Unformatted text preview: Preface’
The present volume is an outgrowth of a Conference on Mathematical
held at Cambridge,
Mass., on September 15-16, 1954, under the auspices of the
National Science Foundation and the Massachusetts Institute of Technology.
purpose of the meeting was to evaluate the need for mathematical
tables in the light
of the availability
of large scale computing
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
of problems before programming for machine operation.
For those without easy access to machines, such tables are, of course, indispensable.
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
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
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
To carry out the goal set forth by tbe Ad Hoc Committee,
it has been
necessary to supplement the tables by including the mathematical
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
are greatly appreciated.
of these and
other individuals are acknowledged at appropriate places in the text. The sponsorship of the National
for the preparation
of the material is
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
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
Bureau of Standards
has long been turning out
tables and has had under consideration,
for at least IO years, the
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
plans for such an undertaking,
indicated the need for technical advice and financial support.
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”
exercised by a Committee
of the Division.
to the NBS Conference
on Tables in 1952 the attention
National Science Foundation
was drawn to the desirability
of financing activity in
With its support a z-day Conference on Tables was called at the
15-16, 1954, to discuss the
needs for tables of various kinds.
persons attended, representing
and engineers using tables as well as table producers.
reached consensus on several cpnclusions
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
task of table making but definitely did not remove the need for tables”.
also agreed that “an outstanding
need is for a Handbook
of Tables for the Occasional
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
of such a Handbook
and that the NSF contribute
The Conference elected, from its
the following Committee:
P. M. Morse (Chairman),
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
as the Committee
Tables of the Mathematics
Division of the National Research Council.
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
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.
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
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.
M. C. GRAY
N. C. METROPOLIS
J. B. ROSSER
H. C. THACHER. Jr.
‘C. B. TOMPKINS
J. W. TUKEY. Handbook of Mathematical Functions with Formulas,
by Milton 1. and Mathematical
Abramowitz and Irene A. Stegun Introduction The present Handbook
has been designed to
with a comprehensive and self-contained
summary of the mathematical functions that arise in physical and engineering problems.
Funct.ions by E. Jahnke and F. Emde has been
invaluable to workers
in these fields in its many
the past half-century.
present volume ext,ends the work of these authors
by giving more extensive
and more accurate
numerical tables, and by giving larger collections
of the tabulated
The number of functions covered has
also been increased.
of functions and organization
of the chapters in this Handbook
is similar to
that of An Index of Mathematical
A. Fletcher, J. C. P. Miller, and L. Rosenhead.
In general, the chapters contain numerical tables,
of the tabulated functions,
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
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
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.
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
books and papers in which proofs of the mathematical properties
stated in the chapter may be
Also listed in the bibliographies
Comprehensive lists of tables are given in the Index mentioned above, and current information
tables is to be found in the National
Tables and Other Aids
The ma.thematical notations used in this Handbook are those commonly
adopted in standard
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,
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
polation near x=0.
The Punctions Ei(x)-In
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.
has been tabulated
decimals for the range 05x<+.
Ei(x) is sufficiently well-behaved to admit direct
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
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.
in which linear interpolation
accurate it is intended
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
methods based on differences and
(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
p= (x--x0>/(x1-~0> 775
E : 89608
g. I ze*El
ix 4 .90227
: 90129 4695
‘453 The numbers in the square brackets mean that
error in a linear interpolate
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;
check is then provided by the multiplier
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
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
register since their sum is unity.
now have the following subtable.
.89774 0379 fn=.3(.89772 -2
10620 .; .89775 & 1
7.7 0999 Yn=ze”[email protected])
:89774 Yo, 1.2. I Yo. 1, (I 44034
2 35221 0379) In cases where the correct order of the Lagrange
is not known, one of the prelimina
may have to be performed witT
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
-. 2527 -.
2706 . 89773 71938
G--z1 l/O.” S2fl yo,n=- Yo. 1. .. ., m--l.m.n-- 1
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
when forming the cross-product on a
desk machine, their accumulation
in the multiplier
register is the divisor to be used
at that stage. An extra decimal place is usually
carried in the intermediate
to safeguard against accumulation
of rounding errors.
The order in which the tabular values are used
to some extent, but to achieve the
rate of convergence and at the same
of rounding errors,
we begin, as in this example, with the tabular
argument nearest to the given argument,
take the nearest of the remaining
tabular arguments, and so on.
The number of tabular values required to
achieve a given precision emerges naturally
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, . . . ,,
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 ...
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