Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Table

Despite the increasing use of computers, the basic need for mathematical tables continues. Tables serve a vital role in preliminary surveys of problems before programming for machine operation, and they are indispensable to thousands of engineers and scientists without access to machines. Because of automatic computers, however, and because of recent scientific advances, a greater variety of functions and a higher accuracy of tabulation than have been available until now are required. In 1954, a conference on mathematical tables, sponsored by M.I.T. and the National Science Foundation, met to discuss a modernization and extension of Jahnke and Emde's classical tables of functions. This volume, published 10 years later by the U.S. Department of Commerce, is the result. Designed to include a maximum of information and to meet the needs of scientists in all fields, it is a monumental piece of work, a comprehensive and selfcontained summary of the mathematical functions that arise in physical and engineering problems. The book contains 29 sets of tables, some to as high as 20 places: mathematical constants; physical constants and conversion factors (6 tables); exponential integral and related functions (7); error function and Fresnel integrals (12); Bessel functions of integer (12) and fractional (13) order; integrals of Bessel functions (2); Struve and related functions (2); confluent hypergeometric functions (2); Coulomb wave functions (2); hypergeometric functions; Jacobian elliptic and theta functions (2); elliptic integrals {9); Weierstrass elliptic and related functions; parabolic cylinder functions {3); Mathieu functions (2); spheroidal wave functions (5); orthogonal polynomials (13); combinatorial analysis (9); numerical interpolation, differentiation and integration (11); probability functions (ll); scales of notation (6); miscellaneous functions (9); Laplace transforms (2); and others. Each of these sections is prefaced by a list of related formulas and graph
UNITED STATES DEPARTMENT OF COMMERCE Luther H. Hodges, Secretary NATIONAL BUREAU0 F STANDARDS●AV.人哪i, Director Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables Edited by Milton Abramowitz and Irene A Stegun NT OF ArEs°5 National Bureau of Standards Applied Mathematics Series.55 lssued une 1964 Tenth Printing, December 1972, with corrections For sale by the Superintendent of Documents, U.S. Government Printing ofce Washington, D.C. 20402 DOCUMENTS DEPT, Ref 0 21a M0.55 SAN FRANCISO。 PUBLIC LIBRARY The text relating to physical constants and conversion factors (page 6) has been modified to take into account the newly adopted systeme Interna tional d'Unites(SI). ERRATA NOTICE The original printing of this Handbook June 1964)contained errors that have been corrected in the reprinted editions. These cor rections are marked with an asterisk(")for identification. The errors occurred on the following pages: 23, 68, 10, 15, 1920, 25, 76, 85, 91, 102 187,189197218,223,225,288,250,255,260268,268271278,292,802, 328,382,333387,362,365,415,423,488440,443,445,447,448,451,484, 498,505506,5089510,543,556,558,562,571595,599,600,7227283,739, 742,744,746,752,756,760765,774,777785,790,797,801,8228283,882, 835,844,86889,897,914915,920,980931,936,940941,944950,953, 960.968.989990、10101026, 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: 6460036 reface 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 Insti tute of Technology. The purpose of the meeting was to evaluate the need for mathematical tables in the light of the availability of large scale com puting 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 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 informa tion within the limits of a moderately large volume, with particular atten tion 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 y including the mathematical properties that are important in compu b tation 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 par ticular 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 isers 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 mathe matical tables that began when John Napier published his tables of loga rithms in 1614. Only four and onehalf 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 100,00oth 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 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. 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 y Dr. Astin in his Preface is being amply fulfilled LEWIS M. BRANSCOMB, Director National Bureau of Standards November 1970 ia Foreword This volume is the result of the copperative eftort of many persons and a number of organizations. The National Bureau of Standards has long been turning out mathematical tables and has had urder 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.Abramo witz 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, "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 Massachusetts Institute of T pport a 2day Conference on Tables was called at the table production, with its Tochnology on Soptombor 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 conclusions and recommendations, which were set forth in tbe published Report of the Conference, There was general agreement, for example, "that the advent of highspeed 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 tedhniques 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 Committed: P. M. Morse(Chairman), M.Abramowitz J. H. Curtiss, R. W. Hamming D.H. Lohmer,CB Tompkins,JWTukey,to help implement these and other recommendations. The Bureau of Standards undertook to produce the mended tables and the National Science Foundation made fupds 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 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. 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 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 Te echnology 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 uBers 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.卫RD工YI M, C. GRAY N. C. METROPOLIS B. RossER H. C. THACHER. Jr JORN TODD C. B. ToMPKINS J.w. TuKEY Contents P reword Introduction I 1. Methematical Constant DaVID S. LIEPMAN 2. Physical Constants and Conversion Factors A. G. MCNIBH 3. Elementary Analytical Methods 9 MILTON ABRAMOWITZ 4.巩l tary Trar dental functi 65 Logarithmic, Exponential, Circular and Hyperbolic Functions RUTH ZUCKER 5. Exponential Integral and Related Functions 227 WALTER GAUTSCH and WILLIAM F. CAHILL 6. Gamma Function and Related Functiong 268 PHIlIP DAVIs 7. Error Function and Fresnel Integrals 295 WALTER GAUTECHI 8. Legendre Functions 331 IRENE A STEGUN 9. Bessel Functions of Integer Order 355 F.W.J].OLⅴER 10. Bessel Functions of Fractional Order 485 H. A. ANTOSIEWICZ 11. Integrals of Bessel Functions 479 YuDELL L, LUkE 12. Struve Functions end Related Functions MILTON ABRAMOWITZ 13. Confluent Hypergeometric Functions 503 Lτ CY JOAN SLATET 14. Coulomb Waye Functions MILTON ABRAMO WITZ 15. Hypergeometric Functions FRiTz OBERHETTINGERI 16. Jacobian Elliptic Functions and Theta Functions L M. MILNE THOMBON 17. Eliptic Integrals 687 L. M. MILNETHOMSON 18. Weierstrass Eliptic and Related Functions 627 THOMAB H. SOUTHARD 19. Parabolic Cylinder Functions 685 J. C. P. MIller VIII CONTENTS Pag 20 M &thieu functions 721 GERTRUDE BLANCH 21. Sphe Wave Functie 751 arnold N. LowAN 22. Orthogonal polynomials 771 Urs W. HocHSTRASSER 23. Bernouli and euler polynomials, Riemann Zeta Function 808 EMIzⅴ. HAYNSWORTH乱 nd 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 MARⅴ IN ZELeN nd norman o.SEⅴERo 27. Miscellaneous functions 997 ⅠRENA. STEGUN 28. Scales of notation 1011 S.PEAⅴ Y and a. ScOPY 29. Laplace Transforms 1019 Subject index Index of notations l044
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