 1.58299 ( 90)1.02102 51 10 06 19 98 4.10265 4.11904 4.13517 4.15105 4.16667 47481 81907 72229 02388 52388 1.00136 1.00134 1.00132 1.00130 1.00128 704 0.00821 912 498 0.00808 619 362 0.00795 750 292 0.00783 284 286 0.00771 203 ( 91)6.68774 4.44735 3.00196 2.05634 1.42915 73)1.26964 3.96082 75166 3.94159 62103 3.92198 82858 3 97969 96734 50 04 15 36 88 4.18205 4.19721 4.21213 4.22684 4.24133 98542 13693 67425 26248 53785 1.00126 1.00124 1.00122 1.00120 1.00119 341 0.00759 489 455 0.00748 125 623 0.00737 096 845 0.00726 388 118 0.00715 986 1.00755 70 4.25562 10927 7.20403 24 4.26970 55998 31188 3.83884 35 5.22292 87 4.28359 44887 4.29729 930 (108)2.85994 23 4.31080 66323 1.00117 1.00115 1.00114 1.00112 1.00111 439 0.00705 878 807 0.00696 052 675 220 0.00686 495 0.00677 197 172 0.00668 148 03354 71 (100 1.19785 71670 (-101 8.34824 074 72 i 101I 8.50478 58857 i -102 I 1.17580 856 :i 105 4.47011 58377 103 6.12344 54615 -106 1.63306 868 -104 2.23707 744 75 (107)3.30788 54415 :; 78 2 40811 109)2.48091 116)8.94618 21308 (-108)3.02307 -117)1.11779 526 6.43131 87 126)3.31424 01346 128)2.81710 41144 4.38826 36140 1.00102 933 0.80618 554 1.00100 452 1.00101 677 1.00099 255 1.00098 087 0.00603 619 0.00610 995 0.00596 419 0.00589 389 '129)2.60868 05 4.44852 20756 2.25650 86 4.46014 99825 1.74738 50 4.47164 42354 1.97444 38 4.48300 78718 it 1.56390 85 4.49424 38268 90 72 36 60 22 98 101 (157)9.33262 (II-l) 15444 (-158)1.07151 ! ,I != (2T)fd’+~~-~~fl (ttj *see p*ge II. 1.00108 709 0.00659 337 1.00109 283 0.00650 756 1.00106 894 0.00642 395 1.00105 220 0.00634 247 540 1.00104 0.00626 302 4.32425 79 4.41280 44150 5.24152 47 4.40060 92931 3.61075 53 4.42485 26078 127)3.05108 83 4.43675 73697 81 E 84 85 (110)2.15925 64 4.32413 99657 1.65183 12 4.33729 78604 1.28016 92 4.35028 48734 7.98921 57 4.36310 36140 1.00493 28 4.37576 53862 0:OOSll 846 029 (158)9.36756 l/(+1)! r (t,) = (2r) f,,J~--,>-,y, (1,) 4.50535 4.51634 4.52721 4.53796 4.54860 49379 39489 35142 62023 45002 79 4.61016 18527 (1/-i)! *-& Zn(tr-l)! +()I) =ln ef:~(~l) 1.00096 946 0.00582 522 1.00095 831 0.00575 814 1.00094 741 0.00569 258 1.00093 676 0.00562 850 1.00092 635 0.00556 584 1.00091 1.00090 1.00089 1.00088 1.00087 0.00550 0.00544 0.00538 0.00532 0.00527 617 620 646 691 757 1.00082 542 [C-37)2] (2+2.50662 457 463 598 858 239 0.00495 866 ['Ff'l] 82746 31001 274 GAMMA FUNCTION LOGARITHMS Table 6.4 log,,r (11) 0.00000 000 0.00000 000 0.30103 000 0.77815 125 1.38021 12 log,, r (,l&) AND RELATED OF THE FUNCTIONS GAMMA log,, r ()I+;) -0.04915 +0.07578 0.44375 0.96663 1.60345 851 023 702 576 79 -0.05245 +0.12363 0.52157 1.06564 1.71885 FUNCTION log,, -0.04443 +0.17741 0.60338 1:167&S 1.83666 477 398 271 41 09 2.07918 2.85733 3.70243 4.60552 5.55976 11 12 13 14 15 f,N r (n-t+) 506 620 621 43 68 12 25 05 05 30 2.33045 3.13208 3.99739 4.91820 5.88824 66 89 04 91 59 2.45921 3.27213 4.14719 5.07661 6.05433 95 28 41 30 66 2.58998 3.41389 4.29850 5.23635 6.22163 86 73 39 60 27 6.55976 7.60115 8.68033 9.79428 10.94040 30 57 70 03 8 6.90248 7.95684 9.04792 10.17286 11.32921 63 40 45 3 0 7.07552 8.13622 9.23313 10.36346 11.52483 59 37 38 8 6 7.24966 8.31660 9.41927 lOI 11.72126 15 83 06 3 5 1.00000 0.96027 0.94661 0.93972 0.93558 ' 000 923 646 921 323 0.93281 0.93083 0.92934 0.92819 0.92726 466 524 980 400 910 0.92651 0.92588 0.92534 0.92488 Oi92444 221 137 753 990 327 12.116500 13.320620 14.551069 15.806341 17.085095 12.514847 13.727922 14.966804 16.230045 17.516352 12.715167 13.932651 15.175689 16.442861 17.732896 12.916241 14.138090 15.385245 16.656311 17.950042 0.92414 0.92383 0.92356 0.92332 0.92310 619 993 769 409 485 18.386125 19.708344 21.050767 227412494 23.792706 18.824561 20.153619 21.502573 22.870550 24.256751 19.044649 20.377088 21.729270 23.100338 24.489504 19.265313 20.601105 21.956492 23.330629 24.722740 0.92290 0.92272 0.92256 0792241 0.92227 649 615 149 055 169 25.190646 26.605619 28.036983 29.484141 30.946539 25.660444 27.080949 28.517642 29.969940 31.437301 25.896045 27.319290 28.758623 30.213468 31.683290 26.132109 27.558078 29.000035 30.457412 31.929681 0.92214 0.92202 0.92191 Oi92181 0.92171 350 481 460 198 621 32.423660 33.915022 35.420172 36.938686 38.470165 32.919221 34.415228 35.924878 37;447757 38.983473 33.167590 34.665900 36.177784 37.702829 39.240648 33.416347 34.916950 36.431055 37.958255 39.498167 0.92162 0.92154 0.92146 0.92138 0.92131 661 262 371 944 942 40.014233 41.570535 43.138737 44.718520 46.309585 40.531658 42.091963 43.664060 45.247636 46.842397 40.790876 42.353169 43.927200 45.512661 47.109258 41.050429 42.614701 44.190658 45.777995 47.376420 0.92125 0.92119 0.92113 0.92107 0.92102 329 073 146 524 182 47.911645 49.524429 51.147678 52.781147 54.424599 48.448061 50.064362 51.691044 53.327866 54.974597 48.716713 50.334761 51.963150 53.601639 55.249999 48.985659 50.605448 52.235536 53.875686 55.525670 0.92097 0.92092 0.92087 0.92083 0.92079 101 262 648 244 035 4487 49 50 56.077812 57.740570 i9:412668 61.093909 62.784105 56.631014 58.296908 59.972075 61.656322 63.349462 56.908011 58.575464 60.252157 61.937899 63.632504 57.185269 58.854276 60.532491 62.219723 631915788 0.92075 0.92071 Oi92067 0.92063 0.92060 010 156 462 919 518 51 64.483075 65.051318 65.335796 65.620510 0.92057 250 3: 33 3354 41 42 t43 45 46 log,, (+f) ! log,, (n-fj ! log,, (n-l) ! log,, (n-+) ! In r(n)=ln (n-l)!=(+) In n+l+fi(?t) In 10=2.30258 509299 log,, r(n) compiled from E. S. Pearson, Table of the logarithms of the complete r-function, arguments 2 to 1200. Tracts for Computers No. VIII (Cambridge Univ. Press, Cambridge, England, 1922) (with permission). GAMMA FUNCTION LOGARITHMS log,, r (11) AND RELATED OF THE 275 FUNCTIONS GAMMA FUNCTION Table 6.4 log,, r (?I++) log,, r (n+t) log,, r (n++) 64.483075 66.190645 67.906648 69.630924 71.363318 65.051318 66.761717 68.480496 70.207494 71.942561 65.335796 67.047603 68.767762 70.496116 72.232512 65.620510 67.333720 69.055256 70.784961 72.522683 0.92057 0.92054 0.92051 0.92048 0.92045 250 108 084 173 367 73.103681 74.851869 76.607744 78.371172 80.142024 73.685548 75.436313 77.194720 70.960637 80.733936 73.976805 75.728854 77.488522 79.255677 81.030194 74.268279 76.021606 77.782531 79.550922 81.326654 0.92042 0.92040 0.92037 0.92035 0.92032 661 051 530 095 741 81.920175 83.705505 85.497896 87.297237 89.103417 82.514493 84.302190 86.096910 87.898542 89.706978 82.811950 84.600825 86.396705 88.199479 90.009038 83.109604 84.899655 86.696691 88.500604 90.311284 0.92030 0.92028 0.92026 0.92024 0.92022 464 261 127 061 057 90.916330 92.735874 94.561949 96.394458 98.233307 91.522113 93.343845 95.172075 97.006708 98.847650 91.825280 93.648101 95.477405 97.313096 99.155080 92.128629 93.952538 95.782913 97.619659 99.462684 0.92020 0.92018 0.92016 0.92014 0.92012 115 231 401 625 900 .f&) 100.07841 101.92966 103.78700 105.65032 107.51955 100.69481 102.54810 104.40744 106.27274 108.14393 101.00327 102.85758 104.71791 106.58420 108.45636 101.31190 103.16722 \ 105.02855 106.89582 108.76895 0.92011 0.92009 0.92008 0.92006 0.92004 223 593 008 465 964 109.39461 111.27543 113.16192 115.05401 116.95164 110.02091 111.90363 113.79200 115.68594 117.58540 110.33430 112.21797 114.10727 116.00214 117.90250 110.64785 112.53246 114.42269 116.31848 118.21976 0.92003 0.92002 0.92000 0.91999 0.91998 502 078 690 338 019 118.85473 120.76321 122.67703 124.59610 126.52038 119.49029 121.40056 123.31614 125.23696 127.16296 119.80830 121.71946 123.63591 125.55760 127.48445 120.12646 122.03850 123.95583 125.87838 127.80610 0.91996 0.91995 0.91994 0.91993 0.91991 733 479 254 059 892 128.44980 130.38430 132.32382 134.26830 136.21769 129.09407 131.03025 132.97143 134.91756 136.86857 129.41642 131.35344 133.29545 135.24239 137.19421 129.73891 131.67676 133.61959 135.56735 137.51999 0.91990 0.91989 0.91988 0.91987 0.91986 752 638 550 486 446 138.17194 140.13098 142.09477 144.06325 146.03638 138.82442 140.78505 142.75041 144.72044 146.69511 139.15086 141.11228 143.07842 145.04923 147.02467 139.47743 141.43964 143.40657 145.37815 147.35435 0.91985 0.91984 0.91983 0.91982 0.91981 428 433 459 505 572 148.01410 149.99637 151.98314 153.97437 155.97000 148.67435 150.65813 152.64639 154.63909 156.63619 149.00467 150.98920 152.97820 154.97164 156.96946 149.33511 151.32039 153.31013 155.30430 157.30285 0.91980 0.91979 0.91978 0.91978 0.91977 659 764 887 028 186 158.63763 158.97163 159.30574 0.91976 361 157.97000 log,, (n-l) ! ln r(n)=ln log,, (14) log,, (n-f) ! (n-l)!=(n-+) In n-?L+.fz(n) ! log,, (n- 8 ! In 10=2.30258 509299 / 276 Table GAMMA 6.5 1 o.xoi15 AUXILIARY FUNCTION AND FUNCTIONS fl FOR RELATED GAMMA (x> FUNCTIONS AND DIGAMMA f2 (xl FUNCTIONS <X> 7”; ;; f3 (xl 0.014 0.013 0.012 0.011 1.00125 1.00116 1.00108 1.00100 1.00091 077 735 391 050 708 0.92018 0.92010 0.92002 0.91993 0.91985 852 519 186 853 520 0.00751 0.00701 0.00651 0.00601 0.00551 875 633 408 200 008 0.010 0.009 0.008 0.007 0.006 1.00083 1.00075 1.00066 1.00058 1.00050 368 028 689 350 012 0.91977 0.91968 0.91960 0.91952 0.91943 186 853 520 187 853 0.00500 0.00450 0.00400 0.00350 0.00300 833 675 533 408 300 100 111 125 143 167 0.005 0.004 0.003 0.002 0.001 1.00041 1.00033 1.00025 1.00016 1.00008 675 339 003 668 334 0.91935 0.91927 0.91918 0.91910 0.91902 52C 187 853 520 187 0.00250 0.00200 0.00150 0.00100 0.00050 208 133 075 033 008 200 250 333 500 1000 0.000 1.00000 (1 c -;)l 0.91893 853 0.00000 000 00 000 c1 i-y In r(x)=ln +(x) =h (x-l)!-(x-3) In x- -z+f2(2) . x-f3f3(x) (2r)*=2.50662 n 100 6.6 82746 31001 <x>=nearest Table integer to x. FACTORIALS n! 91 FOR LARGE ARGUMENTS n! 72316 22543 07425 21544 39441 52682 6;10 1408 1.2655 200 78673 64790 50355 700 1689 2.4220 40124 75027 21799 75122 16440 63604 800 1976 7.7105 30113 35386 00414 300 400 52284 66238 95262 900 2269 6.7526 80220 96458 41584 500 36825 99111 00687 1000 i 2567 I 4.0238 72600 77093 77354 l?(n+l) Un+l> A tame of the factorial numbers and their reciprocals Compiled from Ballistic Research Laboratory, from l! to lOOO! to 20significant digits, Technical Note No. 381, Aberdeen Proving Ground, Md.(1951) (with permission). 9.3326 7.8865 3.0605 6.4034 1.2201 GAMMA GAMMA FUNCTION AND RELATED FUNCTION FOR COMPLEX 277 FUNCTIONS Table ARGUMENTS 6.7 z=l.O Jln ~~~ 0:2 - 0.00000 00000 00 0.00819 77805 65 - 0.03247 62923 18 00:: - 0.12528 93748 00 0.07194 62509 21 - 0.00000 0.05732 0.11230 0.16282 0.20715 r(z) &Tln r(z) !I 00000 00 2940417 22226 44 0672168 58263 16 Yin r(z) :*: 5:3 5.4 - 6.13032 6.27750 6.42487 6.57242 6.72016 41445 24635 30533 85885 21547 53 84 35 29 03 3.81589 3.97816 4.14237 4.30850 4.47650 85746 38691 74050 21885 25956 15 88 86 83 68 Pi 5:9 - 6.86806 7.01613 7.16436 7.31275 7.46128 72180 48 75979 76 7442106 12034 30 36194 29 4.64634 4.81799 4.99142 5.16659 5.34347 42978 41933 03424 19085 91013 70 05 89 37 53 - 7.60995 7.75877 7.90772 8.05680 8.20600 96929 46746 40468 35089 89631 51 55 98 04 00 5.52205 5.70228 5.88415 6.06762 6.25267 31255 61315 11702 21500 37967 15 35 39 13 05 76 12 44 02 93 5.0 o"*z - 0.19094 549918714 0.26729 00682 - 0.24405 82989 91 0.27274 38104 05 0:7 - 0.35276 86908 60 - 0.44597 87835 49 - 0.54570 51286 05 - 0.29282 6351187 - 0.30422 56029 76 - 0.30707 43756 42 :*: 1:4 - - 0.30164 0.28826 0.26733 0.23921 0.20430 1.5 - 1.23448 30515 47 ::: ::g" - 1.3593122484 1.48608 96127 65 57 - 1.61459 53960 00 1.74464 4276174 + 0.16293 97694 0.11546 87935 0.06219 86983 0.0034166314 0.0606128742 80 89 29 77 95 - 8.35533 65025 11 8.50478 2399125 8.65434 30931 23 8.8040151829 10 8.95379 54158 79 6.43928 6.62742 6.81707 7.00820 7.20081 16159 18579 14837 81345 01014 21" - 2.00876 41504 31 1.87607 87864 71 2'2 214 - 2.14258 42092 96 2.27743 81922 04 - 2.41323 81411 84 0.12964 63163 0.20345 94738 0.28184 56584 0.3646140489 0.45158 81524 10 33 26 50 41 - 9.10368 06798 9.25366 79950 9.40375 45067 9.55393 74783 9.7042142849 32 15 08 21 72 7.39485 7.59032 7.78719 7.98545 8.18508 62984 36 6235184 99928 77 82004 68 20125 03 22:; - 2.54990 61537 95 2.68737 68424 50 0.54260 44058 0.6375109190 0.73616 63516 0.83843 89130 0.94420 54730 52 46 79 96 39 - 9.85458 24074 -10.00503 94267 -10.15558 30186 -10.3062109489 -10.45692 10687 86 90 86 48 39 8.38605 8.58835 8.79196 8.99687 9.20305 30880 35709 60705 36442 97799 89 62 87 29 25 1.05335 07710 1.16576 67132 1.28135 17459 1.4000102965 1.52165 22746 69 86 32 76 73 -10.6077113103 -10.75857 96829 -10.90952 42693 -11.06054 32217 -11.21163 47589 15 95 78 92 48 9.41050 83803 9.61920 37472 9.82913 05671 10.04027 38971 10.2526191518 12 42 62 80 09 1.64619 26242 69 1.77355 09225 91 1.903651019019 2.03642 0709693 2.17179 14436 05 -11.36279 -11.51402 -11.66532 -11.81669 -11.96812 71628 87756 79970 32818 31369 04 02 81 48 01 10.46615 10.68085 10.89672 11.11373 11.33188 20903 24 88047 12 5708177 9524157 72758 53 2.30969 2.45007 2.59287 2.73802 2.88548 0.65092 31993 02 0.76078 39588 41 0.87459 04638 95 1.11186 27669 26 0.99177 45664 59 22'87 - 2.96448 564119189 2.82558 14617 2:9 - 3.1040154399 01 33:; - 3.24414 90223 90 3.38482 42995 77 :*: 314 - 3.52603 43067 09 3.66772 81104 88 - 3.80988 12618 23 ::: - 3.95246 712618951 4.09546 13204 zl 3:9 - 4.38258 69752 71 4.23884 14660 28 - 4.52667 88647 16 21" - 4.67109 95934 96 4.81583 29197 09 i-3' 414 - 5.10617 37766 87 4.96086 81606 63 - 5.25176 30342 30 44:; 4;i - 5.54369 62389 84 5.39760 64183 04 - 5;69002 29483 73 i:; 5.0 03204 68 66142 39 05805 81 67844 65 0724149 80565 85299 37713 74148 56389 73 47 19 20 27 -12.1196161192 -12.27117 08338 -12.42278 59312 -12.57446 01059 -12.72619 20940 81 67 81 08 29 11.55115 11.77153 11.99300 12.21556 12.43920 62762 41183 86662 80464 06390 - 5.83657 58655 54 5.98334 58764 32 3.03519 69999 3.1871122793 3.34118 43443 3.49736 80186 3.6556199647 22 89 27 15 12 -12.87798 -13.02982 -1ji18172 -13.33367 -13.48567 06720 46547 28939 42765 77234 44 89 51 47 95 12.66389 12.88964 13.11642 13.34423 13.57307 5070128 02037 08 51346 66 91814 77 18794 55 - 6.13032 41445 53 3.81589 85746 15 -13.63773 21882 47 10.0 02 09 85 79 90 13.80291 29742 30 Linear interpolation will yield about three figures;eight-point interpolation will yield about eight figures. For z outsidethe range of the table, seeExamples S?ln r(z)=ln Ir(z) / 5-8. .P In r (2)=arg r (2) 278 GAMMA Table 6.7 FUNCTION GAMMA AND FUNCTION FOR RELATED FUNCTIONS COMPLEX ARGUMENTS .r=l.l - gin r(z) 0.04987 24412 0.05702 02290 0.07824 35801 0.1129143470 0.16008 21257 60 38 68 17 99 - Jln 0.00000 0.04206 0.08230 0.11905 0.15086 r(z) 00000 65443 97383 06275 79240 00 76 98 18 09 - 0.21858 0.28718 0.36464 0144978 0.54157 96764 99839 38731 83131 54093 09 43 53 87 11 - 0.1766611398 0.19566 16788 0.20740 35526 0.21167 10325 0.20843 91333 43 64 60 55 00 :G! 1:2 1.3 1.4 - 0.63908 0.74153 0.84825 0.95868 1.07235 78153 80620 85646 73364 26519 48 74 26 97 67 - 0.1978178257 0.18000 55175 0.15525 33222 0.12383 93047 0.08605 08957 :-z 1:7 - 1.18885 84815 1.30787 15575 1.4291103402 1.55233 58336 1.67734 40572 22 95 04 11 49 - 0.04217 34907 + 0.00751 65191 0.06275 56777 0.12329 53847 0.18890 25358 11 79 30 15 69 ::: 2:: - 1.80395 1.93203 2.06142 2.19203 2.32375 99248 22878 99239 82866 68617 63 13 46 29 01 0.25935 0.33446 0.41402 0.49786 0758582 z 214 2.5 - 2.45649 2.59018 2.72473 2.86010 2.99622 70097 01959 65306 35591 52529 26 43 67 81 98 0.67775 04868 0.77349 56148 0.87292 80949 0.97592 26515 1.0823617859 Z 29" 3.0 - 3.13305 11644 3.27053 57144 3.40863 75892 3.5473192273 3.68654 63804 50 30 32 03 17 ;:: ;:: 3.5 3.6 3.7 - 3.82628 77368 3.9665145962 4.10720 05882 4.24832 14278 4.38985 47017 E 4.0 - 4.53177 96812 4.67407 71584 4.81672 93009 4.9597195242 5.10303 23779 2: 413 4.4 44:;--t-i -4:9 5.0 514 Vln 5.96893 6.11415 6.25959 6.40526 6.55114 r(z) 91493 43840 93585 53566 41480 55'2 517 5.8 5.9 - 6.69722 6.84350 6.98998 7.13663 7.28347 - 7.43047 7.57764 7.72498 7.87247 8.02011 67 74 12 38 00 00~~ 0:2 0.3 0.4 - 4 In r(z) 3.96198 63258 4.12446 68364 4.28888 73284 4i4552112743 4.62340 34819 60 90 80 47 04 7953189 94110 69 15495 70 77586 96 17659 19 4.79343 4.96525 5.13885 5.31419 5.49124 00232 81683 63238 39750 16322 04 67 91 77 40 76136 96383 24519 09237 01645 25 95 72 38 61 5.66997 5i85035 6.03236 6.21597 6.40116 07803 94 3832146 40835 50 56726 90 35407 92 - 8.16789 55118 88 8.31582 25159 69 8.46388 6927117 8.61208 46838 95 8.760411902172 6.58790 6.77617 6.96594 7.15720 7.34992 33956 16773 55256 27497 17993 - 8.90886 9.05744 9.20613 9.35494 9.50386 Y E :-; 6.5 93780 23 29085 79 4032150 66085 82 64745 04 52 05 61 40 20 67 32 30 24 20 48649 00129 39357 33637 51603 60 63 92 73 25 7.54408 7173966 7.93664 8i13500 8.33473 17375 09 2215113 34464 25 61862 70 17082 71 - 9.65289 63148 - 9.80203 39359 - 9.95127 52455 -10.100617572694 -10.25005 83482 29 83 81 21 8.53580 8.73819 8.94190 9.14690 9.35317 17842 86648 50606 41251 94376 76 33 84 84 01 50997 54469 70966 78390 55435 80 17 06 24 72 9.56071 9.76949 9.97950 10.19072 10.40315 49872 51583 47158 87913 28704 49 85 43 49 84 09 91 66 07 08 7.5 1.19213 1.30513 1.42127 1.54045 1.66258 51297 05 8858177 51595 43 17547 76 1463194 8.0 2: -10.39959 -10.54922 -10.69894 -10.84875 -10.99865 25 20 64 81 40 1.78758 1.91537 2.04588 2.17904 2.31478 18092 46664 59340 52440 56943 68 26 24 32 26 8.5 8.6 8.7 8.8 8.9 -11.14863 -11.29870 -11.44885 -11.59907 -11.74937 8155138 36905 72 02353 71 59405 42 90196 53 10.61676 10.83154 11.04748 11.26457 11.48278 27802 46772 50362 06394 85664 52 22 14 86 18 84 70 83 44 21 2.45304 2.59375 2.73687 2.88232 3.03007 36058 83010 19016 91437 72080 25 13 54 48 09 E! 9:2 -11.89975 77460 43 -12.050210450183 -12.20073 5517188 -12.35133 13844 58 -12.50199 65394 43 11.70212 11.92257 12.14411 12.36673 12.59043 61836 11355 13354 49565 04241 32 62 15 33 06 5.24665 34450 28 5.3905692519 72 5.53476 7188164 5.67923 54339 89 5.82396 2896129 3.18006 3.33224 3.48657 3.64299 3.80148 55643 58288 16324 84993 37357 29 43 07 84 79 -12.65272 95175 33 -12.80352 89000 52 -12.95439 33123 60 -13.10532 14220 44 -13.25631 19372 14 12.81518 13.04099 13.26783 13.49570 13.72459 64072 18113 57709 76423 69974 43 65 12 49 44 -13.40736 13.95449 36168 27 - 5.96893 91493 52 3.96198 63258 60 E E t:: 9.3 9.4 Xi Lx 9:9 10.0 36048 74 GAMMA FUNCTION GAMMA FUNCTION AND FOR RELATED 279 FUNCTIONS COMPLEX ARGUMENTS Table 6.7 1=1.2 Y In T(z) ?I tiln r(z) 0.0 - 0.08537 40900 03 8:: - 0.09169 75124 13 0.11050 89067 86 8:: - 0.18352 09532 62 0.14135 07443 57 0.5 - 0.23614 32688 51 0":; 0.8 0.9 - 0.36884 98509 35 0.29824 83560 49 - 0.44697 73864 90 - 0.53174 22756 96 1':: :*: - 0.71803 46814 87 0.62233 95313 44 - 0.81823 34133 78 0.92237 79303 20 1:4 - 1.0300106294 1.5 II*! - 1.14073 5234162 - 1.25421 01536 37 1.37015 22047 39 1:8 1.9 z 55'; 5:4 - 5.8073152672 85 5.95057 66519 39 6.09410 4721191 6.23788 94064 81 6.38192 11972 10 4.10609 4.26883 4.43349 4.60005 4.76847 87 13 28 16 71 5.5 5.6 5.7 5.8 5.9 - 6.52619 11003 6.67069 06038 6.8154116425 6.96034 65682 7.10548 81209 82 24 98 97 15 4.9387143339 5.11075 23127 5.28455 29803 5.46008 61980 5.63732 28266 56 64 68 02 55 0.10119 48344 0.07868 85726 0.04983 92764 0.01483 57562 0.026111520147 90 52 14 65 E! 6:2 - 7.25082 7.39636 7.54208 7.68798 7.83406 94030 38562 52390 76072 52949 54 29 70 47 57 5.81623 5.99679 6.17897 6.36275 6.54810 46788 44733 57929 30441 14200 41 73 16 11 83 61 38 01 48 35 6.5 6.6 - 1.48829 83245 09 - 1.60844 01578 57 0.07278 23932 0.12495 51937 0.1824121090 0.24494 25273 0.31234 49712 - 7.9803128978 8.12672 52570 8.27329 74450 8.42002 47512 8.56690 26702 26 99 10 17 20 6.73499 6.92341 7.11333 7.30473 7.49759 68651 60416 62984 56416 27064 55 24 34 32 69 0.38442 0.46101 0.54192 0.62700 0.71610 80719 09100 29484 37140 23338 73 87 31 16 39 7.0 x-32 214 - 1.73038 79144 93 1.85397 78680 87 - 1.97906 72374 32 - 2.23325 56848 33 2.10553 0137117 - 8.71392 8.86109 9.00839 9.15583 9.30340 74 24 89 37 98 7.69188 7188759 8.08470 8728319 8.48303 67310 75313 54778 14724 69297 43 86 77 22 94 2.5 2.6 - 2.36214 55727 43 - 2.4921123232 46 El 2:9 - 2.62307 77928 95 2.75497 19177 39 - 2.88773 16568 77 0.80907 0.90579 1.00612 1.10996 1.21718 69945 69 43715 71 9056143 29987 33 49784 62 - 9.45110 - 9.59892 - 9.74686 - 9.89492 -10.04309 37743 60 47746 01 64719 23 5764138 96669 84 8.68422 8.88673 9.09055 9.29565 9.50203 37525 82 4317155 14530 96 84265 39 89238 50 Z:! - 3.15562 00992 07 3.02130 57049 65 E 314 - 3.29066 53170 00 3.42636 16590 56 - 3.56269 77297 54 1.32769 1.44137 1.55815 1.67794 1.80064 01044 93510 91278 08829 07379 18 29 68 56 67 -10.19138 -lo:33977 -10.48828 -iO:63688 -10.78559 53082 31 9922146 08443 04 55067 01 1433166 9.70967 9.91855 10.12866 10.33998 10.55250 70361 72443 44054 37387 08134 08 36 34 77 40 3:: - 3.69962 90860 85 3.83710 32317 24 5-i 3:9 - 3.97512 5174107 4.11364 37264 61 - 4.25263 90859 57 1.92617 2.05448 2.18547 2.31908 2.45526 91533 49 0621184 33836 08 91746 67 29835 70 -10.93439 -11108329 -11.23229 -11.38138 -11.53055 62350 76070 33237 12352 92646 38 93 11 53 46 10.76620 10.98107 11.19709 11.41426 11.63257 15360 21389 91694 94790 02129 05 38 76 19 90 4.0 - 4.39208 75003 42 - 4.67225 69393 70 4.53196 69332 23 - 4.81293 84293 30 - 4.95399 36651 50 28374 96019 67976 04316 88424 55 03 01 86 26 -11.67982 54041 -11.82917 77123 -11.978614311170 -12.12813 33832 -12.27773 31694 57 44 44:: 4.3 4.4 2.59393 2.73503 2.87852 3.02434 3.17242 78 04 11.85198 12.07251 12.29413 12.51683 12.74059 88011 29482 06252 00607 97329 32 35 48 77 36 4.5 :-; 4:8 4.9 - 3.32274 3.47523 3.62985 3.78657 3.94533 25560 41545 81537 08902 04167 43 72 79 31 32 -12.42741 -12i57716 -12.72700 -12187690 -13.02688 19659 29 81225 64 0040142 61685 35 50046 68 12.96542 13.19130 13.41821 13.64615 13.87511 83615 49005 85311 86543 48849 35 92 47 64 16 5.0 - 5.80731 52672 85 -13.17693 50906 38 14.10507 70446 23 86 5.09540 60548 36 5.23716 00880 20 5.37924 58863 97 5.52163 1239193 5.66433 12381 00 - 0.00000 0.02865 0.05586 0.08025 0.10066 00000 84973 39903 91592 05658 00 21 67 09 03 5.0 5.1 - 0.11610 0.12588 0.12948 0.12663 0.11720 77219 00935 68069 80564 77278 + 9 In r(z) x In r (2) Y 4.10609 64053 70 6.3 6.4 fx 6:9 ;:: 8.8 8.9 10.0 68896 32795 78818 69016 66975 64053 70 00575 53 40204 01 23089 91 02,339 50 280 GAMMA Table 6.7 GAMMA FUNCTION AND RELATED FUNCTION FOR FUNCTIONS COMPLEX ARGUMENTS .x=1.3 Y :1 In r(z) E - 0.10817 61080 85 0.11383 48095 08 0:2 - 0.13070 20636 90 0":; - 0.19649 1008149 0.15843 12771 78 0.5 0:8 E - 0.24420 93680 45 - 0.30082 34434 02 - 0.36553 39002 27 0.43754 53407 19 0.9 - 0.51609 74046 40 ::; - 0.69006 45154 05 0.60048 62005 12 :*: 1:4 - 0.78427 03001 02 0.88259 13601 03 - 0.98458 61322 90 1.5 :*; - 1.08986 76158 16 - 1.30898 54162 82 1.19809 86148 04 1:s 1.9 - 1.42227 1923714 - 1.53773 44011 63 2.0 - 1.65517 68709 10 2Il - 1.77442 7143191 s-23 - 2.01776 34239 34 1.89533 14331 28 4ln r(z) din r(z) Y 4 In r(z) 5.0 - 5.645414138133 5.78673 23355 5.92835 35606 6.07026 64370 6.21246 02140 37 66 51 03 4.24823 90621 27 4.41126 31957 95 4.57620 66023 67 4.74303 39118 17 4.9117110050 12 0.06126 78750 55 0.06229 79103 48 0.05805 28252 04 0.04820 73993 35 0.03257 37450 94 5.5 - 6.35492 47217 66 6ii9765 03105 97 6.64062 79133 72 617838488113 55 6.92730 48028 21 5.08220 49501 77 5.25448 39434 72 5.4285172533 50 5.60427 51684 12 5.78172 89485 09 - 0.01107 52190 48 + 010162790894 04 0.04941 70710 23 0.08822 25250 96 0.13255 01649 50 6.0 6.1 - 7.07098 80742 52 7.21489 11938 62 7.35900 70872 13 7.50332 90147 58 7.64785 0551098 5.96085 07788 45 6.1416137268 52 6.32399 1701649 6.50795 94158 99 6.69349 23498 81 0.18223 70479 17 Oi2371109920 47 0.29699 65855 44 Oi3617193463 93 0.43110 85022 51 6.5 - 7.79256 55658 27 7.93746 82058 02 8.08255 28787 24 8.2278142379 13 8.37324 7168176 6.88056 67176 38 7.06915 94350 45 7.25924 80896 76 7.45081 09123 38 7.64382 67501 64 7.0 7.1 7.2 7.3 7.4 - 8.51884 67726 68 8.66460 83606 78 8.81052 74362 48 8.95659 96875 66 9.10282 09770 73 7.83827 50411 67 8.03413 57901 50 8.23138 95458 91 8.4300173795 19 8.63000 08640 04 7.5 - 9.24918 73322 19 9.39569 49368 29 9.54234 01230 14 9.68911 93636 11 9.83602 92650 88 8.83132 20546 97 9.03396 34708 43 9.23790 80780 23 9.44313 92714 58 9.64964 08601 22 - 0.00000 00000 00 0.0167199199 34 0.03225 84033 35 0.04549 95427 81 0.05544 82296 06 55': 5:s 5.9 z*: 6:4 2:4 - 2.14159 19646 87 0.50499 87656 67 0.58323 13926 09 0.66565 47394 67 0.75212 4475930 0.84250 35670 42 2.5 2.8 2.9 - 2.2667188222 04 2.39304 70725 18 2.52049 15659 37 2.64897 56799 18 2.77843 02497 03 0.93666 21049 03 1.03447 70464 53 1.13583 18965 15 1.2406163628 56 1.34872 60013 87 3-f - 2.90879 60402 26 3.04000 26554 06 312 - 3.1720186387 60 ;:: - 3.43825 31979 94 3.30478 64765 05 1.46006 18633 96 1.57453 01525 07 1.69204 18960 57 1.81251 26335 69 1.93586 21235 97 - 9.98306 65608 89 -10.13022 8105196 -10.27751 08670 60 -10.4249119248 88 -10.57242 84612 54 9.85739 70516 25 10.06639 24378 12 10.2766119810 47 10.48804 10011 24 10.70066 51627 91 ;*87 - 3.57239 88099 07 - 3.70717 37325 19 - 3.84254 9534695 3.97848 76469 59 3:9 - 4.11497 07016 98 2.0620140693 37 2.19089 58627 45 2.32243 83465 44 2.45657 55932 86 2.59324 47004 59 -10.72005 -10.86779 -11.01564 -11.16359 -11.31165 77580 15 71917 09 42292 16 64236 64 14105 63 10.91447 04638 39 11.12944 32237 30 11.34557 00727 24 11.56283 79415 00 11.78123 40512 20 2: - 4.38944 64012 12 4.25196 45543 38 t:: 4.4 - 4.52739 32778 30 4.66578 37904 84 - 4.80459 79774 65 2.73238 56006 34 2.87394 08855 80 3.01785 56433 48 3.16407 73073 22 3.31255 55163 23 -11.45980 -11.60806 -11.75641 -11.90485 -12.05339 6904159 06939 74 06415 49 46773 52 0797849 12.00074 59040 23 12.22136 12739 31 12.44306 8198138 12.66585 49686 64 12.88971 01243 51 4.5 t:: - 4.9438171850 33 - 5.22340 39564 94 5.08342 19323 42 -12.20201 7062734 -12.35073 15923 02 -12.49953 2564949 -12.64841 82148 10 -12779738 68295 12 13.11462 2443199 13.34058 09350 03 13.56757 48342 95 13.79559 35935 62 14.02462 68767 33 -12.94643 6748034 14.25466 45529 28 t:," - 5.36373 57615 52 5.5044110199 31 3.46324 19848 78 3.61609 03828 59 3.77105 62237 32 3.92809 67607 19 4.08717 08902 55 5.0 - 5.64541 41381 33 4.24823 90621 27 :*; 7:8 7.9 9.4 9.5 10.0 99*! 9:8 9.9 GAMMA GAMMA Y din r(z) i:: E - 0.12473 21357 76 0.119612914172 - 0.14000 5955189 0.16515 01552 88 0:4 - 0.19978 93616 12 22 - 0.24337 34438 09 0.29530 16779 62 8G 0:9 - 0.42158 4616110 0.35492 20669 55 - 0.49462 85345 46 1.0 1.1 :2 1:4 - 1.5 - 1.03605 77156 27 ::; 1.8 1.9 - 1.24542 54742 88 1.13933 63479 49 - 1.35407 41615 64 - 1.46505 2600714 3:: - 1.57816 32702 85 1.69322 14562 19 f-32 2:4 - 1.92859 03838 54 1.81008 23663 09 - 2.04863 37884 08 2.5 22:; - 2.17009 23032 73 - 2.29286 69947 58 2.41686 69570 17 2.8 2.9 - 2.5420100734 84 - 2.66822 19640 86 33:: - 2.79543 50784 95 2.92358 79116 75 E 314 E 3:7 0.57345 1292103 0.65748 16506 41 0.74620 06322 98 FUNCTION AND RELATED FUNCTION FOR COMPLEX Pin r(z) 0.00000 0.00597 0.01097 0.01405 0.01439 00 43 66 03 49 5.0 5.1 - 0.01124 72025 - 0.0040177865 + 0.00775 78473 0.02441 65124 0.04618 11610 18 38 84 32 42 5.5 5.6 2: 5:4 :*zI 5:9 Table ilRGUMENTS Xln r(z) Y 00000 40017 08056 93840 47989 - 281 FUNCTIONS 6.7 Xln r(z) - 5.48319 5.62258 5.76231 5.90236 6.04272 80511 51037 08530 26637 85898 50 75 59 68 90 4.38842 4.55177 4.71703 4.88416 5.05313 59888 72808 54898 59286 51119 87 10 14 80 86 - 6.18339 6.32435 6.46560 6.60711 6.74889 73257 81614 09417 60288 42683 62 11 01 99 24 5.2239106968 5.3964614275 5.57075 70829 5.74676 84279 5.92446 71670 84 35 41 33 92 0.07317 0.10545 0.14300 0.18575 0.23362 82199 58409 11986 57618 80933 73 92 37 52 40 6.0 - 6.89092 7.03320 7.17572 7.31847 7.46144 69567 58135 29534 08625 23750 80 18 78 98 25 6.10382 59013 94 6.2848180874 01 6.46741 79988 09 6.65160 0690196 6.83734 19628 28 0.28650 0.34425 0.40674 0.47383 0.54537 41540 53337 45404 07041 20299 26 92 87 21 26 22 --i-87-6:9 7.60463 7.74802 7.89163 8.03543 8.17942 06520 91624 16647 21899 50262 25 64 23 02 34 7.0246183323 7.21340 69984 7.40368 58155 7.59543 32663 7.78862 84351 73 03 67 20 12 0.62122 0.70126 0.78534 0.87333 0.96512 82885 81 23803 49 13608 50 70735 61 6499100 7.0 7.1 7.2 7.3 7.4 - 8.32360 47045 8.46796 59849 8.61250 38438 8.7572134627 8.90209 02169 82 44 82 90 54 7.98325 8.17928 8.37669 8.57548 8.77562 09839 11291 96196 77156 71692 40 83 29 28 98 1.06059 1.15962 1.26210 li36794 1.47704 19035 08468 60952 54704 16591 92 95 18 02 47 7.5 7.6 - 9.04712 9.19232 9.33767 9.48318 9.62883 96653 75409 97419 23233 14889 17 21 53 58 78 8.97710 9.17988 9.38397 9.58934 9.79598 02057 95050 81856 97875 82575 23 80 34 68 76 19987 82510 63729 62885 16678 43 60 35 89 10 8.0 - 3.05262 43245 92 3.18249 29542 71 - 3.31314 67001 61 1.58930 1.70463 1.82296 1.94420 2.06828 - 9.77462 - 9.92055 -10.06662 -10.21282 -10.35915 35841 50889 26112 28810 27447 76 05 05 76 20 10.00387 10.21300 10.42335 10.63490 10.84764 7934191 35337 97 01372 94 31773 72 84263 58 - 2.1951197123 13 2.32465 09517 70 2.45680 90502 77 2.59153 06235 98 2.72875 5067188 -10.50560 -10.65218 -10.79888 -10.94570 -11.09264 9159110 91868 81 99915 05 88327 39 30623 27 11.06157 11.27666 11.49290 11.71027 11.92878 19846 02694 00045 82098 21916 19 74 92 57 70 -11.23969 -11.38684 -11.53411 -11.68148 -11.82895 01199 75293 28946 38972 82920 39 27 97 65 01 12.14839 95336 12.3691180877 12.59092 59658 12.81381 15312 13.03776 33912 59 89 40 39 29 39045 86282 04214 73048 73587 38 47 52 02 55 13.26277 13.48882 13.71590 13.94401 14.17313 53 45 03 46 16 87212 03 0.93588 32199 04 0.83914 04638 21 3.44454 22757 38 3.70940 25331 00 3.57663 98160 21 3.84279 64130 02 3.97678 99482 49 44'10 - 4.24646 39012 69 4.11135 10946 79 2.86842 3.01048 3.15487 3.30155 3.45047 43947 56 30870 18 7950177 79836 24 42563 18 97913 94580 98712 92973 75662 2: 3-i 7:9 2: ::i 9.0 9.1 4: 2 - 4.38208 62246 51 44:: - 4.65479 74683 47 4.51820 56949 75 22 4:7 29" - 4.79184 09340 18 4.9293167880 70 - 5.06720 69267 30 - 5.20549 30732 30 5.34416 43497 23 3.60157 3.75482 3.91017 4.06758 4.22701 33 13 52 81 27 ~~~ 9:7 Z -11.97653 -12.12420 -12.27198 -12.41984 -12.56780 5.0 - 5.48319 8051150 4.38842 59888 87 10.0 -12.71585 ;*: 9:4 03893 15982 63127 40430 45087 14.40325 7632142 GAMMA 282 Table 3 111r(z) Y 0.0 0.1 3 0:4 0":: x*e7 0:9 1':: :*; 1:4 1.5 :*; 1:8 1.9 2.0 2.1 2.2 22:: E 217 ::t ;:: 3-c 314 3.5 ;:; GAMMA 6.7 FUNCTION FUNCTION AND FOR BELATED FUNCTIONS COMPLEX ARGUMENTS .f In r(z) 9 In r(z) 9 In r(z) - 0.12078 22376 35 0.12545 0392811 0.13938 53175 79 0.16238 37050 76 0.19412 35254 45 0.00000 00000 00 0.00378 68415 10 0.00839 39012 17 0.01460 80536 11 0.02315 34211 15 - 5.32063 00229 09 5.45809 92990 12 5.59594 21987 69 5.73414 48816 77 5.87269 42552 05 4.52667 02683 19 4.69038 4659451 4.85599 23475 89 5.02345 93914 30 5.19275 29984 42 - 0.23418 6347470 0.28208 36136 63 0.33728 34790 33 0.39923 5430120 0.46739 0870408 0.03466 89612 75 0.04969 46638 36 0.06866 64150 66 0.09191 83319 43 0.11969 06415 60 - 6.01157 79223 61 6.15078 41337 33 6.2903017435 55 6.43012 0169396 6.57022 9355139 5.36384 14702 24 5.53669 41510 65 5.71128 1379495 5.88757 44426 18 6.06554 55330 63 - 0.5412188685 47 0.6202170896 71 0.7039184698 97 0.79189 44573 28 0.88375 56946 74 0.15214 09934 52 0.18935 7309101 0.23137 07067 73 0.27816 75270 32 0.32969 99180 52 - 6.7106197369 14 6.85128 22117 36 6.99220 81085 67 7.13338 91616 09 7.27481 7485607 6.24516 77083 65 6.42641 48526 40 6.6092616403 83 6.79368 35022 65 6.97965 65928 01 - 0.97915 0939181 1.07776 48736 47 1.179315306181 1.28355 01134 19 1.39024 41643 92 0.38589 47712 67 0.44666 1020149 0.51189 54441 75 0.58148 71805 09 0.65532 1161093 - 7.41648 55529 97 7.55838 61727 29 7.7005124706 26 7.84285 7871149 7.9854160804 40 7.16715 77597 60 7.35616 45152 22 7.54665 50081 65 7.73860 79984 87 7.93200 28323 86 - 1.49919 6372585 1.61022 69592 23 1.72317 49667 28 1.83789 60327 96 1.95426 04180 71 0.73328 0681691 0.81524 92850 60 0.90111 2111692 0.99075 68430 94 1.08407 4337092 7.2 7.3 7.4 - 8.12818 10705 51 8.27114 70647 52 8.41430 85238 40 8.55766 01333 52 8.70119 67916 34 8.1268194190 02 8.32303 82082 45 8.52064 01697 48 8.71960 67728 67 8.9199199676 60 - 2.0721512706 83 2.19146 3106138 2.31210 04795 77 2.43397 68277 27 2.55701 34593 17 1.18095 90329 08 1.28130 91860 05 1.38502 69784 97 1.4920185397 98 1.60219 39035 70 7.5 7.6 7.7 7.8 7.9 - 8.84491 3598681 8.98880 5845698 9.13286 90053 22 9.27709 87224 65 9.42149 08057 13 9.12156 21668 12 9.3245162284 17 9.52876 54395 97 9.73429 35008 92 9.94108 45113 82 - 2.68113 86746 74 2.80628 69972 89 2.93239 85022 62 3.05941 82284 63 3.18729 56630 57 1.71546 69204 67 1.83175 5141118 1.95097 96800 61 2.07306 50684 28 2.19793 9101106 8.0 - 9.56604 12192 67 - 9.71074 60753 60 - 9.85560 1627136 -10.00060 42619 46 -10.14575 04950 41 10.14912 29545 01 10.35839 36845 06 10.56888 19135 53 10.78057 31993 69 10.99345 34334 60 - 3.31598 42885 64 3.44544 11840 65 3.57562 66733 10 3.70650 40135 44 3.83803 91197 27 2.32553 26824 38 2.45577 96733 92 2.588616742182 2.72398 32197 35 2.86182 0960836 8.5 -10.29103 69636 22 -10.43646 04212 40 -10.5820177325 09 -10.72770 5868109 -10.87352 19000 77 11.20750 88298 51 llii2272 59143 12 11.63909 15140 53 11.85659 27478 60 12.0752170166 56 - 3.97020 03195 93 4.10295 81356 26 4.23628 50905 75 4.37015 55336 09 4.50454 54845 89 3.00207 42115 08 3.14468 94828 47 3.2896154314 23 3.43680 2746151 3.58620 40415 07 9.0 9.1 -11.01946 -11.16552 -11.31170 -11.45800 -11.60442 29973 44 64215 28 95229 33 97367 84 45796 38 12.29495 19944 46 12.51578 56196 58 12.73770 60868 20 12.96070 18385 99 13.18476 1558147 - 4.63943 24943 00 4.77479 55187 51 4.9106148059 11 5.04687 17934 63 5i18354 90163 32 3.73777 37568 62 3.89146 80616 79 4.04724 47663 05 4.20506 32380 55 4.36488 43223 09 9.5 -11.75095 -11.89758 -12.04433 -12.19118 -12.33813 16459 94 86050 76 31977 78 32337 59 65886 95 13.40987 41617 61 13.63602 87918 31 13.86321 48100 75 14.09142 1791027 14.32063 95157 82 - 5.32063 00229 09 4.52667 02683 19 10.0 -12.48519 12016 51 14.55085 79659 84 2: 24' 2: 29" z*: 9:4 Z E GAMMA FUNCTION GAMMA FUNCTION 283 AND RELATED FUNCTIONS FOR COMPLEX ARGUMENTS Table 6.7 .r=1.6 Y tiln I?(z) o":! - 0.11687 93076 97 0.11259 17656 07 8.: 0:4 - 0.15085 38452 13 0.12968 70233 14 - 0.18012 29875 82 Go*: 0:7 - 0.21715 7659172 - 0.26155 99560 69 0.31289 07142 50 0":: - 0.43448 83847 40 0.37068 55339 80 1':: - 0.50382 21960 58 0.57825 58588 66 :s 1:4 - 0.74076 95833 44 0.65736 82809 61 - 0.82810 01661 20 ::'b - 0.91903 27864 52 1.01326 10002 05 119 :*i - 1.11052 43845 66 - 1.21057 08228 70 1.3131811150 50 22': - 1.41815 60399 85 1.525315986147 2:2 - 1.63449 89215 98 24' - 1.74555 85219 99 1.85836 24696 22 2.5 - 1.97279 09238 15 22:; - 2.20609 63358 24 2.08873 51557 10 f:! - 2.32478 44606 95 2.44471 74052 94 33:; S In r(z) %ln r(z) xln r(z) 0.00000 00000 0.01272 17953 0.02614 08547 0.04092 98346 0.0577147266 00 11 67 69 93 - 5.15767 38696 5.29324 00046 5.4292138858 5.56558 05247 5.70232 57347 89 70 50 67 10 4.66298 4.82709 4.99309 5.16092 5.33057 63139 40 8942123 00410 26 64732 77 61938 29 0.07705 74009 0.09944 39491 0.12527 90746 0.15488 59553 0.1885104588 90 75 90 99 87 - 5.83943 5.97689 6.11470 6.25283 6.39128 60752 88014 18170 36319 33226 49 04 24 59 66 5.50200 5.67519 5.85009 6.02670 6.20497 8200133 24850 30 99922 08 25740 71 29518 79 0.22632 0.26845 0.31495 0.36583 0.42110 8363144 42738 89 11405 00 95580 78 63293 75 - 6.53004 6.66909 6.80843 6.94806 7.08795 04959 52554 81708 02492 29088 33 28 20 33 41 6.38488 46780 6.56641 21003 6.74953 03284 6.934215201179 7.12044 32570 25 0.48071 0.54459 0.61268 0.68490 0.76114 2003131 72874 22 83586 73 11588 51 48080 60 - 7.22810 79544 7.3685175545 7.50917 42208 7.65007 07879 7.79120 03956 00 64 19 17 68 7.30819 7.49743 7.68816 7.88034 8.07395 17047 83963 18010 09808 55670 52 44 64 67 43 0.84132 0.92534 1.01310 1.10450 1.19945 42695 23984 14934 44515 56127 09 61 56 88 07 - 7.93255 8.07413 8.21592 8.35792 8.50012 64719 27171 30888 17887 32493 90 08 20 32 99 8.26898 8.46541 8.66321 8.86237 9.06288 57380 21983 61583 93155 38358 27 05 45 10 78 1.29786 1.39963 1.50467 1.61290 1.72424 13618 05453 47448 84436 91120 36 39 81 93 48 - 8.64252 21222 8.7851132665 8.92789 17384 9.07085 27813 9.21399 18168 97 62 38 87 02 9.26471 9.46784 9.67227 9.87797 10.08493 23369 78710 39098 43290 33943 30 61 48 61 44 - 2.56582 37258 46 2.68802 00865 40 - 2.81126 51983 53 - 3.06063 64586 59 2.93548 4033169 1.8386172327 21 1.95593 62824 65 2.07613 26817 55 2.19913 5722155 2.32487 74784 17 - 9.35730 9.50078 9.64443 9.78824 9.93220 44352 63884 35813 20648 80292 92 89 39 48 58 10.29313 10.50256 10.71321 10.92505 11.13808 57475 63937 06886 43268 33302 61 51 60 31 08 3.; 3:8 3.9 - 2.45329 27106 2.5843187608 2.71789 54457 2.85396 49506 2.9924717222 82 00 96 80 46 -10.07632 77975 -10.22059 78196 -10.3650146660 -10.50957 50232 -10.65427 56879 98 20 67 55 66 11.35228 11.56764 11.78414 12.00178 12.22054 40372 30924 74364 42963 11767 42 55 58 80 06 4.0 - 3.82845 04545 47 2: - 3.95888 63415 67 4.08994 07464 23 t:: - 4.35379 66759 90 4.22158 61190 32 3.13336 3.27658 3.42209 3.56983 3.71976 23649 55399 18672 38320 56948 89 89 73 36 92 -10.79911 -10.94408 -11.08918 -11.23442 -11.37977 35626 56506 90522 09602 86562 11 53 76 86 21 12.44040 12.66136 12.88341 13.10652 13.33070 58504 89 63509 22 09632 56 8217040 68786 75 34062 45248 81404 48000 64388 62 92 46 81 72 9.5 -11.52525 95066 -11.67086 09597 -11.81658 05418 -11.9624158543 -12.10836 45707 64 45 21 24 60 13.55593 13.78220 14.00950 14.23781 14.46714 59442 46327 23791 88286 38298 10.0 c: 3:4 3.5 3.18665 85710 48 3.31351 44463 00 3.44115 94046 31 3.56955 42495 22 3.69866 25626 62 4.5 - 4.48654 82548 65 44'7b - 4.6198181847 33 4.75358 51673 38 4:8 4.9 - 4.88782 91705 81 - 5.02253 13317 74 3.87184 4.02602 4.18226 4.34053 4.50078 5.0 - 5.15767 38696 89 4.66298 63139 40 E E -12.25442 44338 60 37 90 11 57 06 60 23 57 14.69746 74295 03 284 GAMMA GAMMA Table 6.7 FUNCTION FUNCTION AND RELATED FUNCTIONS FOR COMPLEX ARGUMENTS .x=1.7 J In I'(z) x In r(z) Y %ln r(z) Y 9 In r(z) - 4.99429 5.12797 5.26209 5.39663 5.53159 42740 31077 29486 77210 21994 24 01 79 79 12 4.79738 4.96193 5.12834 5.29658 5.46661 98064 49448 25830 04404 72692 83 06 82 85 29 - 5.66694 5.80267 5.93877 6.07522 6.21202 19505 32805 31855 93070 98903 53 14 28 61 76 5.63842 5.81196 5.98722 6.16416 6.34275 28098 55 7748103 36749 88 30480 45 91548 66 91243 32455 44205 88915 05624 57 42 39 80 82 - 6.34916 6.48662 6.62438 6.76246 6.90082 37463 02160 91385 08200 60067 25 75 04 42 27 6.52298 6.70481 6.88823 7.07320 7.25970 60784 86640 24881 38287 96365 05 24 89 20 25 0.57124 0.63832 0.70935 0.78428 0.86303 72307 60866 84280 36123 23052 84 03 02 89 04 - 7.03947 7.17840 7.31759 7.45705 7.59675 58582 98 1924147 61209 77 07120 18 82876 82 7.44772 7.63723 7.82821 8.02063 8.21449 75087 56630 29137 86480 28045 22 84 39 35 37 36116 09 46369 05 8541153 87389 10 18623 15 0.94552 1.03169 1.12144 1.21470 1.31138 91079 51 46541 37 7259194 42030 73 27144 41 - 7.7367117475 7.87690 42834 8.01732 93640 8.15798 07198 8.29885 23295 34 81 69 22 23 8.40975 8.60640 8.80443 9.00381 9.20452 58520 87697 30279 05701 37958 62 25 13 63 73 2.5 2.6 2.7 - 1.8719164452 44 - 1.98459 17246 80 - 2.09876 6557199 ;:t - 2.21434 84448 82 2.33125 26629 53 1.41140 1.51467 1.62113 1.73069 1.84327 07152 73744 35114 18813 73680 26 45 76 34 71 - 8.43993 84073 80 8.58123 33910 02 8.72273 1930122 8.86442 88760 30 9.0063192716 38 9.40655 9.60988 9.81450 10.02039 10.22753 55438 90763 80646 65738 90498 14 93 38 46 84 3.0 3.1 3.2 3.3 3.4 - 2.44940 2.56872 2.68915 2.81062 2.93309 14805 34658 28670 90603 60594 61 89 03 59 79 1.95881 2:07724 2.19847 2.32246 2.44914 7107134 0553198 95064 74 81077 41 28100 87 - 9.14839 9.29066 9.43310 9.57572 9.71851 8342151 14862 98 42680 75 24089 73 17806 54 10.43592 10.64552 10.85634 11.06835 11.28154 03060 55107 01750 01418 15743 85 28 59 23 00 3.5 3.6 3.7 3.8 3.9 - 3.05650 3.18079 3.30594 3.43189 3.55860 20770 24 9134133 27115 93 14379 84 68105 24 2.57844 2171030 2.84468 2i98150 3.12073 23336 16 76079 67 17064 22 97744 80 8955142 - 9.86146 -10.00458 -10.14786 -10.29130 -10.43489 83980 84128 81072 38884 22827 47 32 85 74 58 11.49590 09457 89 11.7114150295 52 11.92807 0889158 12.14585 58692 46 12.36475 75866 47 4.0 4.1 f-G - 3.68605 29448 47 - 3.81419 63503 82 - 4.07245 57284 59 3.94300 17902 13 414 - 4.20250 70933 22 -10.57862 -10.72251 -10.86654 -11.01070 -11.15500 99305 35816 00900 64100 95918 96 27 14 32 83 12.58476 12.80586 13.02804 13.25129 13.47560 39218 30109 32377 32259 18323 4.5 4.6 - 4.33314 58930 01 - 4.46434 40087 52 :*s7 4:9 -11.29944 67777 -11.44401 51979 -11.5887121674 -11.73353 50824 -11.87848 14J72 28 25 47 91 43 13.70095 13.92735 14.15477 14.38320 14.61264 81399 16 14505 47 1279190 73474 23 95775 51 5.0 Oil 0.2 - 0.09580 76974 0.09977 01624 0.1116135203 0.13120 82417 0.15834 67099 07 55 43 20 43 0.00000 0.02095 0.04250 0.06524 0.08970 00000 00 5310147 9978199 48506 20 54480 34 0.5 0.6 0.7 0.8 0.9 - 0.19275 0.23410 0.28203 0.33614 0.39604 43 11 30 35 33 0.11638 0.14573 0.17810 0.21382 0.25312 82473 09476 70108 42284 66649 1.0 - 0.46133 2644119 2: - 0.53162 06562 78 0.60653 43029 30 ::: - 0.68572 93610 37 0.76884 05552 19 0.29619 0.34317 0.39413 0.44912 0.50817 1.5 1.6 1.7 1.8 1.9 - 0.8556148134 0.94573 52538 1.03895 26210 1.13503 13039 1.23375 66975 32 42 76 83 90 2.6 2.1 2.2 2.3 2.4 - 1.3?493 1.43838 1.54394 1.65147 1.76084 0.0 44989 41754 01468 32007 36829 3.2623183125 3.40619 87555 3.55233 29614 3.70067 53013 3.8511817677 99 93 33 46 02 5.0 52 5:3 5.4 6.9 :*: 7:2 7.3 ta: 9:3 9.4 9.5 - 4.72832 87027 47 4.59607 85697 79 - 4.86107 34372 26 4.00380 99034 45 4.1585187339 90 4.31526 87017 23 4.47402 1603194 4.63474 05290 18 - 4.99429 42740 24 4.79738 98064 85 10.0 99:; E -12.02354 j?f208 09 85 28 88 97 91 81 93 08 06 86 14.84308 80868 68 GAMMA GAMMA FUNCTION AND RELATED FUNCTION FOR COMPLEX 285 FUNCTIONS ARGUMENTS Table 6.7 .r=1.8 .#!‘I Y .f In r(z) r(z) 0":: - 0.07108 38729 86 0.07476 57386 14 t: 0:4 - 0.08577 55297 09 0.10400 76857 32 - 0.12929 22486 30 265 - 0.20006 82029 53 0.16140 31015 52 Kl 0:9 0.00000 0.02858 0.05769 0.08782 0.11946 tiln r(z) 00000 00 6333136 29209 31 58538 91 40495 57 - 4.83045 4.96226 5.09454 5.22728 5.36046 ./In r(z) 6845113 53555 54 72216 70 53433 89 35143 73 4.92989 5.09490 5.26176 5.43043 5.60088 76263 86275 50781 56009 97905 84 80 04 62 12 63619 92920 84380 06145 32737 68 13 56 63 64 5.77309 81726 5.94703 21669 6.12266 40498 6.29996 69207 6.478914668158 78 16 86 68 - 6.16796 44658 6.30383 28019 6.44003 74202 6.57656 79546 6.7134145046 02 05 92 04 23 6.65948 6.84164 7.02537 7.21065 7.39746 19384 41059 72437 80966 40550 99 65 42 53 43 - 0.24498 08149 51 0.29581 07721 71 - 0.3522150054 25 0.15304 83729 0.18897 35429 0.22758 31014 0.2691673612 0.31396 39650 82 70 17 58 50 ??I --- 5.49406 5.62807 5.76248 5’78 6.03244 - 5.89728 5:9 - ::i! 1.2 1.3 lI4 - 52 74 74 43 03 0.36216 0.41389 0.46927 0.52836 0.59119 05120 86472 90315 66950 63857 09 00 88 54 23 6.0 6.1 ::z :*7a - 0.8751145440 80 0.78884 75850 57 - 0.96452 30468 26 1.05684 8311180 1:9 - 1.15188 37223 02 0.65777 0.72809 0.80213 0.87984 0.96117 76436 94297 42229 15616 10434 6fi 11 48 08 30 - 6.85056 76090 92 6.9880182204 65 7.12575 76814 17 7.26377 77029 87 7.40207 0344198 7.58577 31298 7.77556 39290 7.9668156346 8.15950 79813 8.35362 12360 85 39 11 46 30 f-01 2:2 ;:i - 1.24943 97659 29 1.34934 28469 99 - 1.45143 40669 35 - 1.55556 80105 11 1.661611576122 1.04606 1.13445 1.22628 1.32148 1.41997 48267 65 96865 98 8684172 25078 65 05387 49 - 7.54062 79930 7.67944 33488 7.81850 94055 7.9578194361 8.09736 69787 63 49 06 78 03 8.54913 8.74603 8.94429 9.14390 9.34484 15 54 74 64 25 2.5 2;6 2.7 - 1.76944 28703 84 - li87895 01786 38 - 1.9900310163 61 22:: - 2.10259 12619 95 2.21654 43688 12 1.52168 li62654 1.73449 1.84544 1.95935 16884 50508 04020 85788 17594 90 69 35 28 45 - 8.23714 58220 8.37714 99935 8.51737 37469 8.6578115513 8.79845 80804 35 16 39 42 75 9.54709 9.75064 9.95547 10.16156 10.36890 7034142 48917 54 27618 74 49130 30 59844 02 3.0 Z - 2.3318106516 27 - 2.56599 45147 78 2.4483166432 13 2.07613 36663 2.19572 97074 2.31807 70690 2.4431147704 2.57078 36890 29 49 52 17 62 8.0 - 8.93930 9.08035 9.22159 9.36303 9.50464 82029 08 69727 14 96207 08 1546181 8309120 10.57748 10.78727 10.99827 11.21046 11.42383 09733 52232 44124 45427 19293 12 56 32 62 59 2.70102 65631 2.83378 79764 2.9690143304 3.10665 38058 3.24665 63186 50 90 05 79 51 8.5 31904 52376 52667 07487 94213 38 37 34 37 31 0.48036 0.41384 0.55143 0.62673 0.70596 32669 67690 15880 30272 59713 ;:'4 - 2.8046197009 41 2.68478 15548 53 ;*56 3:7 - 3.04723 78253 19 2.92545 51190 42 - 3.16992 13469 31 E - 3.29346 24159 89 3.41782 06949 39 4.0 4.1 - 3.54295 85286 89 - 3.66884 07212 13 5.; 4:4 - 3.92270 43338 26 3.79543 85028 21 - 4.05063 42744 24 4.5 - 4.17918 44552 05 2: 6.4 7.8 7.9 i:: 8.3 8.4 27” 29” - 9.64644 56228 - 9.788419347163 - 9.93056 54816 -10.07288 01596 -10.21535 96421 63 43 06 85 11.63836 11.85404 12.07086 12.28881 12.50786 61778 40794 66893 62145 53042 3.38897 3.53355 3.68036 3.82935 3.98047 34693 93 84906 21 61916 47 29025 75 6418131 -10.35800 -10.50079 -10.64375 -10.78685 -10.93010 03128 86719 13321 50132 65382 01 24 05 67 43 12.72802 12.94927 13.17160 13.39499 13.61944 92806 69 85734 79 57894 90 9654143 91215 87 44'76 - 4.30833 34763 48 4.43805 72703 06 4:8 - 4.56833 31585 96 4.9 - 4.69913 97495 61 4.13369 4.28897 4.44626 4.60554 4.76676 59419 20315 65448 25879 44644 -11.07350 28285 -11.21704 09003 -11.3607178605 -11.50453 09034 -11.64847 73069 39 12 47 33 06 13.84494 14.07147 14.29902 14.52758 14.75715 33679 17848 39730 97368 90776 5.0 4.92989 76263 84 - 4.83045 6845113 14 17 66 92 38 10.0 -11.79255 44293 69 42 17 75 21 29 14.98772 21889 61 286 Table GAMMA GAMMA 6.7 FUNCTION FUNCTION AND FOR RELATED FUNCTIONS COMPLEX ARGUMENTS .r=1.9 Y tiln I‘(z) ::"I - 0.03898 42759 23 0.04242 16648 18 ia; 0:4 - 0.06974 4359613 0.05270 5307116 - 0.09340 38158 25 0.5 ::"7 - 0.12349 16727 26 - 0.15978 08372 82 0.20201 20244 30 ::9" - 0.24990 35004 09 0.30315 95035 34 1.0 - 0.36147 78527 10 ::: ::: - 0.49209 86372 39 0.42455 6462111 - 0.5638171504 98 0.63943 71834 20 :2 - 0.71869 54698 30 0.80135 82795 42 1:7 1.8 1.9 - 0.88717 97447 03 - 0.97595 80247 42 - 1.06749 27687 53 2.0 2.1 - 1.16160 13318 68 - 1.25811 51641 83 z-z 2: 4 - 1.35687 89195 14 1.45774 95259 72 - 1.56059 52554 63 2.5 - 1.66529 48176 11 ;:; - 1.77173 64947 51 1.8798173280 00 ;:: - 2.10052 39332 80 1.98944 23595 16 3.0 3.1 - 2.21298 10520 42 - 2.32673 87919 77 ;:; 3.4 - 2.44172 77675 72 2.55788 36468 15 - 2.67514 6711148 3.5 3.6 3.7 ;:9" - 2.79346 2.91277 3.03304 3.27625 3.15421 14569 62346 29224 54337 66305 4.0 4.1 4.2 - 3.39912 3.52277 3.64718 3.77231 3.89813 4.5 416 4.7 4:8 4.9 - 4.02462 4115174 4.27948 4.40780 4.53669 5.0 - 4.66612 81728 77 X In r(z) %ln r(z) 9 In r(z) 0.00000 0.03569 0.07184 0.10889 0.14726 00000 47077 49288 51730 87453 00 36 73 33 39 - 4.66612 81728 4.79608 44074 4.92654 53878 5.05749 30552 5.1889102823 77 24 64 47 51 5.06052 5.22603 5.39337 5.56250 5.73340 77830 70297 36626 72499 82679 38 75 27 47 93 0.18735 0.22952 0.27407 0.32128 0.37139 90383 28050 56544 97690 36389 60 02 06 64 55 - 5.32078 5.45308 5.58582 5.71896 5.85249 0812105 92008 98 07663 21 15389 41 82177 50 5.90604 6.08039 6.25643 6.43412 6.61345 80662 88340 35684 60432 07797 49 38 02 49 49 0.42457 34706 0.48097 58618 0.5407113247 0.60385 82827 0.67046 72268 81 37 70 52 81 - 5.9864181289 6.12070 91879 6.25535 98637 6.39035 91465 6.52569 65169 78 56 85 66 71 6.79438 6.97689 7.16097 7.34658 7.53371 30179 86894 43917 73625 54565 35 96 16 14 59 0.74056 0.81415 0.89123 0.97177 1.05574 4797147 76239 52 58296 55 6140147 45936 43 - 6.66136 6.79734 6.93363 7.07023 7.20711 19179 75 57285 54 87392 01 2129112 74449 04 7.72233 7:9124j 8.10397 8129695 8.49134 71224 13866 78029 64920 80626 13 57 64 80 65 1.14309 1.23379 1.32776 1.42496 1.52533 88592 01934 50714 65323 52787 34 57 39 75 28 - 7.34428 7.48173 7.61944 7.75742 7.89565 65807 17598 55170 06825 03667 56 49 18 11 87 8.68713 8.88429 9.08281 9.28267 9.48385 36229 47573 35092 23655 42409 72 07 45 74 11 1.6288105662 1.73533 09179 1.84483 46926 1.95726 05315 2.07254 77068 06 80 69 67 08 - 8.d3412 8.17284 8.31180 8.45098 8.59039 79462 70499 15468 55343 33269 62 43 79 75 14 9.68634 9.89012 10.09517 10.30148 10.50903 24629 07585 32398 43916 90590 88 45 33 76 64 2.19063 2.31146 2.43498 2.56113 2.68985 63887 78475 46022 05263 09205 13 36 00 98 60 - 8.7300194457 8.86985 86090 9.00990 57226 9.15015 58714 9.29060 43111 32 10 31 69 75 10.71782 24352 lo:92782 00504 11.1390177608 11.35140 17379 11.56495 84588 78 91 39 39 29 24 38 14 96 10 2.82109 2.95480 3.09093 3.22943 3.37025 25566 37012 41220 50808 93162 19 40 91 91 16 - 9.43124 9.57207 9.71309 9.85429 9.99566 23 85 13 97 75 11.77967 11.99553 12.21253 12.43065 12.64988 46963 75096 42358 24807 01110 13 87 42 06 27 01294 40173 27007 39057 73167 42 08 49 84 71 3.51336 3.65869 3.80622 3.95589 4.10767 10185 57993 06560 39339 52859 24 21 50 63 66 -10.137213025172 -10.27892 94790 52 -10.4208115703 58 -10.56285 5789126 -10.70505 87350 54 12.87020 13.09161 13.31410 13.53765 13.76225 52464 62520 17307 05165 16677 75 42 41 78 85 44269 84023 39577 72434 57418 53 59 56 44 38 4.26152 4.41740 4.57528 4.73511 4.89687 56312 71132 30577 79308 72979 41 72 67 60 01 -10.8474171130 -10.98992 77287 -11.13258 74849 -11.27539 33771 -11.41834 24904 13.98789 14.21456 14.44226 14.67096 14.90067 44603 83815 31243 85811 48382 16 73 75 36 65 5.06052 77830 38 -11.56143 64604 78935 43338 16464 58330 08 64 48 93 66 19955 88 15.13137 21707 60 GAMMA GAMMA FUNCTION FUNCTION AND FOR RELATED 287 FUNCTIONS COMPLEX ARGUMENTS Table 6.7 .,-=2.0 %ln Y 0.0 r(z) 9 In r(z) - 0.00000 0.00322 0.01286 0.02885 0.05107 00000 26151 59357 74027 93722 00 39 41 79 62 Ei :c!i - 0.11354 0.07937 0.19863 0.15338 77183 37235 06626 06308 30 40 81 31 0:9 - 0.24904 17059 66 ::!l - 0.36428 0.30434 96090 77010 22 76 :*: 1:4 - 0.49700 0.42859 0.56926 21701 14442 99322 42 52 58 1.5 1.6 0.64515 0.72443 0.80688 55533 19760 50339 76 33 42 :*: 119 - 0.98053 0.89231 03476 37613 69 78 2.0 - 1.07135 98302 14 00000 57120 33372 61223 05507 00 74 06 10 97 0.21958 0.26767 0.31789 0.37051 0.42574 93100 56897 96132 53392 0726144 95 80 02 47 - 4.50127 - 4.62939 - 4.75805 - 4.88723 - 5.01690 5.5 5.6 5.7 2:: 0.48375 78429 0.5447146524 0.60872 74700 0.67588 39160 0.74624 61166 30 35 17 88 63 0.81985 0.89672 0.97687 1.06028 1.14693 67 63 07 26 53 39537 82178 35612 11909 12720 6.0 6.1 66’; 6:4 6.5 6:8 6.9 42 82 - 5.14705 5.27767 5.40874 5.54024 5.67217 - 5.80450 5.93722 6.07033 6.20381 6.33765 76 33 5.18929 5.35533 5.52318 5.69281 5.86418 93415 82031 54439 16137 81052 60 27 62 11 00 75299 60518 39987 66615 00274 57 81 03 82 24 6.03728 6.21208 6.38854 6.56665 6.74637 71248 16640 54709 30238 95048 73 30 43 56 97 07366 60439 37820 23278 05713 29 25 31 98 36 6.92770 7.11059 7.29503 7.48100 7.66847 07748 33491 43738 16040 33815 95 13 76 81 76 - 6.47183 78858 6.60636 41013 6.7412194789 6.87639 46872 7.01188 07803 22 16 19 45 50 7.85742 8.04784 8.23970 8.43299 8.62768 86143 67567 77898 22035 09788 76 00 07 86 99 8.82375 9.02119 9.21999 9.42011 9.62155 55706 78914 02960 55664 68973 27 05 14 09 45 1.23679 5034104 1.32983 65907 1.4260144920 1.52528 30352 1.62759 33595 26 94 04 36 7.0 - 7.14766 7.28375 7.42012 7.55676 7.69368 9177118 16419 02668 74543 59017 82 81 62 46 1.73289 43555 1.84113 34120 1.95225 70264 2.0662112994 2.18294 23322 35 22 63 71 91 - 7.83086 7.96830 8.10599 8.24393 8.38211 85862 87511 98924 57468 02798 69 38 36 08 83 9.82429 10.02832 10.2336151072 10.44016 10.64794 78825 25016 04128 34810 87 83 54 09 35 2.30239 2.42452 2.54926 2.67657 2.80639 65434 09185 32043 20582 71597 67 18 52 60 50 - 8.5205176753 8.65915 23247 8.79800 88177 8.93708 19330 9.07636 66296 67 82 87 47 28 10.85694 11.06716 11.27857 11.49116 11.70492 97125 48351 48933 62386 55194 60 59 86 10 45 2.93868 3.07340 3.21048 3.34989 3.49158 92920 03990 3622188 33215 50837 59 47 9.21585 9.35555 9.49544 9.63552 9.77579 80388 14566 23361 6281184 90392 55 37 92 16 57 - 11 11.9i983 12.13589 12.35308 12.57138 12.79079 96725 59137 17297 48693 33364 52 86 01 62 76 57202 32567 69172 71023 53645 41 78 45 23 81 - 9.91625 -10.05689 -10.19770 -10.33869 -10.47985 64956 46678 96994 78553 55166 49 12 20 49 49 13.01129 13.23287 13.45553 13.67924 13.90401 53818 94959 44022 90499 26078 23 63 19 21 95 -10.62117 91758 -10.76266 54322 -10.904310988175 -11: 04611 26442 -11.18806 72959 12 81 29 27 14.12981 14.35664 14.58449 14.81334 15.04319 4458193 41900 15940 66565 95540 46 42 09 92 -11.33017 27 15.27404 06485 34 - 1.2602188108 96040 1.16463 76 42 1.45772 1.35795 76568 6696157 48 2.5 2.6 c87 -- 1.55940 1.66288 1.76806 1.87483 61080 49866 06566 80234 2:9 - 1.98312 8963102 2.09285 17530 93 33-i! 3:2 ::'4 -- 2.20393 2.31629 2.54462 2.42987 48844 05460 26813 92551 77 64 37 03 ;:2 2; - 2.77736 2.66046 3.01410 2.89525 83499 32717 62029 79709 84 73 30 78 3.9 - 3.13386 46968 42 44:! - 3.25449 3.37595 2871145 81 29213 2: -- 3.62122 3.49820 74039 88720 59 03 4:4 - 3.74497 69383 89 3.63551 3.78164 3.92992 4.08032 4.23280 t:; 4.9 - 3.86942 77912 3.99455 19873 4.12032 31366 4.2467163216 4.37370 79930 99 65 90 20 a7 4.38732 4.54384 4.70234 4.86276 5.02509 43808 79226 08252 89562 91831 43 20 48 20 32 9:9 5.0 - 4.50127 42 5.18929 93415 60 10.0 58755 50755 88796 7022252 13522 38831 i-76 2: 22:: 61 52 65 17 X In r(z) 9 In r(z) 0.00000 0.04234 0.08509 0.12863 0.17335 ;:‘6 x 19298 288 GAMMA Table 6.8 FUNCTION DIGAMMA -A(z) AND FUNCTION x-1.0 .“*(z) 0.00000 -0.57121 56649 -0.56529 -0.53073 -0.47675 -0.40786 77902 04055 48934 79442 0.16342 0.32064 0.46653 0.59770 b3572 65809 0.71269 0.81160 0.89563 0.96655 1.02628 -0.32888 -0.24419 -0.15733 -0.07088 +0.01345 61258 34022 20154 RELATED FOR s ::i :-: FUNCTIONS COMPLEX .lti(z) 1.61278 1.63245 1.65175 0.44469 0.50420 0.56072 0.61448 0.66570 1.79408 1.81053 1.82672 1.84265 1.85833 08018 44105 28842 45939 75219 1.48746 1.48883 1.49015 1.49143 1.49267 92858 71602 80964 87422 54582 1.49387 1.49504 1.49617 1.49727 1.49833 43346 12062 10799 45959 1.49937 1.50037 1.50135 1.50230 1.50323 2.06820 71585 35171 82397 57159 01717 1.50413 1.50501 1.50586 1.50669 1.50751 is 814 2.08074 2.09313 2.10537 2.11746 2.12941 56749 61434 53524 69410 44191 1.50830 1.50907 1.50982 1.51056 1.51127 88:; 2.14122 2.15289 11731 04718 287 8:9 2.16442 54716 2.17582 2.18710 46687 1.51197 1.51266 1.51332 1.51398 1.51462 2.19825 2.20928 2.22018 2.23097 2.24165 46616 19555 92160 90229 38740 1.51524 1.51585 1.51645 1.51703 1.51760 2.25221 2.26266 2.27301 2.28325 2.29338 61882 83093 25085 09877 58823 1.51816 1.51871 1.51925 1.51978 1.52029 2.30341 92637 1.52080 5.8 5.9 1.23772 1.25843 1.27675 1.29306 1.30766 6.5 2 6:8 6.9 1.81377 1.88898 1.90396 1.91872 1.93327 :*i 712 7.3 7.4 1.94761 1.96175 1.97569 1.98944 2.00300 0.80607 0.84899 0.89021 54079 42662 1.32081 1.33271 1.34353 1.35341 1.36246 0.92985 0;96803 1.00485 1.04039 1.07474 78387 70243 21252 40175 51976 1.37080 1.37849 1.38561 1.39222 1.39838 1.10798 1.1401b 1.17137 1.20164 1.23104 07107 89703 24783 84581 94107 1.40413 1.40951 1.41455 1.41928 1.42374 1.25962 1.28741 1.31446 1.34081 1.36649 36033 54995 61381 34679 26435 1.42794 1.43191 1.43566 1.43922 1.44259 1.39153 1.41597 1.43983 1.46314 1.48593 62879 41255 61892 70060 17620 1.44580 1.44885 1.45175 1.45452 1.45716 34505 1.50821 1.53001 1.55135 1.57224 1.59272 24197 88550 07370 1.45969 1.46210 1.46441 1.46663 1.46876 1.61278 48446 1.47080 36052 1.47464 1.47646 1.47820 1.47989 1.48151 1.48308 1.48459 1.48605 79402 34618 00645 06554 39172 14328 84807 0.71459 0.76132 1.47276 25175 90807 25988 32133 28134 65515 1.47080 1.70751 1.72543 1.74303 1.76034 1.77735 :*z 517 b.0 6.1 20906 20861 1.68926 1.07667 1.11938 1.15580 [email protected] 1.21413 03206 05426 48446 b9889 <f*(z) 42228 67162 5:4 1.67068 15154 0.09465 0.17219 0.24588 0.31576 0.38196 ARGUMENTS t-G 6:4 ::2 x 719 X:! x 9:2 ;:i 8:: 99.87 9:9 10.0 2.01638 2.02959 2.04262 2.05549 21681 16663 92217 +(z) to 5D, computed by M. Goldstein, Los Alainos Scientific Laboratory. AUXILIARY 0.09 0.08 0.07 0.06 .54(Y) 0.00100 0.00083 0.00067 0.00053 0.00040 0.00030 FUNCTION FOR :&(l+iy) J;(Y) <Y> 956 417 555 368 853 011 9 1110 :: 0.03 0.02 0.01 <y>=nearest integer to y. 0.00020 0.00013 0.00007 0.00003 0.00000 0.00000 CY> 839 335 501 333 833 000 :so 53: 100 GAMMA FUNCTION DIGAMMA FUNCTION AND RELATED FUNCTIONS FOR C( IMPLEX x=1.1 m(z) .fGZ) 1.61498 1.45097 1.63457 1.45332 1.65378 1.45557 ;.p; . 1.45774 1.45983 0.63764 ::2 0.73229 :*I: 0.81484 0.88630 0.94792 5:9 1.0 0.14255 1.00102 0.21327 1.04687 2: 2-7 0.28131 1.08660 6:2 1.12119 ::i F%% 1.15146 . 0.46829 1.17810 0.52507 1.20169 22 0.57930 1.22269 0.63111 1;24148 2 1:9 0.68067 1.25839 6:9 0.72813 1.27368 7.0 0.77363 1.28755 7.1 0.81730 1.30021 7.2 0.85928 1.31179 7.3 214 0.89967 1.32243 7.4 1.72718 1.70933 1.46184 ;.;z;;; * 1.46565 1.46378 1.89025 1.47996 1.87508 1.47857 1.91992 1.48132 1.90519 1.93443 1.48263 1.48391 ;-;z;;; 1.48515 1.48635 1197675 1.48752 1.99047 1.48866 2.00401 1.48977 ::i 2.5 ::76 1.33224 1.34131 7.5 ::; 1.34972 1.35753 1.36482 ::: 1.37162 1.37800 i-1" 1.38398 812 1.38960 1.39489 88:: 1.49085 2.01736 2.04356 2.03054 1.49190 1.49292 1.39989 1.40461 8.5 1.40907 1.41331 8:8 it; 1.41732 8.9 1.42114 1.42478 Z:! 1.42824 1.43154 z-3 1.43469 9:4 1.43771 1.44059 ;:z 1.44335 Z-I: 1.44600 1.44854 9:9 1.50024 2.15363 2.14198 1.50106 ;:2 2.17654 1.50265 2.16515 1.50186 ::; 2.18780 1.50341 3.9 ::2 2 Z:! $2 ::98 3.0 ::: ;:: ::: :::, 3.9 i-t 4:2 ::: 4.5 0.93858 0.97610 1.01234 1.04736 1.08124 1.11405 1.14586 1.17671 1.20667 1.23578 1.26409 1.29164 1.31847 1.34461 1.37010 1.39496 1.41924 1.44294 1.46611 1.48876 1.51092 1.53261 t-t 1.55384 418 1.57463 4.9 1.59501 5.0 1.61498 1.45097 10.0 [ c-y] 1.77893 1.46746 1.46921 1.79561 1.47090 1.81201 1.47253 1.82815 1.47411 1.84404 1.47565 1.85968 1.47713 ARGUMENTS Table 6.8 x=1.2 %qz) .W(z) .eJ(z) !I .feC(Z) ?/ 0.0 -0.28964 o.ooodo 5.0 1.61756 1.43125 0.1 -0.28169 0.12620 5.1 1.63705 1.43396 1.65617 1.43658 0.3 -0.22578 0.36640 5.3 0.2 -0.26014 0.24926 5.2 1.67494 1.43910 0.4 -0.18064 0.47552 5.4 1.69336 1.44152 H -0 42ji5 0.00000 5.0 -0.41451 0.14258 5.1 -0.38753 0.28082 -0.34490 0.41099 :-: -0.28961 0.53042 5:s -0.22498 -0.15426 -0.08023 -0.00509 +0.06954 289 0.5 -0.12710 0.57530 -0.06753 0.66517 ;; -0.00412 0.74519 +0.06130 0.81589 0.12730 0.87806 4; . 5.5 5.6 5.7 5.8 5.9 1.71146 1.72924 1.74672 1.76390 1.78079 1.44386 1.44612 1.44829 1.45039 1.45243 0.19280 0.93260 6.0 1.79740 1.81375 1.82983 1.84567 1.86126 1.45439 1.45629 1.45813 1.45991 1.46164 1.46331 1.46493 1.46651 1.46803 1.46952 11.1" 0.31960 0.98046 6.1 . 0.25707 1.02252 6.2 ::: 0.38012 1.05960 6.3 1.4 0.43846 1.09240 6.4 0.49459 0.54851 0.60028 0.64999 0.69774 1.12153 1.14752 1.17082 1.19179 1.21074 6.5 6.6 6.7 6.8 6.9 1.87661 1.89173 1.90663 1.92132 1.93579 0.74362 0.78775 212 0.83022 5:: 0.87114 0.91060 1.22794 1.24362 1.25796 1.27112 1.28323 7.0 7.1 7.2 7.3 7.4 1.95006 1.47096 1.96413 1.47236 %E 1.47372 1.47505 0.94868 0.98546 1.02103 1.05546 219 1.08881 z-i 1.12113 ::1" 1.15250 3.2 1.18295 1.21254 ::4' 1.24132 1.29442 1.30478 1.31441 1.32337 1.33173 7.5 7.b 7.7 7.8 7.9 2.01852 2.03167 2.04465 2.05746 2.07012 1.47760 1.47882 1.48001 1.48117 1.48230 1.33955 1.34688 1.35377 1.36024 1.36635 8.0 8.1 8.2 8.3 8.4 2.08262 2.09496 2.10716 2.11921 2.13111 1.48341 1.48448 1.48553 1.48656 1.48756 1.26932 1.29659 1.32315 1.34905 1.37432 1.37211 1.37756 1.38272 1.38761 1.39226 8.5 8.6 8.7 8.8 8.9 2.14288 2.15451 2.16601 2.17738 2.18862 1.48853 1.48949 ii44042 1.49133 1.49222 2.23161 1.50631 2.22084 1.50561 2'3 2.24228 1.50699 4.4 1.39898 1.42306 1.44659 1.46959 1.49209 1.39667 1.40088 1.40489 1.40871 1.41236 9.0 9.1 9.2 9.3 9.4 2.19973 2.21073 2.22160 2.23236 2.24301 1.49310 1.49395 1.49478 1.49560 1.49640 2.26326 1.50832 2.25283 1.50766 22 2.28382 2.27360 yw& 2; 1.51410 1.41920 9.6 1.53565 1.41586 9.5 1.57743 1.42547 9.8 1.55676 1.42240 9.7 2.26397 1.49718 2.25354 1.49794 2.28450 1.49869 2.27429 1.49943 2.29395 1:51021 4.9 1.59769 1.42842 9.9 2.29461 1.50015 2.05640 1.49489 2.06908 1.49392 2.09397 2.08160 1.49584 2.10619 1.49676 1.49767 2.13019 1.49940 2.11826 1.49855 zi 1:9 :-i 22 2.20995 1.50489 2.19893 1.50416 2: 2.30397 1.51082 5.O [‘-;)5] [‘-;)I] 2:00519 1.47634 1.61756 1.43125 10.0 2.30462 1.50085 [(-p] [‘-;)“I [c-y] [f-2”“] 290 GAMMA Table 6.8 DIGAMMA FUNCTION AND FUNCTION RELATED FUNCTIONS FOR COMPLEX ARGUMENTS x-1.4 x=1.3 m(z) Y 0.0 -0.16919 -0.16323 :*: -0.14567 013 -0.11748 0.4 -0.08009 A(z) 0.00000 0.11303 0.22372 0.32997 0.43011 -0.03520 +0.01541 0.07003 E 0.12718 0:9 0.18561 0.52298 0.60796 0.68491 0.75404 0.81582 :*!? 0.24434 0.30262 1:2 0.35994 6:2 g 6.3 6.4 6.5 6.6 6.7 6.8 6.9 22 9e4 ;a'4 . 5.5 ;$ 5:8 5.9 W(Z) 1.62052 1.63990 1.65891 1.67758 1.69591 1.41163 1.41472 1.41769 1.42055 1.42331 1.71392 1.73161 174900 1'76611 1:78292 1.79947 Y 5.0 3 ?J 0.0 9@(z) 4e4 Y*(z) 0.00000 0.10223 0.20269 0.29974 0.39204 Y 5.0 5.1 5.2 5.3 5.4 m(z) 1.62386 1.64311 1.66200 1.68055 1.69878 1.39213 1.39559 1.39891 1:4021i 1.40519 0.1 0.2 0.3 0.4 -0.06138 -0.05646 -0.04192 -0.01844 +0.01295 1.42597 1.42853 1743101 1.43340 1.43571 0.5 0.6 0.7 0.8 0.9 0.05100 0.09436 0.14171 0.19183 0.24367 0.47862 0.55886 0.63250 0.69957 0.76033 5.5 5.6 5.7 5.8 5.9 1.71668 1.73428 1.75158 1.76860 1.78533 1.40817 1.41103 1.41380 1.41648 1.41907 1.0 0.29635 ;*;;;;+ 1:84754 1.86308 1.43794 1.44011 1.44220 1.44423 1.44619 0.81517 0.86457 0.90903 0.94907 0.98517 6.0 6.1 6.2 6.3 6.4 1.80180 1.81800 1.83395 1.84966 1.86513 1.42157 1.42399 1.42634 1':: 0.34918 0.40163 0.45331 0.50395 1.87837 1.89344 1.90829 1.92293 1.93735 1.44810 1.44995 1.45174 1.45348 1.45517 1.5 1.6 1.7 1.8 1.9 0.55336 0.60144 0.64811 0.69337 0.73722 1.01778 1.04730 1.07409 1.09849 1.12075 6.5 6.6 6.7 6.8 6.9 1.88036 1.89537 1.91017 1.92475 1.93912 1.43294 1.43502 1.43702 1.43898 1.44087 7.0 7.1 7.2 7.3 7.4 1.95330 1.44271 1.96727 1.44450 1.98106 1.44625 1.99467 1.44794 2.00809 1.44959 1.3 1.4 0.41593 0.47035 0.87085 0.91983 0196341 1.00227 1.03698 1.5 1.6 1.7 ::: 0.52310 0.57409 0.62333 0.67084 0.71667 1.06809 1.09605 1.12126 1.14409 1.16483 2.0 2.1 225 0.76087 0.80353 0.88447 0.84470 1.18373 7.0 1.20102 7.1 1.23148 7.3 1.21688 7.2 1.95158 1.96560 1.97944 1.45681 1.45841 ;.;fJy;; 2.0 2.1 f-3 2:4 0.92290 1.24495 7.4 2.00655 1.99309 i46294 2:4 0.77968 1.14113 0182078 1.15984 0.86058 1.17707 1.19296 0.89913 1.20768 0.93647 2.5 2.6 0.96007 0.99604 1.25743 1.26900 7.5 7.6 2.01984 2.03296 1.46438 1.46577 z-i 2:9 1.03088 1.06464 1.09739 1.27976 1.28980 1.29918 7.8 7.7 7.9 2.04591 2.07131 2.05869 1.46845 1.46713 I.46974 2.5 2.6 2.7 2.8 2.9 0.97265 1.00775 1.04179 1.07484 1.10693 1.22133 7.5 1.23402 7.6 li24585 7.7 1.25689 7.8 1.26723 7.9 3.0 1.12917 1.16004 1.19005 1.21923 1.24763 1.30797 1.31621 1.32396 1.33126 1.33814 8.0 8.1 8.2 8.3 8.4 2.08378 2.09610 2.10827 2.12029 2.13217 1.47100 1.47223 1.47342 1.47459 1.47573 3.0 3.1 3.2 3.3 3.4 1.13813 1.16846 1.19797 1.22670 1.25469 1.27693 1.28604 1.29461 1.30269 1.31032 8.0 8.1 8.2 8.3 8.4 2.08510 1.45862 2.09739 1.46000 2.10952 1.46134 2.12151 1.46266 2.13337 1.46394 1.27529 1.30223 1.32851 33'87 1.35413 319 1.37915 1.34464 1.35080 1.35663 1.36216 1.36742 8.5 8.6 8.7 8.8 8.9 2.14391 2.15552 2.16700 2.17834 2.18956 1.47685 1.47794 1.47900 1.48004 1.48106 3.5 E 3:s 3.9 1.28196 1.30855 1.33450 1.35983 1.38456 1.31753 1.32436 1.33084 1.33699 1.34283 8.5 8.6 8.7 8.8 8.9 2.14508 2.15666 2.16811 2.17943 2.19063 1.46519 1.46641 1.46760 1.46877 1.46991 4.0 4.1 4.2 4.3 4.4 1.40357 1.42744 1.45077 1.47358 1.49590 1.37242 1.37718 1.38172 1.38606 1.39020 9.0 9.1 9.2 9.3 9.4 2.20066 2.21163 2.22249 2.23323 2.24386 1.48205 1.48302 1.48397 1.48490 1.48582 4.0 4.1 4.2 4.3 4.4 1.40873 1.43235 1.45546 1.47806 1.50019 1.34840 1.35370 1.35876 1136359 1.36821 9.0 9.1 9.2 9.3 9.4 2.20170 2.21265 2.22349 2.23421 2.24481 1.47103 1.47212 1.47319 1.47423 1.47525 4.5 4.6 1.51775 1.53914 1.56010 1.58064 1.60078 1.39416 1.39795 1.40258 1.40507 1.40841 9.5 9.6 9.7 9.8 9.9 2.25437 2.26478 2.27508 2.28528 2.29537 1.48671 1.48758 1.48844 1.48927 1.49010 4.5 4.6 4.7 4.8 4.9 1.52185 1.54307 1.56387 1.58425 1.60425 1.37263 1.37686 1.38092 1.38481 1.38854 9.5 9.6 9.7 9.8 9.9 2.25531 1.47626 2.26570 1.47724 2.27598 1.47820 2.28616 1.47914 2.29623 1.48006 1.62052 [C-,,2] 1.41163 10.0 2.30537 1.49090 [‘-p”] 5.0 1.62386 [C-y] 1.39213 10.0 2.30621 1.48096 ['-;'"I ['-;'"I ['-['"I ;:: 3:: 3.5 3.6 f-i 4:9 5.0 [(-$W] [(-y-j ::2' 2.02134 2.03442 2.04733 2.06008 2:07267 ?%: . 1.45119 1.45276 1.45428 1.45576 1.45721 GAMMA DIGAMMA FUNCTION AND RELATED FOR COMPLEX FUNCTION 291 FUNCTIONS x= x=1.5 3$(Z) ye.4 0.00000 Y -@ tic4 %ti(z) -e+(z) Y ad44 Y+(z) 3995 O.l-..- 0.15687 0.17976 5.0 5.1 5.2 5.3 5.4 1.63162 1.65057 1.66919 1.68748 1.70546 1.35357 1.35773 1.36173 1.36558 1.36930 0.5 0.6 0.7 0.8 0.9 0.20790 0.24050 0.27674 0.31581 0.35697 0.40789 0.47942 0.54642 0.60875 0.66642 5.5 5.6 5.7 5.8 5.9 1.72313 1.74051 1.75760 1.77441 1.79095 1.37289 1.37635 1.37969 1.38293 1.38605 1.40528 1.40796 1.41055 1.41306 1.41549 1.0 1.1 :.g 0.39957 0.44305 yyw& 0.71957 6.0 o*7684o 0.81319 0.85423 0157445 0.89183 22 "6:: . 1.80724 1.82327 1.83906 1.85460 1.86992 1.38908 1.39200 1.39484 1.39759 1.40025 1.88258 1.89752 1.91225 1.92677 1.94109 1.41786 1.42015 1.42237 1.42453 1.42663 1.5 1.6 1.7 1.8 1.9 0.61757 0.66001 0.70167 0.74244 0.78228 0.92629 0.95790 0.98693 1.01363 1.03824 6.5 6.6 6.7 6.8 6.9 1.88501 1.89989 1.91455 1.92900 1.94326 1.40284 1.40534 1.40778 1.41014 1.41244 1.95521 1.96914 1.98287 1.99643 2.00981 1.42866 1.43065 ii43257 1.43445 1.43628 2.0 2:3 2.4 0.82115 0.85905 0.89597 0.93193 0.96694 1.06096 1.08197 1.10144 1.11953 1.13635 7.0 7.1 7.2 7.3 7.4 2.02301 2.03604 2.04891 2.06162 2.07417 1.43805 1.43978 1.44147 1.44312 1.44472 2.5 2.6 2.7 2.8 2.9 1.00102 1.03421 1.06653 1.09801 1.12867 1.15204 1.16668 1.18039 1.19324 1.20530 3.0 3.1 3.2 1.15856 1.18770 33:: 1.21611 1.24383 1.27089 1.21664 1.22733 1.23741 1.24693 1.25594 3.5 3.6 1.29731 1.32311 1.42065 1.44373 1.4b632 1.48844 .%$(Z) 1.37278 1.37658 1.38025 1.38378 1.38719 nn n i3m5 -,,5 0.1395’ 0.27432 0.35978 1.62756 1.64667 1.66543 1.68386 1.70196 0.2 0.3 0.4 0.13189 0.16935 0.21064 0.25479 0.30091 0.44066 0.51640 0.58668 0.65144 0.71078 5.5 5.6 5.7 5.8 5.9 1.71976 1.73725 1.75445 1.77137 1.78801 1.39047 1.39364 1.39670 1.39965 1.40251 0.34824 0.39614 0.44411 0.49175 0.53878 0.76494 0.81424 0.85907 0.89980 0.93684 6.0 6.1 6.2 6.3 6.4 1.80439 1.82051 1.83638 1.85201 1.86741 1.4 1.5 1.6 1.7 0.58497 0.63018 0.67432 0.71732 0.75916 0.97054 1.00127 1.02932 1.05500 1.07855 6.5 t-7" . 1.8 1.9 2.0 3:: 5:: 92 2:7 2.8 2.9 3:: 0.09325 0.18511 “:87772 1.10020 o*ZE1.13857 li12015 0 0.98634 1.02050 1.05370 1.08598 1.18618 1.19990 1.21271 1.22469 1.23592 ;.; 7:7 1.24647 1.25639 1.26574 1.27457 1.28290 8.0 z.; 2.08657 2.09882 8:3 8.4 ;-;;;;; 2:13470 8.5 8.6 8.7 8.8 8.9 2.14638 2.15794 2.16936 2.18065 2.19182 2.22460 2.23530 2.24588 2.25635 2.26672 2.27698 2.28714 2.29720 1.46582 1.46691 1.46798 1.46902 1.47004 1.47105 (-;I2 1.14794 1.17769 1.20667 1.28931 131552 1:34112 1.36612 1.39055 1.29080 1.41443 1.43779 1.46065 1.48302 1.50493 1.32464 1.52639 1.54742 1.56804 1.58826 1.60810 1.35128 1.35594 1.36041 1.36470 1.36882 :*s7 4:9 5.0 22.: 1146242 1.46358 1.46471 9.5 9.6 9.7 9.8 9.9 114 2.20286 2.21379 9.2 9.3 9.4 0.1 1.45355 1.45491 1.45623 1.45753 1.45879 9.0 9.1 “.” 1.44628 1.44781 1.44930 1.45075 1.45217 1.11738 ;*: 3:7 3i8 3.9 4.5 4.6 34 7:3 7.4 1.15563 1.17146 x:51 . 4.2 4.3 4.4 7.0 0.91499 0.95118 z*; 3:4 4.0 4.1 66:: 1.29828 1.30537 1.31212 1.31853 1.33047 1.33603 1.34134 1.34642 ;a; . 6.8 1.6 0.00000 0.08566 0.17023 0.25268 0.33214 Y 5.0 5.1 5.2 5.3 5.4 0.03649 0.04062 0.05284 0.07266 0.09932 Table ARGUMENTS 1.62756 1.37278 lo.0 2.30716 [i-y] ['-:I'] [ C-i)4 3.7 3.8 3.9 2:: 1[ 2; 414 I.&b”“_ 1.34833 li37297 1.39707 li51012 4.5 i-76 418 4.9 1 .f$(1.5+iy) 1.53136 1:552i9 1.57262 1.59265 1.61232 =$ tanh KY-~- 4Y 4&l 1.95731 1.41467 1.97118 1.41684 1.98487 1.99837 2.01169 1.42101 7.5 7.6 7.7 7.8 7.9 2.02485 2.03784 2.05066 2.06332 2.07583 1.42496 1.42686 1.42871 1.43051 1.43227 8.0 2.08819 2.10040 2.11246 2.12439 2.13617 1.43398 1.43565 1.43728 1.43888 1.44043 1.26448 1.27257 1.28026 1.28757 1.29454 2.14782 2.15934 2.17073 2.18199 2.19313 1.44195 1.44344 1.44489 1.44631 1.44770 1.30117 1.30750 1.31354 1.31932 1.32485 2.20415 2.21504 2.22583 2.23650 2.24706 1.44905 1.45038 1.45168 1.45295 1.45420 1.33014 1.33522 1.34009 1.34476 1.34925 2.25751 2.26785 2.27809 2.28822 2.29826 1.45542 1.45661 1.45778 1.45892 1.46005 8.1 8.2 1.41895 1.42301 292 GAMMA Table 6.8 DIGAMMA FUNCTION AND FUNCTION FOR RELATED FUNCTIONS COMPLEX ARGUMENTS 2=1.8 x=1.7 0'0 0 se> 20855 0"G> 00000 5: 163603 9 * (2) 133453 YG> w W(z) 0.0 0.1 0.2 0.3 0.4 0.28499 0.28760 0.29537 0.30809 0.32541 0.00000 0.07358 0.14644 0.21792 0.28740 5.0 5.1 5.2 5.3 5.4 1.64078 1.65939 1.67769 1.69567 1.71336 1.31566 1.32048 1.32513 1.32961 1.33393 0.5 0.6 0.34693 0.37215 0.40053 0.43155 0.46469 0.35437 0.41842 0.47928 0.53675 0.59076 5.5 5.6 5.7 5.8 5.9 1.73076 1.74787 1.76472 1.78130 1.79762 1.33810 1.34213 1.34603 1.34979 1.35344 1.81369 1.82952 1.84511 1.86047 1.87561 1.35697 1.36038 1.36369 1.36690 1.37001 1.89053 1.90525 1.91975 1.93406 1.94817 1.37303 1.37596 1.37881 1.38158 1.38426 w4 Y a+) ‘e(z) 0:l 0.2 0.3 0.4 0:21156 0.22050 0.23511 0.25494 0:07918 0.15747 0.23407 0.30824 5:1 5.2 5.3 5.4 1:65482 1.67328 1.69142 1.70926 1:33902 1.34335 1.34752 1.35154 0.5 0.6 0.7 0.8 0.9 0.27945 0.30803 0.34001 0.37474 0.41161 0.37937 0.44701 0.51086 0.57074 0.62661 5.5 5.6 5.7 5.8 5.9 1.72680 1.74405 1.76102 1.77772 1.79416 1.35543 1.35918 1.36280 1.36630 1.36969 1.0 1.1 1.2 1.3 1.4 0.45005 0.48957 0.52973 0.57018 0.61063 0.67852 0.72661 0.77107 0.81211 0.84996 6.0 6.1 6.2 6.3 6.4 1.81034 1.82627 1.84196 1.85742 1.87266 1.37297 1.37614 1.37922 1.38220 1.38509 0.49947 0.53546 0.57226 0.60955 0.64706 0.64131 6.0 o*68847 0.73237 0.77316 0.81103 2 6:3 6.4 1.5 1.6 1.7 1.8 1.9 0.65085 0.69065 0.72990 0.76849 0.80636 0.88488 0.91710 0.94685 0.97436 0.99982 6.5 6.6 6.7 6.8 6.9 1.88767 1.90246 1.91705 1.93143 1.94561 1.38789 1.39061 1.39326 1.39582 1.39832 0.68455 0.72184 0.75879 0.79528 0.83122 0.84617 0.87877 0.90903 6.5 6.6 6.7 :*z:2 . 2: 2.0 2.1 2.2 2.3 2.4 0.84345 0.87973 0.91519 0.94981 0.98362 1.02342 1.04533 1.06570 1.08468 1.10238 7.0 7.1 7.2 7.3 7.4 1.95961 1.97342 1.98704 2.00048 2.01375 Il*to051: 1:40539 1.40762 1.40980 2.0 2.1 2.2 0.86655 0.90123 0.93523 0.98757 1.01022 1.03136 7.0 7.1 7.2 1.96210 1.97583 1.38688 1.38942 1.39189 2.4 2.3 1.00111 0.96853 1.06957 1.05110 7.4 7.3 2:01598 ;.lf;;;; ;:;;:;: 2.5 2.6 2.7 2.8 2.9 1.01661 1.04879 1.08020 1.11084 1.14075 1.11893 1.13441 1.14893 1.16257 1.17539 7.5 7.6 7.7 7.8 7.9 2.02685 2.03979 2.05256 2.06518 2.07764 1.41191 1.41398 1.41599 1.41794 1.41986 2.5 2.6 2.7 2.8 2.9 1.03299 1.06416 1.09463 1.12442 1.15353 1.08687 1.10310 1.11836 1.13270 1.14622 7.5 7.6 7.7 7.8 7.9 2.02903 2.04191 2.05463 2.06719 2.07960 1.39892 1.40115 1.40332 1.40543 1.40749 3.0 3.1 3.2 3.3 3.4 1.16993 1.19842 1.22625 1.25342 1.27997 1.18747 1.19886 1.20962 1.21981 1.22945 8.0 8.1 8.2 8.3 8.4 2.08996 2.10212 2.11415 2.12603 2.13778 1.42172 1.42354 1.42531 1.42704 1.42874 3.0 1.18200 1.20982 li23703 1.26363 1.28965 1.15898 1.17103 1.18243 1.19322 1.20345 2.09187 2.10399 2.11597 2.12781 2.13952 1.40950 1.41146 1.41338 1.41525 1.41708 3.5 3.6 3.7 3.8 3.9 1.30592 1.33129 1.35610 1.38037 1.40413 1.23859 1.24727 1.25553 1.26338 1.27087 8.5 8.6 8.7 8.8 8.9 2.14939 2.16087 2.17222 2.18345 2.19456 1.43039 1.43200 1.43358 1.43513 1.43664 3.5 1.31511 1.34003 1.36441 1.38829 1.41168 1.21317 1.22241 1.23119 1.23956 1.24754 2.15109 2.16253 2.17385 2.18504 2.19611 1.41886 1.42061 1.42231 1;42398 1.42561 4.0 4.1 4.2 4.3 4.4 1.42738 1.45015 1.47246 1.49432 1.51574 1.27800 1.28481 1.29132 1.29755 1.30351 9.0 9.1 9.2 9.3 9.4 2.20555 2.21642 2.22717 2.23781 2.24834 1.43811 1.43956 1.44097 1.44235 1.44371 1.43459 1.45704 1.47904 1.50062 1.52178 1.25516 1.26243 1.26939 1.27605 1.28242 2.20707 2.21790 2.22862 2;23423 2.24974 1.42720 1.42876 1.43029 1.43178 1.43324 4.5 4.6 4.7 4.8 4.9 1.53675 1.55736 1.57758 1.59742 1.61690 1.30922 1.31470 1.31996 1.32501 1.32986 9.5 9.6 9.7 9.8 9.9 2.25877 2.26908 2.27930 2.28941 2.29942 1.44503 1.44633 1.44760 1.44885 1.45007 4.5 1.54254 1.56292 1.58291 1.60255 1.62183 1.28854 1.29440 1;30004 1.30545 1.31065 2.26013 2.27042 1.43468 1.43608 1.43745 1.43880 1.44012 5.0 1.63603 [C-i"] 1.33453 [I'-;'"] 10.0 * 2.30933 ['-y-j 1.45127 ['-['"I 5.0 1.64078 ['-;'"I 1.31566 ['-;I"] *see page II. 8:'8 019 ;:2' 33:: E 3:8 3.9 44:: 2; 414 i:; t:: 2.28061 2.29069 2.30068 10.0 2.31057 [5’4] 1.44142 ['-;'"I GAMMA FUNCTION DIGAMMA AND FUNCTION RELATED 293 FUNCTIONS FO IR COMPLEX ARG IUMENTS Table 6.8 s=1.9 :!t*(z) 0.35618 0.35847 0.36528 0.37644 0.39169 0.41071 ./tic4 0.00000 0;06870 0.13681 0.20377 0.26908 z*(z) 5yo 1 tic4 0.42583 0.47874 0.52904 z.7" 5:8 5.9 1.81728 1.83300 1.84848 1.86374 1.87878 1.34107 1.34473 1.0 2: 0.57667 0.62165 0.66400 0.70380 0.74116 6.0 Z% 1:35503 0.59465 0.62468 0.65572 0.68751 0.71980 1.35826 1.36140 1.36445 1.36741 1.37029 zl 1:9 0.75239 0.78510 0.81779 0.85033 0.88262 0.77618 0.80899 0.83973 0.86853 0.89551 6.5 6.6 2: 6:9 1.89361 1.90824 1.92266 1.93688 1.95092 x 712 1.96476 1.97843 1.99192 :-::::1" 1137846 ?1" 2:2 0.94617 0.91459 0.97731 2: 2.01838 2.00523 1.38355 1.38104 ;:: 1.03814 1.00798 0.920E: 0.94454 0.96681 0.98775 1.00743 7.5 ;.;;I:; 1.38599 2.5 1.06779 ::; 2:05684 1.39070 1.38838 ::; 1.12548 1.09690 ::; 2.08171 2.06935 1.39518 1.39297 29" 1.18102 1.15352 3.0 1.20798 ::: ::: 1.26034 1.23442 1.31067 1.28575 z-2 317 1.35905 1.33510 1.38254 2.14139 2.12973 1.40546 1.40350 1.32485 ::2 1.34929 1.37324 :-‘B 1.41970 1.39670 3:9 1.18823 1.19798 1.20727 2: 817 88:: 2.15292 2.16432 2.17560 2.18675 2.19778 1.40738 1.40925 1.41108 1.41286 1.41461 Z:i 2.21950 2.20870 1.41800 1.41632 2'3 9:4 2.24077 2.23019 2.25124 1.42124 1.41964 1.42281 2 z-i 2.27186 2.26160 2.29207 2.28202 9:9 10.0 f-i 4:9 1.26810 1.27434 1;28033 1.28610 1.29164 5.0 1.64585 1.29698 [1 C-t)6 1.38522 1.38746 1.38966 1.39180 1.39389 0.51380 0.56594 0.53887 i:: 1.54872 1.56885 1.58861 1.60803 1.62710 2.09613 2.10815 2.12003 2.13178 2.14339 i:; 2: 1.39734 1.39944 1.40149 t:: 1.37313 1.37567 1.37815 1.38056 1.38292 1.32938 1.33730 1.33341 2.09393 2.10600 2.11793 :*:z . 2.03385 2.04661 2.05921 2.07167 2.08397 ihi 5:9 1.76868 1.78513 1.80133 21" 8;2 1.23265 1.24037 ii24775 1.25482 1.26160 1.35937 1.36227 1.36509 1.36784 1.37052 5.5 1.13119 1;14384 1.15583 1.16719 1.17798 1.44226 1.46437 1.48606 1.96761 1.98120 1.99462 2.00786 2.02094 0.31269 0.37042 1.19470 1.22184 1.24841 1.27442 1.29990 :*: 4:3 4.4 1.34358 1.34692 0.47111 0.49110 1.05588 1.07278 1.08868 1.10367 1.11782 4.0 1.89690 1.91143 1.92576 1.93990 1.95385 0.5 1.05008 1.08022 1.10975 1.13867 1.16698 :*::::i . 1.32530 1.32918 1.33295 1.33660 1.34015 1.32522 1.32092 0.98795 1.01932 ::4' 1.82111 1.83671 1.85208 1.86723 1.88217 1.75197 1.73500 0.95338 0.97664 0.99840 1.01879 1.03792 3.0 3.1 3.2 1.30389 1.30846 1.31288 1.31715 1.32129 ::6' 0.89031 z-10 0.92342 2:2 0.95598 z3 2:9 1.73951 1.75633 1.77290 1.78921 1.80528 1.31185 1.30707 1.31647 0.80999 0.84278 0.87335 0.90188 0.92851 2.5 2.6 1.72242 1.27849 1.28394 1.28919 1.29426 1.29916 1.70022 1.68240 1.71775 22 0.71846 0.75338 0.78814 1’2 0.82261 1:9 0.85669 ::i ::y 1.65125 1.66948 2-G1.68742 1.70506 514 2.: 5:4 0.60749 0.65359 0.69677 0.73714 0.77483 ::: 0.42480 0.43081 0.44068 0.45420 0.1 1.30212 1.29698 0.54770 0.58053 0.61431 0.64872 0.68351 1:2 Y [email protected](z) 0.00000 0.06441 0.12833 0.19130 0.25288 1.66428 1.64585 0.43309 0.45842 0.48625 0.51614 :?i X!b(z) 0.4i2i8 5:1 0.33229 0.39306 0.45110 0.50624 0.55838 0:7 i-2 v 010 21" 22 ::: ::2 t-t . 2:: 2; 6:9 7'*1" 712 ::: 1.02597 1.04344 1.05992 1.07548 1.09020 7.5 7.6 7.7 1.10413 1.11733 1.12985 8.0 :%: . ::: 2: 88:: 88’22.15487 2.16623 817 2.17746 :*E:: 1:35639 1.39593 ::i 1.42818 1.40558 1.18379 1.19310 1.20200 4.0 1.45036 1.21050 t:: 44:: 1.47212 1.49348 1.53505 1.51446 1.22643 1.21864 1.24105 1.23389 2.21045 2.22121 2.23187 2.24241 2.25284 1.40548 1.40727 1.40902 1.41074 1.41241 1.42586 1.42435 1.42878 1.42733 2: 2; 1.57514 1.55527 1.59466 1.61385 2.30203 1.43020 4:9 1.63270 1.24792 1.25452 1.26086 1.26696 1.27283 2.26318 2.27340 2.28353 2.29356 2.30349 1.41406 1.41566 1.41724 1.41879 1.42030 2.31190 [ (-;)“I 1.43159 [ (-,121 5.0 1.65125 [(-951 1.27849 [ (-:)“I 2.31332 [ (-35)4] 1.42179 [ (635)3] coth 1r,/--1+3!!? ’ Wl+!f2) 88:: 2.18858 2.19957 :%z 1:40179 1.40366 7. Error Function and Fresnel WALTER GAUTSCHI Integrals l Contents Mathematical Properties . . . . . . . . . . . . . . . . . . 7.1. Error Function . . . . . . . . . . . . . . . . , . . . 7.2.. Repeated Integrals of the Error Function ........ 7.3. Fresnel Integrals . . . . . . . . . . . . . . . . . . . 7.4. Definite and Indefinite Integrals ............ . . . . . . . . . . 297 297 299 300 302 ...................... 7.5. Use and Extension of the Tables . , . . . . . . . . . . . . 304 304 .......................... 308 ...... 310 .... 312 ...... '316 Numerical Methods References Table 7.1. Error Function and its Derivative (2/-l;;) cz2, erf x=(2/&) Table 7.2. Derivative 1 Error Function m 2*r ( :+I > in erfc x=0(.1)5,n=1(1)6, Table 7.5. p2 Dawson’s .. (2 Iz< w). 7D Integrals of the Error Function (0 Ix 15) x=~~+T ~fl)iJ s2 Table 7.4. Repeated 10) e-f2dt, x-2=.25(-.005)0, xez2erfc x= (2/J;;) xez2 n=l(l)lO, (2 5x5 10D ss Table 7.3. Complementary erfc&, ecf2dt, 2=0(.01)2, of the Error Function esz2, 2=2(.01)10, (2/J;;) (O_<z<2) l5D 10, 11, Integral S ‘ef2cZt, x=0(.02) 2, - (t-XP n.I z 317 e-12dt 6s (O<x< a) . . . . . . . . . . . . 319 10D 0 *2 xeez2 ef dt, xm2=.25(-.005)0, S0 1 Guest worker, National Purdye University.) 9D Bureau of Standards, from The American University. (Presently 295 ERROR FUNCTION 296 AND FRESNEL INTEGRALS Page (0<&2.3) ............. 320 7.7. Fresnel Integrals (O<r<5). ...........,.. 321 C(z)=l Table cos (i P) dt, S(r)=lsin(g 7.6. (3/I’(1/3))Jzemt3dt 0 z=O(.O2)1.7(.04)2.3, Table Table 7.8. Auxiliary f(z) =[l- 7D Table 7.10. 15D T&l~,7.11. CC&?%, 2=0(.1)3.9,y=O(.1)3, (l<n< S(z:)=O,n=0(1)5, 325 6D 10) . . . . 329 ..... 329 4D Maxima and Minima of Fresnel Integrals (0 5n55) C(J4n+3), . 8D Complex Zeros of Fresnel Integrals (057~15) z,, z:, C(z,)=O, 7.12. z=z+iy, Complex Zeros of the Error Function z,,erf z,=O,n=l(l)lO, Table x2) Error Function for Complex Arguments (O<zi3.9,O<y13) ft.u(~)=e-~~ erfc (-iz), 323 sin (i x2) cos (i r2)+[-+S(d][email protected] 2=0(.02)1,2-‘=1(-.02)0, 7.9. 7D Functions (O<X< m) . . . . . . . . . . . . . f?(r)] cos G x2)-[&C(x)] g(x)=[&C(z)] Table t2)dt, x=0(.02)5, S(d4n+2), S(J4%+4), n=O(1)5, ... 329 6D The author acknowledges the assistance of Alfred E. Beam, Ruth E. Capuano, Lois K. Cherwinski, Elizabeth F. Godefroy, David S. Liepman, Mary Orr, Bertha H. Walter, and Ruth Zucker in the preparation and checking of the tables. 7. Error Function Mathematical 7.1. Error erf z=Z Integrals Properties Function Series 7.1.5 Definitions 7.1.1 and Fresnel erf z=L 2 & * eet2dt 4 lr s 0 7.1.6 a=0 7.1.3 erfc z=- w(z) =e-“’ ; -“dt=l-erf s zme restriction path. arg t-+a with (cx=: 7.1.7 is permissible (42) is subject lal<z as t-+= along the w (z)=% =Jz ILo For 1,-t(x), Symmetry S m eet2dt s 0 Z2-t2 Relations erf (--z)=-erf 7.1.10 Representation 2iz (-1>“[12n+l/2(22)-12n+3,2(22)1 see chapter 10. 7.1.9 m emt2dt -m z--t’ 22n+1 7.1.8 if L%Q2 remains bounded 7.l.4 ’ [email protected]+l) to the to the left.) Integral 792n+1 2 (l+$J’e’adt)=e-z’erfc In 7.1.2 the path of integration C-1) 2” e-z2 5 n=o 1.3 . . . (2n+l) =2 & 7.1.2 Expansions erf Y=erf 2 7.1.11 V~>O) 2 w(-.z)=2e-z2-w(z) 7.1.12 w(Z)=w(-2) Y FIQUBE 7.1. y=e”’ P=w)6 m e-“dt. s ..I FIQURE 7.2. y=e-” z t' s0 e dt. p=2(1)6 297 ERROR FUNCTION 298 AND FRESNEL INTEGRALS 7.1.15 $Jy e-'2d~-~~~~$. m z-t Y .. (f.. # 0) xp) and El:“) are the zeros and weight factors of the Hermite polynomials. For numerical values see chapter 25. Value at Infinity 7.1.16 erf z-+1 (~-+a M aximum and in jarg z/<:) Inflection Integral F(x) =e-* Points [7.31] for Dawson’s ’ e”dt s0 7.1.17 Ft.92413 88730. . . )=.54104 42246. . . 7.1.18 F(1.50197 52682 . . . )=.42768 66160 . . . Derivatives 7.1.19 dn+’ p erf 2=(-l) * 7.1.20 u1("+2)(z)+2zw("t1) (z)+2(n+l)?P(z)=O (n=O, 1,2, . . .) FIQURE 7.3. Altitude o'(z)=-2zw(z)+~ w’“‘(z)=w(z), Chart of w(z). (For the Hermite polynomials H,(z) see chapter 22.) (7.111, [7.17] Inequalities Relation to Coniluent 7.1.21 (For other inequalities The see [7.2].) Hypergeometric Function (see Mean m and chapter 13) Normal Distribution Standard Deviation Function IJ (see With chapter 26) --(t-v&p 7-1-22 Continued (D s 7.1.14 2ez2 e-z2dt- * ----&~~, e %* dt=a (l+erf Asymptotic Expansion 7.1.23 1 (s)) Fractions ‘I2 ’ -3/2 -.2 .. &zeJ erfc zml+m$I (-1)” (27n-1) ’ *3 * (222)" *’ (z-t-, larg zl<$) ERROR FUNCTION AND FRESNEL Infinite If R,(z) is the remainder after n terms then 299 INTEGRALS Series 7.1.24 erf (x+iy)=erf for Complex Error [7.19] 7.1.29 &(2)=(-l)” Approximation Function 2 l-3 .(2$:-l) B=re-’ 8, (l+$)-‘+ [(l-cos +i emz2 --gl (larg z]<i) dt x+& For x real, R,(x) is less in absolute value than the first neglected term and of the same sign. [fn(x,YY)+ig,(x,Y)l+~(x,y) x cash ny cos 2xy+n sinh ny sin 2xy gn(x, y) =2x cash ny sin Bxy+n sinh ny cos 2xy le(x,y)[ -lo-l61 erf (x+;y)[ fn(x,y)=2x---2 7.2. Repeated Integrals Approximations 2 (0 <z< (ult+azt2+a3t3) eVzz+e (x), aI = .34802 42 p = .47047 of the Error Function 0~ ) 7.1.25 erf x=1- t=1 7.2.1 1+px i” erfc z= m S P Definition in-l erfc t dt (n=O,1,2,...) eez2, io erfc z=erfc a~= - .09587 98 a3= .74785 56 7.1.26 Differential erf x=1- sin 2xy] where PI<1 Rational 2xy)+i Equation ‘&+2z 2 $-2ny=O (alt+azt2+a3t3+a4t4+ast5)e-.Z1+E(x), 7.2.2 t-L 1+pxe(x))1<1.5X10-7 p= .32759 11 al= .25482 9592 a2= - .28449 6736 a,=1.42141 3741 a,=-1.45315 2027 as= 1.06140 5429 y=Ai” erfc z+Bi” erfc (-2) (A and B are constants.) Expression as a Single Integral 7.2.3 7.1.27 Power Series a 7.2.4 al= .278393 a3= .000972 7.1.28 a2= .230389 u4= .078108 Recurrence Relations 7.2.5 * erf x=1-[l+alz+a2& ...+ag26]16+~(x) 2 1 erfc z ++n inm2 ‘n-l erfc z i” erfc z=-; (n=1,2,3,. le(z)(~3XlO-' al= .07052 30784 a3= .00927 05272 aa= .00027 65672 a2= .04228 20123 a4= .00015 20143 aa= .00004 30638 7.2.6 2(n+l) (n+2)i”+2 erfc 2 = (2n+1+2z2)in 2 Approximations7.1.25-7.1.28 are from C. Hastings, Jr., Approximations for digital computers. Princeton Univ. Press, Princeton, N. J., 1955 (with permission). . .) erfc z-f inW2 erfc 2 (n=1,2,3, 8 The terms n+4, n+6, . . .> in this series corresponding to k=ni-2, are understood to be zero. ... ERROR 300 Value FUNCTION AND lelation at Zero 7.2.7 INTEGRALS to the Confluent Hypergeometric (see chapter 13) Function ‘.2.12 1 in erfc 0= FRESNEL (n=-1,0,1,2,. . .) n erfc z=e-@ Relation to Parabolic Cylinder chapter 19) Asymptotic Functions (see Expansion .6 (-1)“(27+n)! n!m!(22y” (~+a, largA<?) 7.3. Fresnel Integrals Definition 7.3.1 7.4. Repeated y=2T(++l) n=o, of the Integrals Error cos 6 ta) dt 7.3;2 FIGURE C(z)=l S(z)=1 sin 6 P) dt Function. The inerfcz following functions are also in use: 7.3.3 1, 2. 4, 8, 14, 22 Cl (x)=4; l zett o4 S cos Pdt, G(x)=-& Derivatives 7.3.4 7.2.8 -$ in erfc z=--in-l erfc z (n=O, 1,2, . . .) S,(+$~ sin Pdt, &(x)=&l ‘@$ dt 7.2.9 Auxiliary g (era erfc 2) = (- 1) n2nn! erzin erfc 2 (n=O, Relation 7.2.11 to Hermite (-1)“i”erfc Polynomials z+i” 1,2, . . .) .f(z)s[i-S(Z)] s(z)+-C(Z)] f&(dW (see erfc (-z)=&J COS 6 sin COS 6 Z2)+[+S(z)]sin 6 22) 6 Z2)-[k-C(Z)] z2) 7.3.6 (see 19.14) in erfc z= @A-$ 7.2.10 Relation to Hh,(a) Functions 7.3.5 chapter *-* Interrelations 22: H,(iz: 7.3.7 C(x>=C1 (x&)4?2 6 x2) ERROR F’UNCTION AND FRESNEL INTEGRALS 301 7.3.14 7.3.8 S(z)=-cos 7.3.9 c(z)=;+f(z)sin (a 2)-g(z) ( 522 go 1 ynT2"+' * . . . (4n+3) > cos G 9) S(Z)=:-j(z)cos G z2)-g(z)& 7.3.15 G z2) c&)=Jl,2(2)+&,2(2)+&,2(2)+ 7.3~6 7.3.10 s2~z>=~3,2~Z~+~,,2~~~+~11,2(~~+ For Bessel functions Series C(z)=$ J,+I,2(z) ~~;~;~;~~)l; CPfl 7.3.17 l. y& ... +sin 6 z2) SO 1 S-l”& /.- C(iz)=iC(z), 7.3.19 24n+l S(S)=--is(z) Value 7.3.20 C(x) -+-7 S(Z)=S(z) at 1 Infinity 1 S(s)+ 2 '(')=go S(-2)=--s(z) C(Z)=C(z), 24n+3 .i Relations C(-2)=--c(z), 7.3.18 C(z) =cos (5 2) go 7.3.13 10. see chapter Expansions 7.3.12 fi ... ... Symmetry 7.3.11 24r+3 (-00) 2 Derivatives (2n+1)!(4n+3) (-1,7+/2)2nf Z4n+3 V A lRelation to Error Function (see 7.1.1, 7.1.3) 7.3.22 C(z)+iS(z)=~ erf [$ =A${ (I--&] l-e+aW [$ 7.3.23 g(z)=$? {J&q ,l,ih]} 7.3.24 f(x)=9 (l+i)z]} (l+i)%l) Relation to Confluent {qL[f Hypergeometric chapter 13) Function (see 7.3.25 .6 1.2 FIGURE I.8 7.5. u=C(z), Fresnel r=w 2.4 Integrals. 3.0 3.6 X Relation to Spherical 7.3.26 C2(+f Bessel Functions (see chapter 10) ERROR FUNCTION 302 Asymptotic AND FRESNEL 7.4.2 Expansions m -(at2+2bt+e)dt f:rfc e =I2 Pe+ ’ S Ja & 7.3.27 nzf(+l+g (4m-1) *3 . . * (az2)2n (-1)“l INTEGRALS (9?a>O) 0 7.4.3 m S 7.3.28 e (4m+l) az$(z) Sri0 l *3($,+i(--lP (ga>o, 0 m= St2?le-atz& LB>01 7.4.4 If Rh’) (z), RAC) are the remainders after n terms (z) in 7.3.27, 7.3.28, respectively, then 7.3.29 1 * 3 . . . (4n-1) e(,’ , (7rZ*)2n - e-rt2n-t 1 p= ,dl(Isrg rW+3) o 1+ 2 ( uz2 > 7.3.30 m 7.3.31 /8”‘[<1, S 2t ( rz2> (S?k>O;n=O, 4<:) 7.4.6 S me-d’ cos (2zt)dt=i 0 d- 12 %e-: zl<z) (9a>O) .* 7.4.7 ,dt (lwz 1,2, . . .) L?Za>O;n=O, 1,2,. . .) 0 e-lt2n+* o 1+ r(n+3> =2a"ff E a m S t*TS+le-d2&=& f$‘(z)=(-1)“l * 3 .(;;;:+l) p, 1 e(g)=r (2n+$) d 7.4.5 Ri”(z)=(-1)” S 1.3...(2n-1) plan 0 S me-d’ sin (2st)dt =i e-I’/a 0 xl vs e%!t o S Wa>O) pP’l<l 7.4.8 t, For z real, R:“(z) and RLg)(x) are less in absolute value than the first neglected term and of the same sign. Rational Approximations ’ (0 <Z 5 m) 7.3.32 1+.92&x f(x~=2+l.792x+3.1042 +4x1 la(z)l52X10-3 -=L 0 &t+z) ear erfc & & [email protected]>o, zfO,larg A<*) 7.4.10 7.3.33 1 g(“)=2+4.142x+3.492~~+6.6702+ e(Z) (For more accurate approximations see [i’.l].) 7.4. Definite and Indefinite Integrals For a more extensive list of integrals see [7.5], 1781, L7.151. . oe-t$-jts - IS 7.4.1 S t+x - e-D’2dtse+2 0 10, 173, 1956.1 J;; [Is 4ar,12dt -k Ei(ax2) 0 1 * (a>% s>O) 7.4.11 erfc 6X x>O) 0t2+2eaI2 S e-a12dLJ!. (a>% 2x 7.4.12 2 4 Approximations7.3.32, 7.3.33 are based on those given in C. Hastings, Jr., Approximations for calculating Fresnel integrals, Approximation Newsletter, April 1956, Note 10. [Seealso MTAC m ma’& e S -“t2dt e’[lu o tZS1=4 1e S m S (erf -\lZ)‘] (a>01 7.4.13 yemt2dt =u 9w(x+iy) -0 (X-t)*+yZ *See page II. (x real, y>O) ERROR FUNCTION AND FRESNEL 303 INTEGRALS 7.4.24 m --(lf sin (t”> dt=; e~ t s0 7.4.15 +; [i-s ?r * [P(x2-y2)]e-tZdt s (g4”+iY) y-ix t"-z(~-y*)t*+(z*+y*)*=2 0 0 n-j j-[&S w(z+;Y> 7.4.17 e-a’ erf bt 1”’ dt=- s0 a sin s0 (@)I sin (ab) } 7.4.26 m ewaldt =‘$ s o &t2+b2) b b {[$S(dF)]cos -[&CT(@)] (2at)rfc bt dt=& e [l-e-‘“‘“‘*](a>O,~b>O) sin (ab)} 7.4.27 m S (ab) e-“la(t)dt=i {[:-A @] -[&C(z)] (D e -ar erfd%dt=i S Wb+b)>o) 0 7.4.28 m s0 7.4.20 dt=a e -2fi (9%>0, L%?b>O) cos (f) 0 7.4.19 Jm eeat erfc .$ (Si’a>O, Wb>O) [email protected] $ (la>O, lw bl<i) 7.4.18 0 (ab) y--ix (x red y>O) 0 (a%>O) { [;-C(,/T)]coi ‘om$$dt=,rg s 7.4.16 2xye-“*dt t4-2($-y2)t*+(22+y*)*=Z (;$)1 7.4.25 (5 real, y>O) m S [+(;$)] e+S(t)dt =i { [$C’(~)] +[& (Wa>O, ab>O) sin (g)} S @] (%z>O) cos (f) sin @} @?a>O) 7.4.29 7.4.21 s,- e-“‘C(~~)dt=ze(~~la)r~~ (9u>O) (Wb>o, ac>o) 7.4.30 7.4.22 m S e --(II cos (t*)dt =&{ Jim e-“‘S(~~)dt=2~~~:a)t~~ [;-S(&/~)]cos@ 0 (S?a>O) -[&C(i #)]sing)} (9a>O) 7.4.31 lm{ [;-C(t)]‘+[;4(t)]*}dt=; 7.4.23 m S e 7.4.32 --atsin (t*)dt= &{ [&?(q)] cos @ 0 S e +[+B (Ed:)] sin (:)} (&%>O) erf (&z+$)+const. (a+3 ERROR 304 FUNCTION AND FRESNEL INTEGRALS 7.4.38 7.4.33 e-&Lb’ S .*dx=g [em” erf (ox+:) S +e- 2oberf ( ax-- i I +const. cos (ax2+2bx+c)dx (a+01 =Jg (ax+b)] { co9 (p)C[Jz 7.4.34 -.~~+gjx= emazz2+S -2 2 [w(i+iax) Se +sin(~)S[JZ(ax+b)]}+const. +w (-$+iax)]+const. Serf xdx=x 7.4.35 erf x+’ (a#O) e-z2+const. fi 7.4.39 S sin(ax2+2bx+c)dx 7.4.36 Seaz erf bxdx=k [em erf bx-ee$ erf (bx-$)I +const . (a#O) -sin(e) C[g (ax+b)]}+const. 7.4.37 Sea erf $dx=a(e- erf .$ +$ e+[w(&%+i -&)+w 7.4.40 (-Jiiz+i (a#@ Numerical Extension 7.4.41 Ss(x)dz=xs(x)+~ +const. -Jk)]} +const. 7.5. Use and SC(x)dx=xC(x)-isin Methods erf.745=.70467 of the Tables Example 1. Compute erf .745 and e-(.‘4a)” using Taylor’s series. With the aid of Taylor’s theorem and 7.1.19 it can be shown that 58247)X 80779+(.5)(.00652 [l-(.005)(.74)+(.00000 83333)(.0952)] = .70792 8920 e--(.,w,~; (.65258 24665) [l-.0074 +(.000025)(.0952)+(.00000 erf (x,+ph) =erf x0 +const. co9 00833)(.74) (1.9048) 1 =.57405 7910. +-$ e-4$ [ 1--phz,+$ p2h2(2$- l)]+e 1-2phx0+p%‘(2~--1) -;p3h3q(2+3) where (e(<l.2~10-~~, IpI4 3. With ~=.74, 1+v Iql<3.2X10-10 if h=10m2, p=.5 and using Table 7.1 As a check the computation was repeated with xQ=.75,p=-.5. Example 2. Compute erfc x to 5s for x=4.8. We have l/22=.0434028. With Table 7.2 and linear interpolation in Table 7.3, we obtain & erfc 4.8=$8 (1.11253)(10-10)(.552669) -2=(1.1352)10-“. ERROR FUNCTION AND FRESNEL z Example 3. Compute e+’ to e%t INTEGRALS 305 From Table 7.1 we have $ e-(1.72)2=.058565. 5s for s0 Thus, x=6.5. With and linear l/22=.0236686 in Table 7.5 6.6 e-WJ’ S i erfc 1.72 = (.058565)(6.0064X interpolation 101*)/1.0087X =3.4873X 1013 10-S i2 erfc 1.72 = (.O58565)(1.292OX1O11)/1.OO87X1O13 =7.5013x (.506143)/(6.5) ==.077868. e%t= 0 10-4 i3 erfc 1.72 = (.058565) (2.6031 X 1O1o)/1.OO87X 1013 Compute i2 erfc 1.72 using recurrence relation and Table 7.1. Example 4. =1.5114x10-4. the Example 6. Compute C(8.65) using Table 7.8. With x=8.65, l/x= .115607 we have from Table 7.8 by linear interpolation By 7.2.1, using Table 7.1, j(8.65) = .036797, g(8.65) = .000159. i-‘erfc 1.72 = .05856 50. Using the recurrence relation i erfc 1.72= - (1.72)(.01499 From Table 4.6 7.2.5 and Table 7.1 72)f (.5)(.05856 50) = .0034873 i2 erfc 1.72= - (.86) (.0034873) f (.25) (.01499 72) = .0007502. sin .9613827 Using 7.3.9 C(8.65)=.5+(.036797)(--961382) -(.000159)(-.275218)=.46467. Example 7. Note the loss of two significant Compute &(l.l) to 10D. Using 7.3.8 and 7.3.10 we obtain by 6-pt interpolation in Table 7.8 digits. Example 5. Compute i” erfc 1.72 for k=l, 2, 3 by backward recurrence. Let the sequence w~(x)(~=m, m-l, . . ., 1, 0, - 1) be generated by backward use of the recurrence relation 7.2.5 starting with u$+~.=O, w:+~= 1. Then, for any fixed k, (see [7.7]), &(l.l)=S ( -=w”x) w!%(x) J;; 2 eZ2ik erfc x (x>O) 30169)=.31865 57172. Example 8. Compute s2(5.24) to 6D. Enter Table 7.7 in the column headed by u. u . 5.20310 5.31898 6.0893801 6.43432 4.97691 With x= 1.72, m= 15 we obtain J3 =s(.87767 Using Aitken’s lim m+w 1.1 scheme of interpolation S:(u) 58 80 .4329l 04 .03689 .41673 97 -. 07803 .45993&3 .16061 70 .39999 44 -. 19432 11 .4699094 .2830889 42 89 99 70 .42732 691 756 674 63 63 .42718 63 60 6 52 79 9 39 .42717 71 61 .42717 67 &(5.24)=.427177 - - - 17 16 :: 0 i.44 4.3834 12 :i 9 13 (1) (2) 2.6399 8 (3) (4) (4) (6) (6) - 2.1011 1.3831 6.4143 9.8096 i L 4.1866 3 I$ ::iE (8) 8.9787 4.9570 (9) (10) 2.6031 :: -1 0 (11) 1.2920 (11) 6.0064 Example 9. Compute 5,(5.24) using Taylor’s series and Table 7.8. Using 7.3.21 we can write Taylor’s series forf, (u) = 306 ERROR FUNCTION AND f2(u)=c0++u0)+~ (u-uo)2+$ (u-uo>“+-*, * FRESNEL By numerical lo 1 +g (u-uo)2+$ (u-Q+ . . .], where clr+2= -cr+ integration, using Table 9.1, sYe(t) g2(u)=-[c~+c2(u-zc,) co=f2c7d, INTEGRALS $=.41826 00. Using the fact that the remainder terms of the asymptotic expansion are less in absolute value than the first neglected terms, we can estimate c1=--g2(u3, 1 * 3 . . . (2&l) t--v J%zo(2uo)k (k=O, 1 +32 ’ 252*72’ g2t-13’2 dt=7,33XlO-‘. 15. 5! 1, 2, . . .). Finally, Consulting Table 7.8 we chose ~=1/.185638 =5.386819, thus having u-u,=5.24-5.386819 =--* 146819. From Table 7.8 5953819 co8 lo-sin -2688ooo m 23107 coslO+sinlO -21504oo qb f2(u,,) = .168270, gz(u,) = .014483. Hence, applying the series above, f,(5.24) = .170436, g2(5.24) =.015030. Using the 4th formula at the bottom of Table 7.8 s2(5.24) = .5- (.170436) (.503471) -(.015030)(-.864012)=.42718. 52(2) 10. =&/2(2) +&2w +&2.(2) +e7,5,2ca + . - - =.49129+.06852+.00297+.00006=.56284. Example 11. Compute s -7 Yo(t> dt by numerical 1 integration using Tables 9.1 and 7.8. [Y,,(t) is the Bessel function of thesecond kinddetkedin9.1.16.1 We decompose the integral into three parts J- YOW ,=jy” Yo(t> $+J,’ sYe(t) I Compute S,(2) using 7.3.16. Generating the values of J,,++(2) as described in chapter 10 we find Example using Tables 7.8 and 4.8. [Yo@> -To(t)1 f +Jy PO(t) $ where represents the first two terms of the asymptotic expansion 9.2.2. OD $=.41826 10 =-.02298 78, Hence 00- .02298 78= .39527 22. The answer correct to 8D is .39527 290 (Table 11.2). Example 12. Compute w(.44$.67i) using bivariate linear interpolation. By linear interpolation in Table 7.9 along the x-direction at y=.6 and y=.7 w(.44+.6i) -.6(.522246+.16788Oi)+.4(.498591 +.2026663)=.512784+.1817943 w(.44+.7i) =.6(.487556+.147975i)+.4(.467521 +.179123i)=.479542+.160434i. By linear x= .44 interpolation w(.44+.67i) along the y-direction at =.3(.512784+.181794i)+.7(.479542 +.160434i)= .489515+.166842i. The correct answer is .489557+ .166889i. Example 13. Compute ~w(z) for z=.44+.61i. Bivariate linear interpolation, as described in Example 12, is most accurate if z lies near the center or along a diagonal of one of the squares of the tabular grid [7.6]. It is not as accurate for z near the midpoint of a side of a square, as in this example. However, we may introduce an auxil- ERROR FUNCTION AND FRESNEL Example 16. Compute Using the second formula iary square (see diagram) which contains z close to its center. Bivariate linear interpolation can then be applied within this auxiliary square. The values of w(z) needed at z=cl, and z=tz are easily approximated by the average of the four the neighboring tabular values. Furthermore parts to be used are given by Izo- &I- w(7+2i)=(-2f7i) ( + w(7+2i). at the end of Table 7.9 44 f;;;;y28i * .05176536 42.27525+28i > =.021853+.075OlOi. 17. Compute erf (2+i). From 7.1.3, 7.1.12 we have H=P,-P2 (Zo-tl(-Pl+Pz’ 307 INTEGRALS Example erf z= 1 -e-Z2w(iz) = 1 -ey2-z2(cos 2xy 4 Using Tables sin 2zy)w(y+iz) (z=x+iy>. 7.9, 4.4, 4.6 erf (2-l-i)=1-e-3 (cos 4--i sin 4)w(l+2i) =1.003606-.011259Oi. where z=zO+ .l (p,+ipJ. tI=.45+.65i, S;=.45+.55i, from Table 7.9 S%‘w(~~)=$(.522246 &!w(s;) =2(.522246 Thus, with zo=.4+ .6i, p,k.4, pz=.l, we get +.498591+.487556+.467521) = .493979 S, ((a+;) From 7.3.22, 7.3.8, 7.3.18 we have Example &(z)=-- Compute 18. ’ 29 [email protected] [(l+i) +.498591+.561252+.533157) = .528812 gw(z)=[l-(.4+.1)]{[1-(.4-.1)].522246 +(.4-.1).528812)+(.4+.1)X {[l-(.4-.1)] .493979+(.4-.1).498591}=.509789. $1 -!$! e-fz2w[(i-l) Jz and making 7.1.12, and Table 4). $1. use of 7.1.11, 7.9 The correct answer is .509756. Straightforward bivariate interpolation gives .509460. Example 14. Compute Yw(.39+.61i) using Taylor’s series. Let z=.39+.61i, zo=.4+.6i. From and using Table 7.9, we have to 6D i ---2 l---i e --2 cos&-isini)w(l+ii) 4 ( 7.1.20, +!$ie2( cosi+ising)w(i+$i) w(z,)=.522246+.167880i w’(z,)=-.21634+.367383, #w”(zo)=-.215--.185i, z-zo=(~-l+i)10-2 (z--~~)~=-2iXlO-~ ~w(z)=.167880-.0021634--0036738 + .0000430= 15. Example From 7.1.11, Compute w(.4-1.3i). 7.1.12 .162086. Example =-.990734-.681619i. m Compute e--(1’4)L2-31 (2t)dt cos s0 19. using Table 7.9. Setting b=y+ix, 7.1.12 we find m S e -at-w cos c=O in 7.4.2 and using 7.1.3, (2xt)(j& 2&L2w(y) 0 (a>O, 2, y real). Hence from Table Using Tables 7.9, 4.4 and 4.6 w(.4-1.3i)=4.33342+8.042013. 7.9 m s e-(1’4)tP-3r cos (2t)dt= 0 &$?w(2+3i) = .231761. ERROR 308 FUNCTION AND FRESNEL INTEGRALS References Texts [7.1] J. Boersma, Computation of Fresnel integrals, Math. Comp. 14, 380 (1960). 17.21 A. V. Boyd, Inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 6, 44-46 (1959). [7.3] 0. Emersleben, Numerische Werte des FehIerintegrals fiir 6, Z. Angew. Math. Mech. 31, 393-394 (1951). [7.4] A. Erdelyi et al., Higher transcendental functions, vol. 2 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1953). (7.51 A. Erdelyi et al., Tables of integral transforms, vol. 1 (McGraw-Hill Book Co., Inc., New York, N.Y., Toronto, Canada, London, England, 1954). [7.6] W. Gautschi, Note on bivariate linear interpolation for analytic functions, Math. Tables Aids Comp. 13, 91-96 (1959). [7.7] W. Gautschi, Recursive computation of the repeated integrals of t.he error function, Math. Comp. 15, 227-232 (1961). [7.8] W. Grobner and N. Hofreiter, Integraltafel (Springer-Verlag, Wien and Innsbruck, Austria, 1949-50). [7.9] D. R. Hartree, Some properties and applications of the repeated integrals of the error function, Mem. Proc. Manchester Lit. Philos. Sot. 80, 85-102 (1936). [7.10] C. Hastings, Jr., Approximations for digital computers (Princeton Univ. Press, Princeton, N.J., 1955). 17.111 Y. Komatu, Elementary inequalities for Mills’ ratio, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 69-70 (1955-57). [7.12] E. Kreyssig, On the zeros of the Fresnel integrals, Canad. J. Math. 9, 118-131 (1957). [7.13] Th. Laible, Hohenkarte des Fehlerintegrals, Z. Angew. Math. Phys. 2, 484-486 (1951). [7.14] F. Losch and F. Schoblik, Die Fakultiit (B. G. Teubner, Leipzig, Germany, 1951). [7.15] F. Oberhettinger, Tabellen zur Fourier Transformation (Springer-Verlag, Berlin, Gottingen, Heidelberg, Germany, 1957). [7.16] J. R. Philip, The function inv erfc 8, Austral. J. Phys. 13, 13-20 (1960). [7.17] H. 0. Pollak, A remark on “Elementary inequalities for Mills’ ratio” by YQsaku Komatu, Rep. Statist. Appl. Res. Un. Jap. Sci. Engrs. 4, 110 (1955-57). [7.18] J. B. Rosser, Theory and application of *&cIz and s,’ e-.1Jdy~’ JI I e-z’dx (Mapleton House, Brooklyn, N.Y., 1948). [7.19] H. E. Salzer, Formulas for calculating the error function of a complex variable, Math. Tables Aids Comp. 5, 67-70 (1951). [7.20] H. E. Salzer, Complex zeros of the error function, J. Franklin Inst. 266, 209-211 (1955). [7.21] F. G. Tricomi, Funzioni ipergeometriche confluenti (Edizioni Cremonese, Rome, Italy, 1954). L7.221 G. N. Watson, A treatise on the theory of Bessel functions, 2d ed. (Cambridge Univ. Press, London, England, 1958). Tables [7.23] M. Abramowitz, Table of the integral J ‘e-u3 du, J. Math. Phys. 30, 162-163 (1951). :=0(.01)2.5, 8D. [7.24] P. C. Clemmow and Cara M. Munford, A table of &&r)e@ l” e-*irA2dX for complex values of p, Philos. Trans. Roy. Sot. London (A}, 245, 189211 (1952). \p\=O(.Ol).S, arg p=O”(10)450, 4D. [7.25] R. B. Dingle, Doreen Arndt and S. K. Roy, The integrals C,(X) = (P!)-~~~ ep(G+z?)-*e-de D,(x) eP(G+d)-*e-#de and = (p!)-lJm and their tabulation, Appl. Sci. Res. B 6, 155-164 (1956). C(z), S(z), z==O(1)20, 12D. [7.26] V. N. Faddeeva and N. M. Terent’ev, Tables of values of the function u)(z) =e+ (l+zKeGdt) for complex argument. Translated from the Russian by D. G. Fry (Pergamon Press, New York, N.Y., 1961). m(z),z=x+iy; z,y=O(.O2)3; 2=3(.1)5, y=O(.1)3; 2=0(.1)5, y=3(.1)5; 6D. [7.27] B. D. Fried and S. D. Conte, The plasma dispersion function (Academic Press, New York, N.Y. and London, England, 1961). i&(z), iJGd(z), z=x+iy; 2=0(.1)9.9, y=-9.1(.1)10; z=var. (.1)9.9, y= -lO(.l) -9.2; 6s. ’ &fx v [7.28] K. A. Karpov, Tablitsy funktsii w(z) =e+ s kompleksnoi oblasti (Izdat. Akad. NaukO SSSR., Moscow, U.S.S.R., 1954). z=x; 2=0(.001)2(.01) 10; 5D; z=pei@; e=2.5°(2.50)300(1.250)350(.6250)400; p=ps(.OOl)p; (.Ol)p;’ (.0002)5, 0 Ipe<p; <p;l 15, 5D; z=iy; y=O(.OO1)3(.0002)5, 55. [7.29] K. A. Karpov, Tablitsy funktsii F(z)= 0z e+fx J v kompleksnoi oblasti (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1958). z=peid; 8=45o(.3125o)48.75o(.625o)55o(1.25o)65o(2.5o)9Oo, p=pe(.OOl)pi(.Ol)p;‘, 0 <pb<,$ &’ 15, 5D; 2=x; 2=0(.001)10, 5s. [7.30] J. Kaye, A table of the first eleven repeated integrals of the error function, J. Math. Phys. 34, 119-125 (1955). inerfc z, s=0(.01).2(.05)1(.1)3, T&=-1(1)11, 6D. [7.31] B. Lohmander and S. Rittsten, Table of the function ,” eta&?, Kungl. Fysiogr. Slillsk. i Lund I FGrh. 28, 45-52 (1958). z=O(.O1)3(.02)5, x-1=0(.005).2, 10D; x=.5(.5)10, 20D. Contains also 20D values for maximum and inflection points. y=e-23 ERROR [7.32] W. FUNCTION AND Lash Miller and A. R. Gordon, Numerical evaluation of infinite series and integrals which arise in certain problems of linear heat flow, electrochemical diffusion, etc., J. Phys. Chem. 35, 2785-2884 (1931). F(z) =e-z’ ( eL2dt; s=O(.O1)1.99, 6D; z=2(.01)4(.05)7.5(.1)10(.2)12, 8s. [7.33] National Bureau of Standards, Tables of the error function and its derivative, Applied Math. Series 41, 2d ed. (U.S. Government Printing Office, Washington, D.C., 1954). 15D; (2/Jr)e-z*, erf 2, s=0(.0001)1(.001)5.6, (2/Jr)e-z2, erfc 2, 2=4(.01)10, 85. FRESNEL 309 INTEGRALS [7.34] T. Pearcey, Table of the Fresnel integral (Cembridge Univ. Press, London, England, 1956). c(g), S (E), 2=0(.01)50, 6-7D. [7.35] Tablitsy integralov Frenelya (Izdat. Akad. Nauk SSSR., Moscow, U.S.S.R., 1953). C(z), S(Z), Z= 0(.001)25, 7D; S(z), r=O(.OOl) .58, 7s; C(z), 2=0(.001) .lOl, 7s. [7.36] A. van Wijngaarden and Fresnel integrals, Verh. Afd. Natuurk. Sec. I, C(z), S(Z), 2=0(.01)20, numerical values of the asymptotic expansions.) W. L. Scheen, Table of Nederl. Akad. Wetensch., 19, No. 4, l-26 (1949). 5D. (Also contains coefficients in Taylor and ERROR FUNCTION 310 Table ERROR 7.1 AND FRESNEL FUNCTION INTEGRALS AND ITS DERIVATIVE 2 (d-2 0. 00 0. 01 0. 02 0.03 0.04 2 ~ \R 1.12837 1.12826 1.12792 1.12736 1.12657 0.05 0. 06 0.07 0. 08 0. 09 1.12556 1.12432 1.12286 1.12118 1.11927 17424 43052 36333 06004 62126 0.05637 19778 0.06762 15944 0.07885 77198 0.09007 81258 0.10128 05939 0.55 0. 56 0.57 0. 58 0. 59 0.83383 0.82463 0.81536 0.80604 0.79666 66473 22395 63461 33431 75911 0.56332 0.57161 0.57981 0.58792 0.59593 33663 57638 58062 29004 64972 0.10 0. 11 0. 12 0. 13 0.14 1.11715 1.11480 1.11224 1.10946 1.10647 16068 80500 69379 97934 82654 0.11246 0.12362 0.13475 0.14586 0.15694 29160 28962 83518 71148 70331 0. 60 0. 61 0. 62 0. 63 0.64 0.78724 0.77777 0.76826 0.75872 0.74914 34317 51846 71442 35764 87161 0.60385 0.61168 0.61941 0.62704 0.63458 60908 12189 14619 64433 58291 0. 15 0.16 0.17 0.18 0.19 1.10327 1.09985 1.09623 1.09240 1.08837 41267 92726 57192 56008 11683 0.16799 59714 0.17901 18132 0.18999 24612 0.20093 58390 0.21183 98922 0. 65 0.66 0.67 0.68 0. 69 0.73954 0.72992 0.72027 0.71061 0.70095 67634 18814 81930 97784 06721 0.64202 0.64937 0.65662 0.66378 0.67084 93274 66880 77023 22027 00622 0.20 0.21 0.22 0.23 0.24 1.08413 1.07969 1.07506 1.07023 1.06522 47871 89342 61963 92672 09449 0.22270 25892 0.23352 19230 0.24429 59116 0.25502 25996 0.26570 00590 0.70 0.71 0.72 0. 73 0.74 0.69127 48604 0.68159 62792 0.67191 88112 0.66224 62838 0.65258 24665 0.67780 0.68466 0.69143 0.69810 0.70467 11938 55502 31231 39429 80779 0.25 0.26 0. 27 0.28 0.29 1.06001 1.05462 1.04904 1.04329 1.03736 41294 18194 71098 31885 33334 0.27632 63902 0.28689 97232 0.29741 82185 0.30788 00680 0.31828 34959 0.75 0.76 0.77 0.78 0. 79 0.64293 10692 0.63329 57399 0.62368 00626 0.61408 75556 0.60452 16696 0.71115 0.71753 0.72382 0.73001 0.73610 56337 67528 16140 04313 34538 0.30 0.31 0.32 0. 33 0. 34 1.03126 1.02498 1.01855 1.01195 1.00519 09096 93657 22310 31119 56887 0.32862 67595 0.33890 81503 0.34912 59948 0.35927 86550 0.36936 45293 0. 80 0. 81 0.82 0.83 0.84 0.59498 57863 0.58548 32161 0.57601 71973 0.56659 08944 3.55720 73967 0.74210 0.74800 0.75381 0.75952 0.76514 09647 32806 07509 37569 27115 0.35 0. 36 0. 37 0. 38 0. 39 0.99828 0.99122 0.98401 0.97665 0.96916 37121 10001 14337 89542 75592 0.37938 20536 0.38932 97011 0.39920 59840 0.40900 94534 0.41873 87001 0. 85 0.86 0.87 0. 88 0. 89 0.54786 97173 0.53858 07918 0.52934 34773 0.52016 05514 0.51103 47116 0.77066 0.77610 0.78143 0.78668 0.79184 80576 02683 98455 73192 32468 0.40 0.41 0.42 0. 43 0.44 0.96154 12988 0.95378 42727 0.94590 06256 0.93789 45443 0.92977 02537 0.42839 23550 0.43796 90902 0.44746 76184 0.45688 66945 0.46622 51153 0. 90 0. 91 0.92 0.93 0.94 0.50196 85742 0.49296 46742 0.48402 54639 0.47515 33132 0.46635 05090 0.79690 0.80188 0.80676 0.81156 0.81627 82124 28258 77215 35586 10190 0.45 0.46 0. 47 0.48 0.49 0.92153 20130 0.91318 41122 0.90473 08685 0.89617 66223 0.88752 57337 0.47548 17198 0.48465 53900 0.49374 50509 0.50274 96707 0.51166 82612 0.95 0.96 0.97 0. 98 0.99 0.45761 0.44896 0.44037 0.43187 0.42345 92546 16700 97913 55710 08779 0.82089 0.82542 0.82987 0.83423 0.83850 08073 36496 02930 15043 80696 0.50 0.87878 25789 C-5513 0.52049 1.00 0.41510 74974 0.84270 .I’ p-12 91671 63348 79057 40827 52040 erf 0.00000 0.01128 0.02256 0.03384 0.04511 2 00000 34156 45747 12223 11061 0.50 0. 51 0. 52 0. 53 0.54 0.87\8758 0.86995 0.86103 0.85204 0.84297 25789 15467 70343 34444 51813 erf 0.52049 0.52924 0.53789 0.54646 0.55493 .I: 98778 36198 86305 40969 92505 [1 98778 .I c(-y1 See Exa~nple 1. $= 0.88622 69255 [(-y1 07929 ERROR FUNCTION ERROR AND FRESNEL FUNCTION 2 G c--r2 AND 311 INTEGRALS Table ITS DERIVATIVE d erf 7 -2 p-2 ,G 7.1 erf .T .l.OO 1. 01 1.02 1.03 1. 04 0.41510 0.40684 0.39867 0.39058 0.38257 74974 71315 13992 18368 98986 0.84270 0.84681 0.85083 0.85478 0.85864 07929 04962 80177 42115 99465 1.50 1.51 1.52 1.53 1. 54 0.11893 02892 0.11540 38270 0.11195 95356 0.10859 63195 0.10531 30683 0.96610 0.96727 0.96841 0196951 0.97058 1. 05 1. 06 1.07 1.08 1. 09 0.37466 0.36684 0.35911 0.35147 0.34392 69570 43034 31488 46245 97827 0.86243 61061 0.86614 35866 0.86977 32972 0.87332 61584 0.87680 31019 1.55 1. 56 1.57 1.58 1.59 0.10210 0.09898 0.09593 0.09295 0.09005 0.97162 27333 0.97262 81220 0.97360 26275 0.97454 70093 0.97546 20158 1.10 1.11 1.12 1.13 1.14 0.33647 0.32912 0.32186 0.31470 0.30764 95978 49667 67103 55742 22299 0.88020 0.88353 0.88678 0.88997 0.89308 50696 30124 78902 06704 23276 1.60 1.61 1.62 1.63 1.64 0.08722 90586 0.08447 34697 0.08178 85711 0.07917 31730 0.07662 60821 0.97634 0.97720 0.97803 0.97884 0.97962 1. 15 1.16 1.17 1. 18 1.19 0.30067 0.29381 0.28704 0.28037 0.27381 72759 12389 45748 76702 08437 0.89612 0.89909 0.90200 0.90483 0.90760 38429 62029 03990 74269 82860 1.65 1. 66 1.67 1.68 1.69 0.07414 0.07173 0.06938 0.06709 0.06487 61034 20405 26972 68781 33895 0.98037 55850 0.98110 49213 0.98181 04416 0.98249 27870 0.98315 25869 1.20 1.21 1.22 1.23 1.24 0.26734 0.26097 0.25471 0.24854 0.24248 43470 83664 30243 83805 44335 0.91031 0.91295 0.91553 0.91805 0.92050 39782 55080 38810 01041 51843 1.70 1.71 1.72 1.73 1.74 0.06271 0.06060 0.05856 0.05657 0.05464 10405 86436 50157 89788 93607 0.98379 0.98440 0.98500 0.98557 0.98613 04586 70075 28274 84998 45950 1.25 1.26 1.27 1.28 1.29 0.23652 11224 0.23065 83281 0.22489 58748 Oi21923 35317 0.21367 10145 0.92290 0.92523 0.92751 0.92973 0.93189 01283 59418 36293 41930 86327 1.75 1.76 1.77 1.78 1.79 0.05277 0.05095 0.04918 0.04747 0:04580 49959 47262 74012 18791 70274 0.98667 0.98719 0.98769 0198817 0.98864 16712 02752 09422 41959 05487 1. 30 1. 31 1.32 1. 33 1.34 0.20820 0.20284 0.19757 0.19241 0.18734 79868 40621 88048 17326 23172 0.93400 0.93606 0.93806 0.94001 0.94191 79449 31228 51551 50262 37153 1.80 1.81 1. 82 1.83 1. 84 0.04419 17233 0.04262 48543 0.04110 53185 0.03963 20255 0.03820 38966 0.98909 0.98952 0.98994 0.99034 0.99073 05016 45446 31565 68051 59476 1. 35 1.36 1.37 1.38 1. 39 0.18236 0.17749 0.17271 0.16802 0.16343 99865 41262 40811 91568 86216 0.94376 21961 0.94556 14366 0.94731 23980 0.94901 60353 0.95067 32958 1.85 1. 86 1.87 1. 88 1.89 0.03681 0.03547 0.03417 0.03292 0.03170 98653 88774 98920 18811 38307 0.99111 10301 0.99147 24883 0.99182 07476 0.99215 62228 0.99247 93184 1.40 1. 41 1. 42 1.43 1.44 0.15894 0.15453 0.15022 0.14600 0.14187 17077 76130 55027 45107 37413 0.95228 0.95385 0.95537 0.95685 0.95829 51198 24394 61786 72531 65696 1.90 1.91 1.92 1.93 1.94 0.03052 0.02938 0.02827 0.02721 0.02617 47404 36241 95101 14412 84752 0.99279 0.99308 0.99337 0.99365 0.99392 04292 99398 82251 56502 25709 1.45 1.46 1.47 1.48 1.49 0.13783 0.13387 0.13001 0.12623 Oil2254 22708 91486 33993 40239 00011 0.95969 0.96105 0.96237 Oi96365 0.96489 50256 35095 28999 40654 78648 1.95 1. 96 1.97 1.98 1.99 0.02517 96849 0.02421 41583 0.02328 09986 0.02237 93244 0.02150 82701 0.99417 0.99442 0.99466 0:99489 0.99511 93336 62755 37246 20004 14132 1.50 0.11893 [(-;)l 1 0.96610 51465 2.00 0.02066 0.99532 22650 02892 [ c-y I 86576 19506 17995 70461 65239 69854 51465 67481 34969 62041 56899 83833 68366 80884 28397 17795 c1 C-5614 I- s = 0.88622 69255 312 ERROR Table 7.2 DERIVATIVE 2 F- z FUNCTION OF THE FRESNEL ERROR INTEGRALS FUNCTION 0 P-J2 4 X - 2)2.0666 r-i X Y= +f 2.00 2.01 2.02 2.03 2.04 AND 985 2.50 2.51 2.52 2.53 2.54 2.05 2.06 2;07 2.08 2.09 2.55 2.56 2.57 2.58 2.59 2;13 2.14 2.60 2.61 2.62 2.63 2.64 3.00 3.01 3.02 3.03 3.04 I- 3 3 3 3I 3 1.6922 1.6079 1.5275 1.4508 1.3777 II 136 137 078 325 304 - 3 1.2416 455 1.1783 764 1.3080 075 1.0607 500 1.1181 090 I- 411.2345 698 3.50 3.51 3.52 3.53 3.54 3.05 3.06 3.07 3.08 3.09 3.55 3.56 3.57 3.58 3.59 3.10 3.11 3.12 3.13 3.14 3.60 3.61 3.62 3.63 3.64 2.15 2.16 2.65 2.66 2.67 2.68 2769 3.15 3.16 3.17 3.18 3.19 3.65 3.66 3.67 3.68 3.69 2.20 2.21 2.22 2.23 2.24 2.70 2.71 2.72 2.73 2.74 3.20 3.21 3.22 3.23 3.24 3:72 3.73 3.74 2.25 2.26 2.27 2.28 2.29 2.75 2.76 2.77 2.78 2.79 3.25 3.25 3.27 3.28 3.29 3.76 3.77 3.78 3.79 2.80 2.81 2.82 2.83 2.84 3.30 3.31 3.32 3.33 3.34 3.80 3.81 3.82 3.83 3.84 (- 3)6.5249 776 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 $4; 2:42 2.43 2.44 - 3)4.5088 - 3)3.7301 (- I- 3 3.5556 3.3886 3 3.2288 3 3.0760 3 I 2.9298 292 092 2.85 2.86 2.87 2.88 2.89 487 700 871 230 098 - 5)X8428 I - 5)1.7242 (- 5)1.6130 397 768 192 3.85 3.86 3.87 3.88 3.89 2.90 2.91 2.92 2.93 2.94 3.37 3.38 3.39 3.90 3.91 3.92 3;93 3.94 3.45 3.46 3.47 3.48 3.49 2.50 (- 3)2.1782 842 3.00 (- 4)1.3925 305 F~O.88622 3.50 692% 3.95 3.96 3.97 3.98 3.99 (- 6)5.3994 268 4.00 (- 6)1.1028 II 445 - 7)4.1221 624 3.0245 867 3.2689 971 3.5324 796 3.8162 013 (- 7)2.7979 (- 7 1.8896 240 I 245 - 7 I 1.6128 557 1.3754 098 1.4895 135 1.7459 458 (- 7)1.2698 235 ERROR FUNCTION AND DERIVATIVE OF THE FRESNEL ERROR FUNCTION 2 ;= e--Z 0 ‘5 & X .t: Q--f 313 INTEGRALS 5.00 5.01 5.02 5.03 5.04 4.55 4.56 4.57 4.58 4.59 5.05 5.06 5.07 5.08 5.09 5.55 5.56 5.57 5.58 5.59 5.10 5.11 5.12 5.13 5.14 5.60 5.61 5.62 5.63 5.64 4.15 (4.16 4.17 4.18 4.19 8)3.7395 5.15 5.16 5.17 5.18 5.19 5.65 5.66 5767 5.68 5.69 -10)7.2909 -10)6.6494 450 435 I 414 4.20 (- 8 2.4632 041 4.21 4.22 4.23 I - 8 I 2.0814 484 1.9127 901 1.7574 463 2.2645 204 4724 4.70 4I71 4.72 4.73 4.74 (-lOj2.8766 4.75 t-lOj1.7934 4176 4.77 4.78 4.79 357 5.25 5.26 5.27 5:28 5.29 4.80 4.81 4.82 4.83 4.84 -10 -10 I -11 I -11 -11 1.1125 1.0105 9.1780 8.3337 7.5656 261 888 821 894 500 5.30 5.31 5.32 5.33 5.34 I -13 I -13 -13 -13 -13 4.35 4.36 4.37 4.38 4.39 4.85 4.86 4.87 4.88 4.89 I -11 -11 -11 -11 -11 I 6.8669 6.2315 5.6537 5.1285 4.6511 377 074 456 259 675 5.35 5.36 5.37 5.38 5.39 (-13 -13 -13 I -13 -13 4.40 4.41 4.42 4.43 4.44 4.90 4.91 4.92 4.93 4.94 5.40 5.41 5.42 5.43 5.44 4.45 4.46 4.47 4.48 4.49 4.95 4.96 4.97 4.98 4.99 866 169 089 820 702 694 4.25 4.26 4.27 4.28 4.29 :;670 1:4178 1.2825 1.1598 1.0487 x 4.50 4.51 4.52 4.53 4.54 4.60 4.61 4.62 4.63 4.64 -11 I -11 I -11 -11 -11 5.45 5.46 5.47 5.48 5.49 4.30 4.31 4.32 4.33 4.34 4.50 I - 8I 1.6143 994 1.1472 445 1.2498 673 1.3614 993 1.4826 974 I - 9 I 9.6595 598 8 8.8608 977 7.4517 102 8.1266 438 1.0528 442 (- 9)1.8113 059 Table 7.2 -2 e--z2 J; 5.00 (-11)1.5670 866 IT-O.88622 5.50 5.51 5.52 5.53 5.54 I -14)8.2233 -14 I 7.3659 160 5.2876 906 6.5967 480 5.9066 265 187 I -14 I 4.7325 917 -14 4.2349 422 3.3891 310 3.0309 585 3.7888 943 5.70 5.71 5.72 5.73 5.74 5.50 5.75 5.76 5.77 5.78 5.79 7.1305 6.4127 5.7660 5.1835 4.6589 505 516 568 412 423 5.80 5.81 5.82 5.83 5.84 I 4.1865 3.7613 3.3786 3.0343 2.7245 979 895 913 233 096 5.85 5.86 5.87 5.88 5.89 t-1312.4458 -13 2.1952 I -13 1.9699 -13 1.7673 -13 I 1.5853 396 336 112 627 234 5.90 5.91 5.92 5.93 5.94 t-1511.5481 468 -i5 liO888 898 i -16 \ 9.6798 241 5.95 5.96 5.97 5.98 5.99 (-14)8.2233 69255 160 6.00 (-16)2.6173 012 314 Table ERROR FUNCTION AND DERIVATIVE 7.2 2 p-2 FRESNEL OF THE INTEGRALS ERROR FUNCTION 2 AL2 6.00 6.01 6.02 6.03 6.04 -16)2%73 -16 2.3211 -16 1 2.0580 -16)1.8243 -16)1.6169 012 058 187 864 533 6.50 6.51 6.52 6.53 6.54 4 e-z’ q?r -19)5.0525 -19 4.4362 -19 3.8942 -19 I 3.4178 -19)2.9990 6.05 6.06 6.07 6.08 6.09 -16)1.4328 -16 1.2693 -16 1 1.1243 -17)9.9575 -17)8.8165 188 992 934 277 340 6.55 6.56 6.57 6.58 6.59 -19)2.6310 -19 2.3078 -19 2.0238 -19 1.7744 -19 I 1.5555 6.10 6.11 6.12 6.13 6.14 -17)7.8047 -17 6.9076 -17 1 6.1124 211 453 570 6.60 6.61 6.62 6.63 6.64 7.10 7.11 7.12 7.13 7.60 7.61 7.62 7.63 7.64 6.15 6.16 6.17 6.18 6.19 6.65 6.66 6.67 6.68 6.69 7.15 7.16 7.17 7.18 7.19 7.65 7.66 7.67 7.68 7.69 6.20 6.21 6.22 6.23 6.24 6.70 6.71 6.72 6.73 6.74 7.20 7.21 7.22 7.23 7.24 7.70 6.75 6.76 6.77 6.78 6.79 7.25 7.26 7.27 7.28 7.29 7.75 7.76 7.77 7.78 7.79 6.80 6.81 6.82 6.83 6.84 7.30 7.31 7.32 7.33 7.34 6.35 6.36 6.37 6.38 6.39 6.85 6.86 6.87 6.88 6.89 7.35 7.36 7.37 7.38 7.39 7.85 7.86 7.87 7.88 7.89 6.40 6.41 6.42 6.43 6.44 6.90 6.91 6.92 6.93 6.94 t-21)2.3751 704 7.40 7.41 7.42 7.43 7.44 7.90 7.91 7.92 7.93 7.94 (-21)1.1883 540 (-22 (-22 565 967 7.45 7.46 7.47 7.48 7.49 (-22)5.9159 630 7.50 2 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 6.34 iI -17 -17 -18 -18 -18 1.2241 1.0801 9.5297 8.4057 7.4128 x 281 812 064 325 421 (-18)1.0982 455 6.45 6.46 6.47 6.48 6.49 (-19)9.6542 574 (-19)5.7534 461 6.95 6.96 6.97 6.98 6.99 6.50 (-19)5.0525 800 7.00 7:8244 6.8042 L 800 038 418 066 603 7.00 7.01 7.02 7103 7.04 921 100 447 651 031 7.05 7.06 7.07 7.08 7.09 T-O.88622 69255 II t-22)2.9304 -22 2.5448 -22 1.9179 1.6645 2.2094 l-23)3.9913 450 491 450 736 057 893 7.55 7.56 7.57 7i58 7.59 t-25)1.9796 -251112573 -25)1.0803 292 541 765 654 185 7.95 7.96 7.97 7.98 7.99 (-25)4.2013 t-26)2.0097 (-28)4.0175 -28 3.4265 -28 2.9219 -28 2.4912 I -28 I 2.1234 202 874 899 008 982 8.00 (-28)1.8097 068 ERROR FUNCTION DERIVATIVE AND OF THE FRESNEL 315 INTEGRALS ERROR FUNCTION Table 7.2 z 8.00 8.01 8.02 8.03 8.04 9.00 9.01 9.02 9.03 9.04 I -36)7.4920 -36 I 6.2572 -36 5.2249 -36 4.3620 -36 3.6409 734 800 519 651 535 9.50 9.51 9.52 9.53 9.54 -32 1.2057 541 -32 1.0155 245 9.05 9.06 9.07 9108 9.09 -36)3.0384 -36j2.5351 -36j2.1147 -36j1.7637 -36)1.4707 441 317 690 559 105 9.55 9.56 9.57 9.58 9.59 8:63 8.64 -33)510997 -33)4.2908 438 734 9.10 9.11 9.12 9.13 9.14 -36)1.2261 -36 1.0219 -37 1 8.5167 -37)7.0959 -37)5.9110 088 837 148 960 925 8.15 8.16 8.17 8.18 8.19 8.65 8.66 8.67 8.68 8.69 -33)3.6095 -33 3.0358 -33 2.5527 -33 2.1461 -33 I 1.8039 760 465 988 817 709 9.15 9.16 9.17 9.18 9.19 -37 4.9230 -37 14.0993 -37)3.4127 -37)2.8406 -37)2.3639 8.20 8.21 8.22 8.23 8.24 8.70 8.71 8.72 8.73 8.74 -33)1.5160 -3311.2737 -33'1.0700 -34 8.9869 ( -34 i 7.5464 228 818 339 668 360 9.20 9.21 9.22 9.23 9.24 -37)1.9668 -37)1.6361 -37 1.3607 -37 1.1314 -38 I 9.4066 8.25 8.26 8.27 8.28 8.29 8.75 8.76 8.77 8.78 8.79 -34)6.3355 -34j5.3178 -34j4.4627 -34)3.7444 -34)3.1411 422 836 957 525 074 9.25 9.26 9.27 9.28 9.29 8.30 8.31 8.32 8.33 8.34 8.80 8.81 8.82 8.83 8.84 8.05 8.06 8.07 8.08 8.09 8.10 8.11 8.12 8.13 8.14 8.50 8.51 t-29)8.1112 334 8.55 (-29)4.2531 077 8i58 8.59 (-29j3.6173 797 (-29j1.8891 933 (-32)2.0157 780 9.30 9.31 9.32 8.45 8.46 8.47 8.48 8.49 8.50 638 9.60 9.61 9.62 9.63 9.64 -40)1.0662 907 619 592 918 437 423 9.65 9.66 9.67 9.68 9.69 (-41)4.0725 570 i-41)1.8788 710 449 251 427 847 395 9.70 9.71 9.72 9.73 9.74 -41)1.5477 017 -38)7.8186 802 -38 6.4974 888 -38 1 5.3984 710 9.75 9.76 9.77 9.78 9.79 -42)5.8524 -42 4.8150 -42 3.9608 I -42 I 3.2574 (-42)2.6784 252 968 401 873 979 (-43)8.2436 338 (-43)3.7428 271 -31)2.5623 -31 2.1658 -31 1.8303 -31 I 1.5465 -31)1.3064 I ( I (-32)4.7280 380 657 736 399 586 -38j1.7684 -38)1.4672 718 880 9.80 9.81 9.82 9.83 9.84 9.35 9.36 9I37 9.38 9.39 8.35 8.36 8.37 8.38 8.39 8.40 8.41 8.42 8.43 8.44 -40)1.2918 -38 1.2171 -38 1.0094 -39 8.3703 -39 I 6.9392 -39)5.7517 545 602 932 997 311 9.85 9.86 9.87 9.88 9.89 -39 4.7664 -39 3.9491 -39 3.2713 -39 I 2.7093 -39)2.2434 456 520 439 286 186 9.90 9.91 9.92 9.93 9.94 -43)3.0708 -43 2.5189 -43 2.0658 -43 1.6939 -43 I 1.3886 096 477 489 130 628 -35)4.4873 -35 3.7552 -35 3.1420 -35 2.6283 -35 I 2.1982 418 711 030 611 476 9.40 9.41 9.42 9.43 9.44 8.95 -3511.8381 516 574 922 315 435 9.95 9.96 9.97 9.98 9.99 -43)1.1381 -3511.0734 -36)8.9687 9.45 9.46 9.47 9:48 9.49 (-39)1.8572 8;98 8.99 139 8.90 8.91 8.92 8.93 8.94 734 9.50 (-40)7.2007 555 10.00 (-44)4.1976 562 9.00 (-36)7.4920 $f=O.88622 69255 316 ERROR Table FUNCTION AND COMPLEMENTARY 7.3 0.250 0.245 0.240 0.235 0.230 0.225 0.220 0.215 0.210 0.205 0.51505 0.51592 0.51681 0.51769 0.51859 55 92 01 83 40 0.200 0.195 0.190 0.185 0.180 0.51949 0.52040 0.52132 0.52225 0.52318 74 85 75 45 98 0.175 0.170 0.165 0.160 0.155 0.52413 0.52508 0.52604 0.52701 0.52799 33 55 63 59 46 0.150 0.145 0.140 0.135 0.130 0.52898 0.52997 0.53098 0.53200 0.53303 25 98 67 35 02 0.125 <x> 22 xez’ erfc x 0.51079 14 0.51.163 07 0.51247 67 0.51332 94 0.51418 90 0.53406 72 C-i)1 x-2 See Example 2. FRESNEL INTEGRALS ERROR FUNCTION 2 0";25 2 : xez2erfc x 0.53406 72 0.53511 47 0.53617 29 0.53724 20 0.53832 23 0.100 0.095 0.090 0.085 0.080 0.53941 0.54051 0.54163 0.54276 0.54390 0.54505 0.54622 0.54740 0.54859 0.54980 51 19 24 69 58 3 3 0.050 0.045 0.040 0.035 0.030 0.55102 0.55226 0.55352 0.55479 0.55608 95 85 32 41 17 ; 0.025 0.020 0.55738 0.55870 0.56005 0.56140 0.56278 65 90 00 99 96 22 t 2 2 : 3 3 3 3 [1 0.015 0.010 0.005 0.000 0.56418 96 C-i)3 II 1 <x> =nearest integer to x. n erfc &G : i 0.00039 27505 88282 0.01218 88821 84803 0.00000 41444 64662 0.00001 05351 02689 5 0.00000 00208 26552 3 41 76 32 11 16 0.075 0.070 0.065 0.060 0.055 <x> 33 33 n 6 87 9 10 0.00000 0.00000 0.00000 0.00000 0.00000 erfc Jn?r 00008 25422 00000 33136 00000 01343 00000 00055 00000 00002 erfc x =72 me-tzdt =1 -erf x dr S z erfc 4% compiled from 0. Emersleben, Numerische Werte des Fehlerintegrals fiir &, Z. Angew. Math. Mech. 31,393-394, 1951 (with permissi&). 6 7 1: 14 m ERROR REPEATED FUNCTION AND FRESNEL INTEGRALS OF THE ERROR FUNCTION 2°F l+l ( o.“o 0.1 0.2 0. 3 0.4 317 INTEGRALS ) i” erfc .t* 71=3 n=l n=2 1.00000 Table 7.4 1.00000 n=4 1.00000 I - 1I 4.28565 7.62409 3.15756 5.74882 1.00000 - 1)5.56938 I - 1)4.46884 (- 1 2.29846 (- 1 1.65244 (- 2)5:68138 1. 0 :-: 1:3 1.4 1. 5 (- 3j8.02626 :*; 1: 8 1. 9 (- 3j1.26566 ;:1" 9; 2: 4 (- 4j2.22250 - 6)9.52500 92 2: 7 2. 8 2.9 22 33.87 3:9 I - 9I 4.58945 8 2.32831 8)5.10148 1.98190 1.04329 4. 0 4.1 II i-f 4: 4 -13 3.19826 -13 4.02809 -14 8.76348 -12 1.14567 2.35705 t-6' 4: 7 4.8 4.9 5. 0 (-14)5.61169 (-13)2.62561 (-1)5.64189 See Examples 4 and 5. 58355 (-1)2.50000 (-14)1.38998 (-15)3.83592 pr (;+I)]-’ 00000 (-2)9.40315 97258 (-2)3.12500 318 ERROR Table 7.4 FUNCTION REPEATED INTEGRALS 1.00000 n=5 2 0”:: AND FRESNEL OF THE 0:4 ERROR FUYCTION 237 ;+I i" erfc 5 ( 1 n=6 11 =I0 1.00000 1.00000 - 1 6.28971 - 1 3.91490 - lj2:41089 1)1.46861 II I i*% INTEGRALS I- I- 111.65569 if: 0: 7 (- I i:! 2)7.95749 2)4.64127 2)2.67626 2)1.52533 3)8.59126 1. 0 1'::. 2: (- 3)1.19278 1.5 (- 4)2.89186 ::; I- - 5j4;04407 5)2.04244 1':: 22'10 2: 2 $1'4 f:'6 ;*i 2: 9 ;:1" E 3:4 - 9)2.47236 3:: 2; 3:9 -~0)1.84200 -11)7.48503 (-11)7.30331 (-11)2.91245 4. 0 4.1 2.3' 4: 4 I 22 4:7 4. a 4. 9 5. 0 13)1.78294 I--1416.31544 -14j2.20038 -13)3.82601 -13)1.37691 -14j4: 87328 -15 6.46126 -17 2.10125 -16 1.95316 -16 16.71719 2.11065 (-15)1.15173 (-16)3.70336 (-18)6.51829 pr t-3)9.40315 97258 (-3)2.60416 66667 (-18)2.61062 (;+1)]-l (-6)8.13802 08333 (-6) 1.69609 66316 ERROR FUNCTION AND FRESNEL DAWSON’S 2: 0.00 0.00000 0.02 0.04 0.06 0.08 0.01999 46675 0.03995 73606 0.05985 62071 0.07965 95389 1.00 1. 02 1.04 1.06 1.08 0.10 0.12 0.14 0.16 0.18 0.09933 59924 fl.11885 46083 0.13818 49287 0.15729 70920 0.17616 19254 0.20 0.22 0.24 0.26 0.28 0.19475 0.21303 0.23099 0.24859 0.26581 319 INTEGRALS Table 7.5 xe-=zet2c/t s <x> INTEGRAL - s0 ’ e 2.2 rt;lt = 0.53807 95069 x-2 44359 71471 50787 57454 0.250 0.245 0.240 0.235 0.230 0.6026080777 0.60046 6027 0.59819 8606 0.59588 1008 0.59351 6018 1.10 1.12 1.14 1.16 1.18 0.52620 66800 0.52291 53777 0.51935 92435 0.51555 55409 0.51152 13448 0.225 0.220 0.215 0.210 0.205 0.59110 0.58865 0.58616 0.58364 0.58109 6724 6517 9107 8516 9080 10334 68833 28865 34747 41727 1.20 1.22 1.24 1.26 1.28 0.50727 0.50282 0.49820 0.49341 0.48847 34964 85611 27897 20827 19572 0.200 0.195 0.190 0.185 0.180 0.57852 0.57593 0.57332 0.57071 0.56809 5444 2550 5618 0126 1778 0.30 0.32 0.34 0.36 0.38 0.28263 16650 0.29902 38575 0.31496 99336 0.33045 04051 0.34544 71562 1.30 1.32 1.34 1.36 1.38 0.48339 0.47820 0.47290 0.46751 0.46204 75174 34278 38898 26208 28368 0.175 0.170 0.165 0.160 0.155 0.56547 0.56287 0.56027 0.55770 0.55516 6462 0205 9114 9305 6829 0.40 0.42 0.44 0.46 0.48 0.35994 34819 Oii~739241210 0.38737 52812 0.40028 46599 0.41264 14572 1.40 1.42 1.44 1.46 1.48 0.45650 0.45091 0.44528 0.43962 0.43394 72375 79943 67410 45670 20135 0.150 0.145 0.140 0.135 0.130 0.55265 0.55018 0.54776 0.54538 0.54305 7582 7208 0994 3766 9774 0.50 0.52 0.54 0. 56 0.58 0.42443 63835 0.43566 16609 0.44631 10184 0.45637 96813 0.46586 43551 1.50 1.52 1.54 1.56 1.58 0.42824 0.42255 0.41686 0.41119 0.40555 90711 51804 92347 95842 40424 0.125 0.120 0.115 0.110 0.105 0.54079 0.53858 0.53643 0.53435 0.53233 2591 5013 8983 5529 4747 0. 60 0.62 0.64 0.66 0.68 0.47476 32037 0.48307 58219 0.49080 32040 0.49794 77064 0.50451 30066 1.60 1.62 1.64 1.66 1.68 0.39993 0.39436 0.38883 0.38335 0.37792 98943 39058 23346 09429 50103 0.100 0.095 0.090 0.085 0.080 0.53037 0.52847 0.52663 0.52484 0.52311 5810 7031 5967 9575 4393 0.70 0.72 0.74 0.76 0.78 0.51050 40576 0.51592 70382 0.52078 93010 0.52509 93152 0.52886 66089 1.70 1.72 1.74 1.76 1.78 0.37255 93490 Oi36725 83182 0.36202 58410 0.35686 54206 0.35178 01580 0.075 0.070 0.065 0.060 0.055 0.52142 0.51978 0.51817 0.51661 0.51508 6749 2972 9571 3369 1573 0.80 0.82 0.84 0.86 0.88 0.53210 0.53481 0.53702 0.53873 0.53996 17071 60684 20202 26921 19480 1.80 1.82 1.84 1.86 1.88 0.34677 0.34184 0.33700 0.33223 0.32756 27691 56029 06597 96091 38080 0.050 0.045 0.040 0.035 0.030 0.51358 0.51211 0.51067 0.50925 0.50786 1788 1971 0372 5466 5903 0.90 0.92 0.94 0.96 0.98 0.54072 0.54103 0.54090 0.54036 0.53941 43187 49328 94485 39857 50580 1.90 1.92 1.94 1.96 1.98 0.32297 43193 0.31847 19293 0.31405 71655 0.30973 03141 0.30549 14372 0.025 0.020 0.015 0.010 0.005 0.50650 0473 0.50515 8078 0.50383 7717 0.50253 8471 0.50125 9494 0.53807 95069 2.00 0.30134 03889 C-l)4 0.000 0.50000 0000 See Example 00000 3. Compiled from J. B. Rosw, 0.53637 0.53431 0.53192 0.52921 [1 <x> =nearest integer Theory and application to 2. of Jo” rp2dx [c-p1 ; ; 2 6 T: :: m and so2 e-@@dy so” c~‘t/.t~. Mapleton House, Brooklyn, N.Y., 1948; and B. Lohmander and S. Rittsten, Table of the function y=ePzz so2rt*dt,Kungl. Fysiogr. Stillsk. i Lund Forh. 28,45-52, 1958 (with permission). 320 Table X ERROR FUNCTION 7.6 - 3 JZe-Qt r1 0 0 0.80000 00 2 AND FRESNEL INTEGRALS 3 -J” e-t3at r! 0 03 3 _ J” e-Pdt r! 0 0 0.‘72276 69 0.73842 49 0.75360 34 0.76829 12 0.78247 88 X 1.40 1.42 1.44 1.46 1.48 0.398973 0.99109 0.99229 0.99335 0.99429 54 36 70 97 49 1.50 1.52 1.54 1.56 1.58 0.99511 0.99583 0.99645 0.99699 0.99746 49 14 52 62 38 42 62 62 89 05 1.60 1.62 1.64 1.66 1.68 0.99786 0.99821 0.99850 0.99875 0.99897 63 16 65 75 03 0.90428 0.91228 0.91979 0.92683 0.93341 86 25 27 11 06 1.70 0.99914 99 1.10 1.12 1.14 1.16 1.18 0.93954 0.94525 0.95054 0.95543 0.95995 56 09 27 76 30 1.70 1.74 1.78 1.82 1.86 0.99914 99 75 05 26 14 28 66 46 80 78 1.20 1.22 1.24 1.26 1.28 0.96410 0.96791 0.97140 0.97457 0.97746 64 62 05 79 66 1.90 1.94 1.98 2.02 2.06 0.99990 0.99993 0.99996 0.99997 0.99998 01 82 24 76 69 0.63775 0.65560 0.67304 0.69006 0.70664 57 39 52 30 18 1.30 1.32 1.34 1.36 1.38 0.98008 0.98245 0.98458 0.98649 0.98820 48 07 18 52 77 2.10 2.14 2.18 2.22 2.26 0.99999 0.99999 0.99999 0.99999 0.99999 25 57 77 87 93 0.72276 69 1.40 0.98973 54 2.30 0.99999 97 0.02239 0.04479 0.06718 0.08957 69 31 72 63 0.70 0.72 0.74 0.76 0.78 0.10 0.12 0.14 0.16 0.18 0.11195 0.13432 0.15667 0.17899 0.20127 67 36 11 22 90 0.80 0.82 0.84 0.86 0. 88 0.79615 0.80932 0.82196 0.83408 0.84567 78 16 48 41 73 0.20 0. 22 0.24 0.26 0. 28 0.22352 0.24571 0.26783 0.28988 0.31184 24 24 80 71 70 0,90 0.92 0.94 0.96 0.98 0.85674 0.86728 0.87730 0.88680 0.89580 0. 0. 0. 0. 0. 30 32 34 36 38 0.33370 0.35544 0.37704 0.39850 0.41979 37 26 82 45 45 1.00 1.02 1.04 1.06 1.08 0.40 0. 42 0. 44 0. 46 0.48 0.4poso 0.46180 0.48248 0.50293 0.52312 07 52 96 51 25 0.50 0. 52 0.54 0. 56 0. 58 0.54303 0.56264 0.58194 0.60090 0.61951 0. 60 0. 62 0.64 0. 66 0. 68 0.70 0. 00 0. 02 0. 04 0. 06 0.08 3 __ J” eddt r! 0 0 [1 55)‘3 [1 c-:,5)7 0.99942 0.99962 0.99975 0.99984 [c-y1 I 95116 Compiled from M. Abramowitz, (with permission). Table of the integral o Szewu3du,J. Math. Phys. 30,162-163,195l ERROR FUNCTION AND FRESNEL 0.00 0.00000 0.02 0.04 0.00062 0.00251 0. 06 0.00565 0.08 0.01005 00 83 33 00 0.00000 00 0.00026 80 95 76 24 38 0.00052 0.00090 0.00143 0.00214 0.00305 36 47 67 44 31 1.10 1.12 1.14 1.16 1.18 1.90066 1.97040 2.04140 2.11366 2.18717 11 29 36 70 56 0.00418 0.00557 0.00723 0.00919 0.01148 76 30 40 54 lb 1.20 1.22 1.24 1.26 1.28 2.26194 2.33797 2.41525 2.49379 2.57359 56 1.30 1.32 1.34 1.36 1.38 0.06283 19 65 0.28 0.09047 0.10618 0.12315 79 58 04 0.19992 0.21987 0.23980 0.25970 0.27957 0.30 0.32 0. 34 0.36 0. 38 0.14137 0.16084 0.18158 0.20357 0.22682 17 95 41 52 30 0.29940 0.31917 0.33888 0.35851 0.37804 10 31 06 09 96 0.01411 0.01712 0.02053 0.02435 11 68 0.02862 55 0.40 0.42 0.44 0.46 0.48 0.25132 74 0.27708 85 0.30410 62 0.33238 05 0.36191 15 0.39748 0.41678 0.43594 0.45494 0.47375 08 68 a2 40 10 0.03335 0.03858 0.04430 94 02 85 0.50 0.52 0.54 0.39269 0.42474 0.45804 91 33 42 0.56 0.58 0.49260 17 0.49234 0.51069 0.52878 0.54656 0.56401 42 69 01 30 31 0.60 0.62 0.64 0.66 0.68 0.56548 0.60381 0.64339 0.68423 0.72633 67 41 82 a9 62 0. 70 0.72 0. 74 0.76 0.78 0.76969 0.81430 0.86016 0.90729 0.95567 02 08 81 20 25 0.80 0.82 0.84 0.86 0.88 1.00530 1.05620 1.10835 1.16176 1.21642 96 35 39 10 47 0.72284 0.73312 0.74251 0.75095 0.75840 42 83 54 79 90 0.90 0.92 0.94 50 20 56 59 28 0.76482 0.77015 0.77436 30 63 72 0.98 1.27234 1.32952 1.38795 1.44764 1.50859 1.00 1.57079 63 0.77989 59 Example 8. 65 33 68 68 36 69 69 35 0.77989 0.77926 0.77735 0.77414 0.76963 34 11 01 34 03 0.43825 0.45824 0.47815 0.49788 0.51736 91 58 08 a4 86 0.76380 0.75667 67 60 0.53649 0.55517 0.57331 79 92 28 0.59079 0.60752 66 74 0.74824 0.73854 0.72759 94 68 68 0.71543 0.70211 77 76 27 0.68769 0.67223 0.65582 2.65464 58 0.63855 2.73695 2.82052 2.90534 2.99142 55 19 49 45 0.62051 0.60181 0.58259 0.56297 3.07876 3.16735 3.25720 3.3483'0 3.44067 08 37 33 95 23 17 78 68 67 33 64 62 0.62340 09 0.63831 47 78 0.65216 0.66484 34 63 0.67626 72 05 0.68633 33 11 95 73 59 0.69495 0.70205 0.70755 0.71139 62 50 67 77 0.54309 0.52310 0.50316 0.48342 0.46407 58 58 23 80 05 0.71352 0.71389 0.71248 0.70928 0.70428 51 77 78 lb 12 0.44526 12 32 99 29 96 0.69750 0.35351 0.34325 0.33481 0.32830 17 20 29 32 61 69 02 83 87 25 0.05056 42 0.05736 63 0.06473 0.08122 0.09037 0.10014 24 89 06 08 09 1.50 1.52 1.54 1.56 1.58 3.53429 3.62916 3.72530 3.82268 3.92133 60 0.11054 0.12157 0.13325 0.14557 0.15853 02 59 28 29 54 1.60 1.62 1. b4 1.66 1.68 4.02123 4.12239 86 79 0.17213 0.18636 0.20122 0.21668 0.23272 65 89 21 lb 88 1.70 1.72 1.74 1.76 1.78 4.86569 87 4.97691 11 0.32382 0.32145 0.32122 0.32318 0.32733 0.24934 14 :%i 5.08938 5.20310 01 58 0.33363 0.34203 29 39 0.26649 22 98 80 55 1:04 1.86 1.88 5.31808 5.43432 5.55182 80 70 25 0.35244 0.36476 96 35 0.37882 93 68 95 0.33977 0.35904 0.37859 0.39836 0.41827 63 93 a1 12 21 1.90 1.92 1.94 1.96 1.98 5.67057 5.79058 5.91184 0.39447 0.41148 0.42963 05 24 33 6.15814 36 91 12 99 0.44866 0.46830 69 56 34 0.43825 91 2.00 6.28318 53 0.48825 34 0.58109 54 0.59777 37 0.61400 94 0.62976 0.64499 25 0.65965 0.67370 0.68709 24 12 20 0.69977 79 0.71171 13 0.77741 0.77926 70 63 1.40 1.42 1.44 1.46 1.48 12 0.07267 0.28414 0.30227 0.32083 [1[1 C-54)2 h 1.57079 1.63425 1.69897 1.76494 1.83217 75 39 67 41 34 0.07602 0.96 1.00 1.02 1.04 1.06 1.08 0.09999 0.11999 0.13998 0.15997 0.17995 0.20 0.22 0.24 0.52841 Table 7.7 31 0.01570 0.02261 0.03078 0.04021 0.05089 0.26 INTEGRALS 42 35 31 81 49 0.00000 0.00003 0.00011 321 INTEGRALS 0.02000 00 0.04000 00 0.05999 98 0.07999 92 0.12 0.14 0. lb 0. 18 0. 10 0.00000 FRESNEL C-i)8 06 99 4.22481 38 4.32848 4.43341 64 56 4.53960 4.64704 4.75574 14 39 30 6.03437 47 0.42717 0.40997 0.39385 0.37895 0.36546 19 56 50 0.68898 88 0.67878 0.66697 0.65363 46 0.63888 0.62286 0.60570 0.58758 77 07 26 04 0.56867 83 0.54919 0.52934 0.50935 60 73 84 0.48946 49 0.45093 0.43280 0.41573 0.39999 0.38579 88 06 97 44 25 0.37334 0.35448 0.34838 73 37 37 30 0.34466 65 0.46990 0.36285 67 13 94 0.34341 57 C-f3 [. 1 322 ERROR FUNCTION Table 7.7 C(.r)=jl" AND FRESNEL INTEGRALS FRESNEL INTIKRALS cos (;13) t/l s f.1.) C(T) 4. 04 4. 06 4.08 C(r) 0.49842 0.51821 0.53675 0.55284 0.56543 60 54 05 04 47 N (.I.) 0.42051 58 0.42301 99 0.43039 00 0.44217 81 0.45764 45 59 11 29 66 34 4.10 4.12 4.14 4.16 4.18 0.57369 56 0.57705 88 0.57527 76 0.56844 74 0.55700 75 0.47579 83 0.49545 71 0.51532 14 0.53405 87 0.55039 41 0.59334 0.58416 0.57161 0.55618 0.53849 95 97 47 06 35 4.20 4.22 4.24 4.26 4.28 0.54171 0.52362 0.50396 0.48411 0.46549 92 06 08 63 61 0.56319 89 0.57157 23 0.57491 03 0.57295 47 0.56582 05 0.51928 0.49936 0.47960 0.46084 0.44393 61 95 04 46 82 4.30 4. 32 4. 34 4.36 4. 38 0.44944 0.43712 0.42946 0.42704 0.43006 12 50 40 39 79 0.55399 0.53831 0.51990 0.50011 0.48041 59 55 77 73 08 0.43849 17 0.45514 37 0.41375 96 0.49348 70 0.51340 62 0.42964 95 0.41864 11 0.41143 69 0.40839 28 0.40967 54 4. 40 4.42 4.44 4.46 4.48 0.43833 0.45123 0.46781 0.48679 0.50671 29 59 05 41 95 0.46226 0.44707 0143599 0.42990 0.42931 80 06 33 86 16 3.50 3.52 3.54 3.56 3.58 0.53257 0.55006 0.56501 0.57668 0.58446 24 11 32 02 43 0.41524 0.42486 0.43808 0.45428 0.47265 80 72 83 17 92 4.50 4.52 4.54 4. 56 4.58 0.52602 0.54318 0.55680 0.56578 0.56936 59 11 46 27 57 0.43427 30 0.44442 34 0.45897 36 0.47676 89 0.49637 56 93 53 79 35 52 3.60 3. 62 3.64 3.66 3.68 0.58795 0.58694 0.58147 0.57178 0.55838 33 64 10 75 18 0.49230 0.51224 0.53143 0.54888 0.56366 95 12 21 15 38 4. 60 4. 62 4.64 4.66 4.68 0.56723 0.55954 0.54691 0.53039 0.51135 67 81 86 13 38 0.51619 23 0.53451 97 0.54999 67 0.56113 28 0.56702 44 0.45291 0.43518 0.42066 0.40798 0.39817 75 98 03 90 24 3.70 3.72 3.74 3. 76 3.78 0.54194 57 0.52334 49 0.50357 70 0.48371 94 0.46487 19 0.57498 0.58220 0.58492 0.58296 0.57641 04 56 61 92 91 4.70 4.72 4. 74 4.16 4.78 0.49142 0.47232 0.45572 0.44308 0.43554 65 71 30 30 28 0.56714 0.56146 0.55044 0.53504 0.51659 55 19 52 16 82 0.46749 17 0.48720 04 0.50717 21 0.52671 66 0.54538 21 0.39152 0.38828 0.38856 0.39238 0.39964 84 41 43 50 80 3.80 3.82 3.84 3.86 3.88 0.44809 0.43434 0.42443 0.41894 0.41822 49 86 43 43 16 0.56561 0.55115 0.53384 0.51466 0.49472 87 74 32 22 45 4.80 4. 82 4. 84 4.86 4.88 0.43319 0.43802 0.44786 0.46244 0.48042 66 47 69 40 90 0.49675 0.47728 0.45995 0.44637 0.43780 02 00 75 74 82 2.90 2.92 2.94 2.96 2.98 0.56237 0.57718 0.58930 0.59830 0.60384 64 78 60 19 56 0.41014 0.42353 0.43941 0.45124 0.47643 06 87 39 45 06 3.90 3.92 3.94 3. 96 3.98 0.42233 0.43105 0.44389 0.46007 0.41863 27 68 17 70 51 0.47520 0.45726 0.44198 0.43032 0.42301 24 13 92 79 17 4.90 4.92 4.94 4.96 4. 98 0.50016 0.51979 0.53747 0.55150 0.56051 10 51 34 25 94 0.43506 0.43843 0.44761 0.46175 0.47951 74 48 56 67 78 3.00 0.60572 08 [C-64)5] 0.49631 [(-y] 30 4.00 0.49842 60 0.42051 58 5.00 0.56363 [ ‘-;)‘I 12 0.49919 14 [C-$)8] %-I S(T) 0.34i4i 0.34467 0.34844 0.35470 0.36334 57 48 87 04 98 3.00 3.02 3.04 3.06 3.08 0.60572 0.60383 0.59823 0.58910 0.57674 08 73 78 11 01 0.49631 0.51619 0.53536 0.55311 0.56880 30 42 29 95 28 0.58156 41 0.59671 75 0.61000 60 0.62117 32 0.62999 53 0.37421 34 0.38730 0.40223 0.41880 0.43673 37 09 45 63 3.10 3.12 3; 14 3.16 3.18 0.56159 0.54421 0.52525 0.50543 0.48552 39 58 53 56 76 0.58181 0.59165 0.59791 0.60033 0.59880 2.20 2.22 2.24 2.26 2.28 0.63628 0.63990 0.64075 0.63879 0.63403 60 31 25 28 83 0.45570 0.41535 0.49532 0.51521 0.53462 46 85 41 11 03 3.20 3.22 3.24 3.26 3.28 0.46632 0.44858 0.43306 0.42040 0.41113 03 96 55 05 97 2.30 2.32 2.34 2.36 2.38 0.62656 0.61649 0.60402 0.58940 0.57293 17 45 69 65 44 0.55315 0.57041 0.58602 0.59964 0.61095 16 28 84 89 96 3.30 3.32 3. 34 3.36 3.38 0.40569 0.40431 0.40709 0.41393 0.42455 44 99 96 66 18 2.40 2.42 2.44 2.46 2.48 0.55496 0.53588 0.51612 0.49614 0.47641 14 11 29 28 35 0.61969 00 0.62562 11 0.62859 38 0.62851 43 0.62535 98 3.40 3. 42 3.44 3.46 3.48 2.50 2.52 2.54 2.56 2.58 0.45741 0.43961 0.42346 0.40939 0.39777 30 32 72 65 93. 0.61918 0.61010 0.59834 0.58415 0.56790 18 76 06 75 42 2.60 2.62 2.64 2.66 2.68 0.38893 0.38312 0.38052 0.38123 0.38525 75 73 80 50 32 0.54998 0.53087 0.51106 0.49110 0.47153 2.70 2.72 2.74 2.76 2.78 0.39249 0.40277 0.41581 0.43125 0.44865 40 39 68 85 46 2.80 2.82 2.84 2.86 2.88 2.:0 2.02 2. 04 2. 06 2.08 0.48825 0.50820 0.52782 0.54681 0.56482 2.10 2.12 2.14 2.16 2.18 For .1>5 iri 3 34 04 73 06 79 = 0.5i ( O.3183O99-T For u>39 $#= 0.51 [ 1[1 (-;I6 (-;I6 +c(2) t@)1c3 x10-7 ERROR FUNCTION AND AUXILIARY 2(sL X * FRESNEL 323 INTEGRALS Table FUNCTIONS n ,y 7.8 f(x)= h(20 0.00000 2 00000 00000 &00062 0.00251 0.00565 0.01005 83185 32741 48667 30964 0.10 0.12 0. 14 0.16 0.18 0.01570 0.02261 0.03078 0.04021 0.05089 0.20 0.22 0.24 0.26 0.28 30718 22872 76462 91487 0.50000 0.49969 0.49880 0.49739 0.49548 00000 41196 88057 07811 44294 00000 39303 20520 66949 00553 0.50000 0.48031 0.46125 0.44281 0.42500 00000 40626 51239 99356 33536 00000 54163 79101 00196 38036 79632 94671 76080 23859 38009 67949 05847 05180 65949 88155 0.49313 0.49037 0.48724 0.48378 0.48002 18256 27777 48761 35493 21268 06624 82254 11561 31728 70713 0.40779 0.39119 0.37518 0.35976 0.34491 85545 72364 98069 55566 28197 29930 96391 99885 09573 39391 0.06283 0.07602 0.09047 0:10618 0.12315 18530 65422 78684 58316 04320 71796 16873 23386 91335 20720 0.47599 0.47172 0.46724 0.46257 0.45774 19056 22205 05176 24293 la508 49140 45221 22164 12303 40978 0.33061 0.31687 0.30365 0.29095 0.27876 91227 13200 57186 81914 42811 69034 89318 36191 92531 44593 0.30 0. 32 0.34 0. 36 0. 38 0.14137 0.16084 0.18158 0.20357 0.22682 16694 95438 40553 52039 29895 11541 63797 77490 52619 89183 0.45277 0.44768 0.44248 0.43721 0.43187 10172 05805 96860 60487 60273 56087 06203 81319 95888 53913 0.26705 0.25582 0.24505 0.23472 0.22483 92929 83796 66166 9OiO3 08578 81728 24420 57772 35799 07150 0.40 0.42 0.44 0.46 0. 48 0.25132 0.27708 0.30410 0.33238 0.36191 74122 84720 61688 05027 14736 87183 46620 67492 49800 93544 0.42648 0.42105 0.41560 0.41013 0.40466 46973 59227 24246 58491 68313 90789 36507 90070 35691 67950 0.21534 0.20626 0.19756 0.18923 0.18126 72003 34704 52322 82774 86555 95520 48744 49727 60398 47172 0.50 0.52 0.54 0.56 0.58 0.39269 0.42474 0.45804 0.49260 0.52841 90816 33267 42088 17280 58843 98724 65340 93392 82880 33803 0.39920 0.39375 0.38833 0.38294 0.37759 50585 93295 76127 71004 42617 25702 63563 15400 26771 52882 0.17364 0.16634 0.15936 0.15269 0.14631 26996 70480 86623 48414 32329 13238 39628 13733 00876 91905 0.60 0.62 0.64 0.66 0. 68 0.56548 0.60381 0.64339 0.68423 0.72633 66776 41080 81754 88799 62215 46163 19958 55190 51857 09960 0.37228 0.36702 0.36181 0.35666 0.35157 48922 41612 66571 64292 70288 35620 87842 25476 98472 80259 0.14021 0.13437 0.12880 0.12347 0.11838 la419 90361 35503 44874 13187 37684 59907 06985 03863 25611 0.70 0.72 0.74 0.76 0.78 0.76969 0.81430 0.86016 0.90729 0.95S67 02001 08158 80685 19583 24852 29499 10474 52885 56732 22015 0.34655 0.34159 0.33670 0.33188 0.32713 15463 26474 26065 33382 64271 82434 67053 33192 57734 72503 0.11351 0.10886 0.10441 0.10016 0.09611 38821 23788 73696 97688 08389 06517 79214 22082 77848 91866 0. 80 0.82 0.84 0. 86 0.88 1.00530 1.05620 1.10835 1.16176 1.21642 96491 34501 38881 09632 46754 48734 36888 86479 97506 69968 0.32246 0.31786 0.31334 0.30889 0.30452 31553 45283 12993 39917 29200 61284 60796 49704 09068 36579 0.09223 0.08852 0.08498 0.08159 0.07836 21832 57381 37656 88446 38619 05037 23702 77045 61614 62362 0. 90 0.92 0.94 0.96 0.98 1.27234 1.32952 1.38795 1.44764 1.50859 50247 20109 56343 58947 21922 03866 99200 55971 74177 53819 0.30022 0.29600 0.29186 0.28780 0.28380 82096 98149 75359 10340 98467 95385 76518 51781 91658 20271 0.07527 0.07231 0.06949 0.06679 0.06420 20035 67451 la433 13255 94813 30280 87932 26312 49021 13093 1.00 1.57079 63267 94897 0.27989 34003 76823 0.06174 08526 09645 sin uyz(u) cos u 0. 00 0. 02 0.04 0.06 0. 08 Lm1 c-y See Exm~ple~ 6, 7, and 1 Cc,)=-+f(x) 2 1 S(X)=--fG) 2 [ c-g I 9. sin ( ix2 ) cos (5”)-g(x) -8(r) cos t TTZ 2 sin (ixz) 1 C&(U)=! 2 s~(~)=A-~~(u) 2 +fi(u) cos u-g2(u) sin II 324 ERROR Table FUNCTION AUXILIARY 7.8 x-1 1.00 0.98 0.96 0.94 0.92 0.63661 0.61140 0.58670 0.56251 0.53883 0.90 0.88 0.86 0. a4 0.82 0.80 0. 78 0.76 0.74 0.72 AND FRESNEL INTEGRALS FUNCTIONS g(4= k?(u) f(1.)=h(U) 0.27989 34003 0.27597 33733 0.27197 11505 0.26788 56989 0.26371 60682 76823 36442 76851 47656 37287 0.06174 0.05933 0.05693 0.05456 0.05220 08526 31378 89827 06112 03510 09645 64174 01255 91100 52931 0.51566 20156 17741 0.49299 a3517 21455 0.47084 39836 43063 0.44919 a9113 a2565 0.42806 31349 39962 0.25946 0.25512 0.25069 0.24618 0.24157 14023 09512 40835 02994 92449 65674 a0091 25766 44393 31459 0.04986 0.04754 0.04525 0.04299 0.04075 06317 39838 30354 05078 92107 93636 94725 03048 69390 68723 0.40743 0.38731 0.36771 0.34861 0.33002 0.23689 0.23211 0.22725 0.22230 0.21726 07256 47216 14019 11393 45250 57089 24632 06110 53995 44609 0.03856 0.03640 0.03428 0.03220 0.03017 20343 19405 19524 51407 46086 27312 75704 44132 19129 88637 34743 48498 18186 74093 45674 19381 36510 13588 32978 44482 97723 96293 87822 72308 49753 66543 94695 15805 29873 36899 67581 ala25 13963 63995 31921 15252 08436 19515 48488 95354 1 <U> 2 : ; ; I <.r> : 1 : 1 : : 0.70 0. 68 0. 66 0. 64 0.62 0.31194 0.29437 0.27731 0.26075 0.24471 36884 29827 15728 94587 66404 60115 42770 43318 61761 98098 0.21214 0.20693 0.20164 0.19627 0.19082 23821 57789 60404 47584 37987 60229 65521 80635 00004 55563 0.02819 0.02626 0.02439 0.02257 0.02082 0.60 0.58 0.56 0.54 0.52 0.22918 0.21415 0.19964 0.18563 0.17214 31180 88914 39606 a3256 19864 52329 24454 14474 22387 48194 0.18529 0.17969 0.17401 0.16827 0.16245 53067 17083 57076 02799 86594 79209 a6674 a9207 47273 19322 0.01913 0.01751 0.01596 0.01448 0.01307 61240 47623 29821 30628 70253 35536 30357 58470 73722 60097 0.50 0.48 0.46 0.44 0.42 0.15915 0.14667 0.13470 0.12324 0.11229 49430 71955 87438 95879 97278 91895 53491 32980 30364 45641 0.15658 0.15065 0.14466 0.13862 0.13253 43216 09597 24548 28400 62592 36302 56320 29603 34552 29647 0.01174 65939 0.01049 31590 0.00931 77420 -_ __.__ .-~ 0.00822 09631 0.00720 30137 24659 42015 66589 . ..~ 52815 00215 0.40 0.38 0.36 0.34 0.32 0.10185 0.09192 0.08250 0.07359 0.06518 91635 78951 59224 32456 98646 78813 29879 98839 85692 90440 0.12640 0.12023 0.11403 0.10780 0.10154 69204 90456 68174 43252 55126 94864 93806 47880 41741 32988 0.00626 0.00540 0.00461 0.00390 0.00327 36346 21018 72197 73235 02912 49122 72942 27002 12822 03254 : :i : 3 :: 15 0.30 0.28 0.26 0.24 0.22 0.05729 0.04991 0.04303 0.03666 0.03081 57795 09901 54966 92988 23969 13082 53618 12048 a8373 a2591 0.09526 0.08896 0.08264 0.07631 0.06997 41276 36786 73969 82087 a7161 74844 39974 33180 00913 16730 0.00270 0.00220 0.00176 0.00139 0.00107 35642 41768 a7922 37442 50825 68526 84885 53708 77909 02743 i 4 4 5 0.20 0.18 0.16 0.14 0.12 0.02546 0.02062 0.01629 0.01247 0.00916 47908 64806 74661 77475 73247 94703 24710 72610 38405 22093 0.06363 0.05727 0.05091 0.04455 0.03819 11887 75644 94597 ala74 47805 04012 30652 59575 32960 44642 o.oooao a6180 a2883 0.00058 99686 10701 0.00041 45999 la234 0.00027 78633 97799 0.00017 50279 00844 0.10 0. 08 0.06 0.04 0.02 0.00636 0.00407 0.00229 o.ooioi O.QO025 61977 43665 la311 a5916 46479 23676 43153 a0523 35788 08947 0.03183 0.02546 0.01909 0.01273 0.00636 00214 44738 a5179 23855 61974 15118 95252 38105 39770 14061 o.oooio 0.00005 0.00002 0.00000 0.00000 13057 18732 la849 64845 08105 0.00 0.00000 00000 00000 0.00000 00000 00000 0.00000 00000 00000 J t-i)6 1 1 [k--1 1 C(r)=-2 +f(x) sin ;22 -g(,r) cos $9 (> () 1 s(X)=--f(x) 2 COS 5.9 (2 -g(r) > sin Fz19 (2 ) 94484 17470 44630 30524 69272 1 Cz(tr)=-+fz(u) sin u-gz(u) cosu 2 sz(u,=&/z(u) <o=nearest integer to 2. co.5 u-g2(u) sin u : 2 : : : 2 z 2 I 2 :'8 76 a 2 109 10 1'; :'o m 157 245 436 982 3927 03 . ERROR FUNCTION ERROR FUNCTION w(t)=e-2’ AND FRESNEL FOR COMPLEX erfc (-iz) 325 INTEGRALS ARGUMENTS Table z=x+iy “w(.z~z~(z) %w(z) .fW(Z) .oAw(z) x=0.3 1.000000 0.896457 0.809020 0.734599 0.670788 0.615690 0.000000 0.000000 0.000000 0.000000 0.000000 0.990050 0.888479 0.802567 0.129331 0.666463 0.112089 0.094332 0.080029 0.068410 0.058897 0.456532 0.000000 0.000000 0.000000 0.000000 0.000000 0.454731 0.030566 0.427584 0.401730 0.378537 0.351643 0.338744 0.000000 0.000000 0.000000 0.000000 0.000000 0.426044 0.400406 0.377393 0.356649 0.337876 0.021934 0.019805 0.017951 0.321585 0.305953 0.000000 0.000000 0.000000 0.000000 0.000000 0.291072 0.013648 0.278035 0.266042 0.011547 0.567805 0.525930 0.489101 0.291663 0.278560 0.266509 0.612109 0.564818 0.218499 0.039064 0.034465 0.320825 0.305284 0.000000 0.000000 0.000000 0.000000 0.000000 0.255396 0.245119 0.235593 0.226742 0.051048 0.044524 0.523423 0.486982 0.027242 0.024392 Iw(z) x-o.4 0.960789 0.864983 0.783538 0.219753 0.913931 0.318916 0.852144 0.185252 0.157403 0.827246 0.752895 0.269600 0.229653 0.712146 0.713801 0.653680 0.134739 0.116147 0.688720 0.632996 0.197037 0.170203 0.655244 0.605295 0.584333 0.147965 0.129408 0.113821 0.100647 0.522246 0.487556 0.456579 0.089444 0.428808 0.601513 0.555974 7.9 0.100782 0.087993 0.541605 0.406153 0.777267 0.344688 0.294653 0.253613 0.219706 0.561252 0.191500 0.167880 0.147975 0.131101 0.116714 0.515991 0.077275 0.503896 0.480697 0.449383 0.068235 0.060563 0.470452 0.440655 0.421468 0.396470 0.373989 0.054014 0.413989 0.048393 0.043542 0.390028 0.368412 0.064510 0.353691 0.039336 0.035671 0.348839 0.331054 0.058329 0.052936 0.314839 0.259186 0.270346 0.053186 0.048931 0.045139 0.335294 0.079864 0.071628 0.403818 0.104380 0.381250 0.360799 0.342206 0.325248 0.093752 0.084547 0.076538 0.069538 0.016329 0.318561 0.032463 0.014905 0.303290 0.289309 0.276470 0.264648 0.029643 0.300009 0.027154 0.286406 0.024948 0.022987 0.273892 0.262350 0.048210 0.044051 0.040377 0.037118 0.034217 0.234251 0.225531 0.217404 0.021236 0.019669 0.018260 0.016991 0.015845 0.251677 0.241783 0.232592 0.031626 0.029304 0.027217 0.248844 0.239239 0.230300 0.038706 0.035968 0.216047 0.023633 0.025335 0.221963 0.214172 0.033498 0.031263 0.012536 0.010664 0.254978 0.244745 0.235256 0.226438 0.008526 0.218224 0.007949 0.009874 0.009165 0.253732 0.243628 0.224033 0.309736 0.295506 0.282417 0.063393 0.057978 0.041748 0.210557 0.007427 0.203387 0.006952 0.184602 0.000000 0.000000 0.000000 0.000000 0.000000 0.196668.0.006520 0.190360 0.006125 0.184429 0.005764 0.209813 0.202710 0.196050 0.189796 0.183912 0.014806 0.013862 0.013002 0.012216 0.011498 0.208582 0.201589 0.195028 0.188861 0.183056 0.022090 0.020687 0.019409 0.018241 0.017172 0.193613 0.187566 0.181868 0.025706 0.024168 0.022759 0.179001 0.000000 0.178842 0.005433 0.178368 0.010839 0.177581 0.016192 0.176491 0.021466 0.527292 x=0.8 0.600412 0.444858 0.210806 0.203613 0.196874 0.190549 x=0.6 x=0.5 0.778801 0.717588 0.478925 0.408474 0.691676 0.651076 0.535713 0.459665 0.612626 x=0.7 0.576042 0.663223 0.350751 0.608322 0.569238 0.396852 0.344645 0.580698 0.549739 0.520192 0.492289 0.497744 0.432442 0.377688 0.331535 0.614852 0.571717 0.533157 0.498591 0.467521 0.439512 0.414191 0.391234 0.370363 0.351335 0.333942 0.303124 0.263563 0.533581 0.501079 0.471453 0.264268 0.233206 0.444434 0.230488 0.202666 0.206787 0.419766 0.397216 0.127202 0.114460 0.103395 0.357637 0.133501 0.340241 0.120838 0.109759 0.100026 0.299804 0.091443 0.083845 0.077096 0.071081 0.065701 0.281214 0.275602 0.264718 0.254554 0.245050 0.241025 0.060876 0.232204 0.223952 0.056534 0.236152 0.227810 0.376571 0.093744 0.324229 0.085288 0.309463 0.303355 0.289866 0.277412 0.265890 0.255205 0.077851 0.071283 0.295820 0.245276 0.283192 0.271479 0.065461 0.060283 0.260598 0.250469 0.055661 0,051521 0.047804 0.184200 0.164793 0.148036 0.219347 0.211800 0.041428 0.038686 0.216219 0.208961 0.052617 0.049073 0.045859 0.204723 0.198074 0.191818 0.185924 0.180361 0.036196 0.033929 0.031859 0.029966 0.028231 0.202139 0.195717 0.189664 0.183950 0.178549 0.042936 0.040271 0.037836 0.035607 0.033561 0.175105 0.026636 0.173437 0.031680 0.227407 SeeExamples 0.466127 0.441712 0.418998 0.179123 0.159087 0.141945 0.318001 0.236031 0.300989 0.044454 0.360200 0.343375 0.327166 0.313273 Ur(-x+iy)-w(x+iy) 0.355082 0.418736 0.405763 0.429418 0.410264 0.391936 0.374518 0.358043 0.314828 0.280290 0.250532 0.224789 0.202429 0.342511 0.292432 0.166660 0.150681 0.136706 0.124435 0.113620 0.182932 0.165868 0.150877 0.137661 0.125971 0.327900 0.314176 0.301294 0.289208 0.457569 0.402194 0.081245 0.271869 0.267228 0.251237 0.247851 0.115594 0.106355 0.098103 0.090710 0.075190 0.239027 0.084068 0.069748 0.064842 0.078085 0.072680 0.067785 0.063342 0.095563 0.088001 y 0.610142 0.536087 0.472773 0.418491 0.371813 0.392021 0.331544 0.377977 0.363957 0.296692 0.266427 0.350182 0.336799 0.323899 0.311537 0.299741 0.288519 0.277865 0.240057 0.217004 0.196783 0.178990 0.163281 0.149370 0.137012 0.249151 0.126002 0.116164 0.107348 0.240586 0.232482 0.092291 0.267166 0.258203 0.099427 0.224813 0.085845 0.217552 0.080009 0.212616 0.056391 0.230724 0.222905 0.215535 0.208581 0.205686 0.052741 0.202013 0.059298 0.210676 0.204160 0.197982 0.074712 0.069894 0.065500 0.199155 0.192992 0.187170 0.195804 0.189928 0.184362 0.179084 0.174074 0.055610 0.052238 0.04915(. 0.046315 0.043708 0.192120 0.176447 0.049417 0.046384 0.043608 0.041064 0.038728 0.181265 0.176237 0.171452 0.061486 0.057811 0.054439 0.051339 0.048485 0.171502 0.036577 0.169315 0.041306 0.166895 0.0.45851 0.219978 0.181662 w(x-iy)=2eY2-z2 erfc 0.469480 0.449244 0.259136 0.230646 0.206155 0.185005 0.104054 0.439421 0.430271 0.489710 0.029234 0.027389 x10.9 0.522932 0.509299 0.060409 w(x)=e-z2+ e-22 [email protected] g S Jr 12-19. w(iy)=ey2 0.397906 0.378341 0.206879 0.200039 w[(l+i)u]=e-2ia2{ (cos 2xy+i sin Bxy)+$+j l+(i-l)[C(&J)+iS(%)]} 0.186554 l 326 ERROR Table FUNCTION ERROR 7.9 .uRw(z) .fw(z) AND FUNCTION Xw(z) FRESNEL FOR COMPLEX ARGUMENTS w(z)=e- z2 erfc (-it) z= x+iy %w(z) Jw(z) %.u(z) .Iw(z) A4l(z) x=1.1 x=1.0 INTEGRALS x=1.3 x=1.2 0.367879 0.373170 0.373153 0.369386 0.363020 U.607158 0.538555 0.478991 0.427225 0.382166 0.298197 0.312136 0.319717 0.322586 0.321993 0.593761 0.532009 0.477439 0.429275 0.386777 0.236928 0.257374 0.270928 0.279199 0.283443 0.572397 0.518283 0.469488 0.425667 0.386412 0.184520 0.209431 0.227362 0.239793 0.247908 0.545456 0.499216 0.456555 0.417491 0.381908 0.140858 0.168407 0.189247 0.204662 0.215711 0.515113 0.476535 0.440005 0.405823 0.374110 0.354900 0.345649 0.335721 0.325446 0.315064 0.342872 0.308530 0.278445 0.252024 0.228759 0.318884 0.313978 0.307816 0.300807 0.293259 0.349266 0.316128 0.286815 0.260847 0.237800 0.284638 0.283540 0.280740 0.276693 0.271752 0.351299 0.319910 0.291851 0.266757 0.244295 0.252654 0.254784 0.254895 0.253461 0.250858 0.349611 0.320368 0.293927 0.270040 0.248462 0.223262 0.228026 0.230578 0.231385 0.230826 0.344868 0.318022 0.293453 0.271015 0.250549 0.304744 0.294606 0.284731 0.275174 0.265967 0.208219 0.190036 0.173896 0.159531 0.146712 0.285402 0.277407 0.269401 0.261476 0.253697 0.217306 0.199046 0.182742 0.168151 0.155066 0.266189 0.260213 0.253985 0.247628 0.241233 0.224168 0.206108 0.189878 0.175271 0.162100 0.247381 0.243266 0.238695 0.233813 0.228733 0.228967 0.211343 0.195398 0.180957 0.167863 0.229205 0.226767 0.223710 0.220192 0.216340 0.231897 0.214902 0.199416 0.185299 0.172423 0.257128 0.248665 0.240578 0.232861 0.225503 0.135242 0.124954 0.115702 0.107361 0.099824 0.246112 0.238752 0.231635 0.224775 0.218176 0.143305 0.132711 0.123147 0.114495 0.106650 0.234870 0.228592 0.222436 0.216428 0.210587 0.150205 0.139441 0.129684 0.120822 0.112760 0.223542 0.218309 0.213086 0.207912 0.202818 0.155975 0.145167 0.135326 0.126353 0.118158 0.212253 0.208014 0.203684 0.199315 0.194947 0.160668 0.149927 0.140103 0.131106 0.122858 0.218493 0.211816 0.205457 0.199402 0.193634 0.092998 0.086801 0.081162 0.076021 0.071324 0.211839 0.205760 0.199935 0.194356 0.189014 0.099523 0.093035 0.087116 0.081706 0.076753 0.204926 0.199452 0.194166 0.189072 0.184165 0.105411 0.098700 0.092562 0.086936 0.081773 0.197827 0.192953 0.188208 0.183599 0.179131 0.110662 0.103795 0.097495 0.091706 0.086378 0.190608 0.186324 0.182112 0.177985 0.173954 0.115286 0.108325 0.101919 0.096015 0.090567 :*; 2:9 0.188139 0.182903 0.177910 0.173147 0.168602 0.067024 0.063080 0.059456 0.056118 0.053041 0.183901 0.179008 0.174324 0.169840 0.165546 0.072208 0.068031 0.064186 0.060639 0.057363 0.179444 0.174903 0.170538 0.166342 0.162310 0.077024 0.072651 0.068617 0.064890 0.061440 0.174805 0.170623 0.166582 0.162681 0.158916 0.081467 0.076933 0.072742 0.068863 0.065266 0.170024 0.166201 0.162487 0.158883 0.155389 0.0'35532 0.080873 0.076557 0.072553 0.068834 3.0 0.164261 0.050197 0.161434 0.054331 0.158435 0.058243 0.155285 0.061926 0.152005 0.5 0.6 it: 0:9 ::: :.'3 1:4 1.5 ::; ::: ::i :*: 2:4 2.5 2.6 x-1.5 Y x=1.6 0.065375 x-1.9 x=1.8 x=1.7 0.105399 0.134049 0.156521 0.173865 0.186984 0.483227 0.451763 0.421076 0.391665 0.363828 0.077305 0.105843 0.128895 0.147272 0.161702 0.451284 0.426168 0.400837 0.375911 0.351803 0.055576 0.083112 0.105929 0.124612 0.139717 0.420388 0.400743 0.380161 0.359313 0.338676 0.039164 0.065099 0.087090 0.105522 0.120793 0.391291 0.376214 0.359721 0.342479 0.324985 0.027052 0.051038 0.071811 0.089592 0.104641 0.364437 0.353066 0.340004 0.325873 0.311161 0.196636 0.203461 0.207990 0.210664 0.211846 0.337720 0.313397 0.290847 0.270016 0.250823 0.172820 0.181177 0.187245 0.191423 0.194049 0.328777 0.306990 0.286517 0.267378 0.249556 0.151751 0.161171 0.168379 0.173725 0.177513 0.318584 0.299261 0.280846 0.263418 0.247012 0.133288 0.143369 0.151366 0.157578 0.162268 0.307609 0.290613 0.274180 0.258431 0.243439 0.117233 0.127644 0.136134 0.142949 0.148310 0.296240 0.281392 0.266823 0.252681 0.239067 0.211837 0.210881 0.209182 0.206902 0.204177 0.233171 0.216954 0.202067 0.1'38403 0.175862 0.195407 0.195734 0.195228 0.194053 0.192347 0.233009 0.217678 0.203494 0.190384 0.178275 0.180002 0.181414 0.181938 0.181733 0.180933 0.231630 0.217253 0.203847 0.191366 0.179762 0.165667 0.167977 0.169373 0.170003 0.169997 0.229244 0.215857 0.203272 0.191471 0.180425 0.152418 0.155452 0.157569 0.158906 0.159585 0.226046 0.213656 0.201914 0.190821 0.180367 0.201115 0.197806 0.194320 0.190717 0.187043 0.164349 0.153773 0.144054 0.135113 0.126883 0.190222 0.187772 0.185073 0.182189 0.179172 0.167092 0.156765 0.147226 0.138412 0.130262 0.179651 0.177983 0.176008 0.173792 0.171390 0.168980 0.158969 0.149674 0.141045 0.133033 0.169465 0.168500 0.167183 0.165579 0.163746 0.170099 0.160457 0.151458 0.143063 0.135234 0.159709 0.159369 0.158641 0.157593 0.156282 0.170534 0.161300 0.152637 0.144516 0.136908 0.183335 0.179623 0.175930 0.172276 0.168674 0.119298 0.112302 0.105842 0.099870 0.094343 0.176064 0.172901 0.169710 0.166513 0.163330 0.122723 0.115744 0.109277 0.103280 0.097713 0.168849 0.166206 0.163493 0.160737 0.157958 0.125590 0.118674 0.112243 0.106260 0.100689 0.161733 0.159580 0.157320 0.154982 0.152591 0.127931 0.121118 0.114761 0.108827 0.103285 0.154757 0.153059 0.151224 0.149281 0.147256 0.129781 0.123108 0.116858 0.111003 0.105519 :*;: 2:9 0.165136 0.161669 0.158281 0.154975 0.151753 0.089222 0.084472 0.080061 0.075960 0.072142 0.160175 0.157060 0.153993 0.150981 0.148030 0.092541 0.087732 0.083254 0.079082 0.075191 0.155175 0.152402 0.149649 0.146927 0.144243 0.095499 0.090660 0.086143 0.081925 0.077982 0.150165 0.147722 0.145274 0.142834 0.140411 0.098107 0.093265 0.088735 0.084493 0.080519 0.145172 0.143045 0.140892 0.138725 0.136555 0.100378 0.095558 0.091037 0.086794 0.082809 3.0 0.148618 0.068585 0.145144 0.071558 0.141602 -. ‘ 0.074293 0.138012 0.076794 0.134391 0.079065 0.0 0.1 i-: 0:4 0.5 E 0:s 0.9 ::i :*: 1:4 1.5 :*; 1:8 1.9 2; :-: 2:4 $2 See Examples 12-19. w(x)=&+ 4 e-22 SD”,@ dt Jr w(-x+iy)-qq w(xpiy)=2eu2-z2 w(iy)=eY2 erfc y w[(l+i)u]=e-2iu2{ (cos 2xy+i sin Bxy)-w(x+iy) l+(i-l)[C($)+iS(~$)]} ERROR FUNCTION AND FRESNEL ERROR FUNCTION FOR COMPLEX ARGUMENTS w(t)=e-z*erfc (-iii) z=x+iy Ww(z) .Uw(z) stw(z) [email protected](z) dw(z) .Yw(z) x=2.1 x=2.2 x=2.3 %w(z) Y-w(z) x-2.0 327 INTEGRALS Table 7.9 .uRw(z) Yw(z) x=2.4 0.018316 0.040201 0.059531 0.076396 0.090944 0.340026 0.331583 0.321332 0.309831 0.297529 0.012155 0.031936 0.049726 0.065521 0.079385 0.318073 0.311886 0.303894 0.294574 0.284327 0.007907 0.025678 0.041927 0.056586 0.069655 0.298468 0.293982 0.287771 0.280232 0.271710 0.005042 0.020958 0.035728 0.049248 0.061473 0.281026 0.277795 0.272968 0.266865 0.259775 0.003151 0.017397 0.030792 0.043211 0.054585 0.265522 0.263201 0.259435 0.254478 0.248566 0.103359 0.113836 0.122574 0.129768 0.135600 0.284786 0.271881 0.259031 0.246396 0.234096 0.091422 0.101765 0.110558 0.117948 0.124081 0.273482 0.262308 0.251016 0.239772 0.228703 0.081182 0.091245 0.099943 0.107383 0.113679 0.262499 0.252844 0.242947 0.232968 0.223037 0.072408 0.082092 0.090585 0.097963 0.104309 0.251953