# M. Abramowitz and I. A. Stegun

*Handbook of Mathematical Functions*

Table of contents :: Index to all pages and sections :: Subject index

This is an electronic copy of the legendary mathematical reference work
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical
Tables* edited by
Milton Abramowitz and
Irene A. Stegun of the U.S. National Bureau of Standards (now the
National Institute of Standards and
Technology).

Since it was originally published (in 1964), the 1046-page Handbook has been
one of the most comprehensive sources of information on special functions,
containing definitions, identities, approximations, graphs, and tables of
values of numerous functions used in virtually all fields of applied
mathematics. The notation used in the Handbook is the *de facto*
standard for much of applied mathematics today.

As the Handbook is the work of U.S. federal government employees acting in their official capacity, it is not protected by copyright. While it can be ordered from the Government Printing Office, it has also been reprinted by commercial publishers, most notably Dover Publications (ISBN 0-486-61272-4), and can be legally viewed and downloaded off the web.

Credits for the electronic source for the page and subject indices goes to Colin B. Macdonald at the University of Oxford, while credits for the scanned electronic master goes to Alan P. Sexton at the University of Birmingham.

**aas_hmf_master.pdf**
[52.12 MB]
The entire *Handbook of Mathematical Functions* in a single PDF
document. From this PDF master, all individual PDF pages and PNG images
were extracted with GhostScript and a Makefile keeping track of all conversions.

UNITED STATES DEPARTMENT OF COMMERCE

Luther H. Hodges, *Secretary*

NATIONAL BUREAU OF STANDARDS

A. V. Astin, *Director*

## Handbook of Mathematical Functions

With

Formulas, Graphs, and Mathematical Tables

Edited by Milton Abramowitz and Irene A. Stegun

National Bureau of Standards

Applied Mathematics Series - 55

Issued June 1964

Tenth Printing, December 1972, with corrections

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 - Price $11.35 domestic postpaid, or $10.50 GPO Bookstore

The text relating to physical constants and conversion factors (page 6) has been modified to take into account the newly adopted Système International d'Unites (SI).

## ERRATA NOTICE

The original printing of this Handbook (June 1964) contained errors that have been corrected in the reprinted editions. These corrections are marked with an asterisk (*) for identification. The errors occurred on the following pages: 2-3, 6-8, 10, 15, 19-20, 25, 76, 85, 91, 102, 187, 189-197, 218, 223, 225, 233, 250-255, 260-263, 268, 271-273, 292, 302, 328, 332, 333-337, 362, 365, 415, 423, 438-440, 443, 445, 447, 449, 451, 484, 498, 505-506, 509-510, 543, 556, 558, 562, 571, 595, 599, 600, 722-723, 739, 742, 744, 746, 752, 756, 760-765, 774, 777-785, 790, 797, 801, 822-823, 832, 835, 844, 886-889, 897, 914, 915, 920, 930-931, 936, 940-941, 944-950, 953, 960, 963, 989-990, 1010, 1026.

Originally issued June 1964. Second printing, November 1964. Third printing, March 1965. Fourth printing, December 1965. Fifth printing, August 1966. Sixth printing, November 1967. Seventh printing, May 1968. Eighth printing, 1969. Ninth printing, November 1970.

Library of Congress Catalog Card Number: 64-60036

## 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 for preliminary surveys 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 the 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 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, *Director*

June 1964, 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 100,000th 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 success of 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 10 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, 1952, Dr. Abramowitz of that 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 2-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 conclusions and recommendations, which were set forth in the published Report of the Conference. There was general agreement, for example, "that the advent of high-speed computing 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 provide the NSF with independent judgments on grants for 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.

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 to 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. ERDÉLYI

M. C. GRAY

N. C. METROPOLIS

J. B. ROSSER

H. C. THACHER, Jr.

JOHN TODD

C. B. TOMPKINS

J. W. TUKEY.

## Contents

1. Mathematical Constants ………… 1

DAVID S. LIEPMAN

2. Physical Constants and Conversion Factors ………… 5

A. G. McNISH

3. Elementary Analytical Methods ………… 9

MILTON ABRAMOWITZ

4. Elementary Transcendental FunctionsLogarithmic, Exponential, Circular and Hyperbolic Functions ………… 65

RUTH ZUCKER

5. Exponential Integral and Related Functions ………… 227

WALTER GAUTSCHI and WILLIAM F. CAHILL

6. Gamma Function and Related Functions ………… 253

PHILIP J. DAVIS

7. Error Function and Fresnel Integrals ………… 295

WALTER GAUTSCHI

8. Legendre Functions ………… 331

IRENE A. STEGUN

9. Bessel Functions of Integer Order ………… 355

F. W. J. OLVER

10. Bessel Functions of Fractional Order ………… 435

H. A. ANTOSIEWICZ

11. Integrals of Bessel Functions ………… 479

YUDELL L. LUKE

12. Struve Functions and Related Functions ………… 495

MILTON ABRAMOWITZ

13. Confluent Hypergeometric Functions ………… 503

LUCY JOAN SLATER

14. Coulomb Wave Functions ………… 537

MILTON ABRAMOWITZ

15. Hypergeometric Functions ………… 555

FRITZ OBERHETTINGER

16. Jacobian Elliptic Functions and Theta Functions ………… 567

L. M. MILNE-THOMSON

17. Elliptic Integrals ………… 587

L. M. MILNE-THOMSON

18. Weierstrass Elliptic and Related Functions ………… 627

THOMAS H. SOUTHARD

19. Parabolic Cylinder Functions ………… 685

J. C. P. MILLER

20. Mathieu Functions ………… 721

GERTRUDE BLANCH

21. Spheroidal Wave Functions ………… 751

ARNOLD N. LOWAN

22. Orthogonal Polynomials ………… 771

URS W. HOCHSTRASSER

23. Bernoulli and Euler Polynomials, Riemann Zeta Function ………… 803

EMILIE V. HAYNSWORTH and KARL GOLDBERG

24. Combinatorial Analysis ………… 821

K. GOLDBERG, M. NEWMAN and E. HAYNSWORTH

25. Numerical Interpolation, Differentiation and Integration ………… 875

PHILIP J. DAVIS and IVAN POLONSKY

26. Probability Functions ………… 925

MARVIN ZELEN and NORMAN C. SEVERO

27. Miscellaneous Functions ………… 997

IRENE A. STEGUN

28. Scales of Notation ………… 1011

S. PEAVY and A. SCHOPF

*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 comprehensive and self-contained summary of the mathematical functions that
arise in physical and engineering problems. The well-known Tables of Functions
by E. Jahnke and F. Emde has been invaluable to workers in these fields in its
many editions^{1} during the past half-century.
The present volume extends 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.^{2}
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 computational 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 mathematical notations used in this Handbook are those commonly adopted in standard texts, particularly Higher Transcendental Functions, Volumes 1-3, by A. Erdélyi, 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.

^{1} *The most recent, the sixth, with F. Loesch added as co-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.*

## 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 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 6/10 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.

## 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 effectively 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 given on the first page of each table. Myrtle R. Kellington 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, checking and editing of the tabular material was received 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 staff 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.