Unformatted text preview: Preface’
The present volume is an outgrowth of a Conference on Mathematical
Tables
held at Cambridge,
Mass., on September 1516, 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 JahnkeEmde.
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 onehalf 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. 68. 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 mostquoted
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 zday Conference on Tables was called at the
Massachusetts
Institute
of Technology
on September
1516, 1954, to discuss the
needs for tables of various kinds.
Twentyeight
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 highspeed 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 selfcontained
summary of the mathematical functions that arise in physical and engineering problems.
The wellknown
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 halfcentury.
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 fivefigure accuracy, which suffices in most
physical applications.
Users requiring higher
1 The most recent, the sixth, with F. Loesch added as ccauthor, was
published in 1960 by McGrawHill,
U.S.A., and Teubner, Germany.
2 The second edition, with L. J. Comrie added as coauthor, was published
in two volumes in 1962 by AddisonWesley,
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 13, by A. ErdBlyi, W. Magnus,
F. Oberhettinger and F. G. Tricomi (McGrawHill, 195355). 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 higherorder interpolation procedures,
described below.
In certain tables manyfigured 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 endfigure 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 xliEi(ln
xr],
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
xr]
has been tabulated
to nine
decimals for the range 05x<+.
For +<x12,
Ei(x) is sufficiently wellbehaved 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 xl. Twentyone entries of
xe“Ei(x), corresponding to xl = .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= (xx0>/(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 lp 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 5point 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)f1+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”[email protected])
:
:
.
. 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
20x
1 Yo
x,x
x.20 Yn
x,x
1 Yo.1
Yo.1 ,n=x,x
Gz1 l/O.” S2fl yo,n= Yo. 1. .. ., ml.m.n 1
~n%n safz
wa l/0.1. . . ., n1.98
Yo.1. . . , m1.n x,x
x,x 1 If the quantities Z.X and x~5 are used as
multipliers
when forming the crossproduct 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=f1f0, 8f3/a=fzf1, . . . ,,
a2/1=sf3iaafiia=fa2fi+fo
~af3~~=~aja~aj~=fa3j2+3fik
8'fa=~aj~fsla6~3~2=f4~f~+~ja4f~+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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 Prof.William
 Spherical Harmonics, ........., Abramowitz and Stegun, Bureau of Standards

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