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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 scien-
tists and engineers. A greater variety of functions and higher accuracy of tabula-
tion are now required as a result of scientific advances and, especially, of the in-
creasing 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 imple-
ment the project, the National Science Foundation requested the National Bureau
of Standards to prepare such a volume and established an Ad Hoc Advisory Com-
mittee, 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,
and Irene A. Stegun. Its success has depended greatly upon the cooperation of
many mathematicians. Their efforts together with the cooperation of the Ad HOC
Committee are greatly appreciated. The particular contributions of these and
other individuals are acknowledged at appropriate places in the text.
The sponsor-
ship 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 mathe-
matical tables that began when John Napier published his tables of loga-
rithms in 1614. Only four and one-half years after the first copy came
from the press in 1964, Myron Tribus, the Assistant Secretary of Com-
merce 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 chang-
ing 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. Abramo-
witz 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, “Mathe-
matical 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
National Science Foundation made funds available.
To provide technical guidance
to the Mathematics Division of the Bureau, which carried out the work, and to pro-
vide 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
V
,/”
sometimes get acted on.
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 appre-
ciation 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, Graphs, and Mathematical
Tables
Edited by Milton Abramowitz
and Irene A. Stegun
1. Introduction
The present Handbook has been designed to
provide scientific investigators with a compre-
hensive and self-contained summary of the mathe-
matical functions that arise in physical and engi-
neering 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 tabu-
lated functions, particularly
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 mathe-
matical properties stated in the chapter may be
found. Also listed in the bibliographies are the
more important numerical tables. Comprehen-
sive lists of tables are given in the Index men-
tioned 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 Hand-
book are those commonly adopted in standard
texts, particularly Higher Transcendental Func-
tions, Volumes
1-3,
by A. ErdBlyi, W. Magnus,
F. Oberhettinger and F. G. Tricomi (McGraw-
Hill, 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.
those of computa-
2. Accuracy of the Tables
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. four-
or five-figure accuracy, which suffices in most
physical applications. Users requiring higher
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 pur-
pose 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 func-
tions, and 1 unit in the case of the higher functions
except in a few caseswhere it has been permitted
to rise to
2
units.
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 Com-
puting Service Ltd., Great Britain.
IX /-
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