M. Abramowitz and I. A. Stegun
Handbook of mathematical functions
 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: 23, 68, 10, 15, 1920, 25, 76, 85, 91, 102, 187, 189197, 218, 223, 225, 233, 250.255, 260263, 268, 271273, 292, 302, 328, 332, 333337, 362, 365, 415, 423, 438440, 443, 445, 447, 449, 451, 484, 498, 505506, 509510, 543, 556, 558, 562, 571, 595, 599, 600, 722723, 739, 742, 744, 746, 752, 756, 760765, 774, 777785, 790, 797, 801, 822823, 832, 835, 844, 886889, 897, 914, 915, 920, 930931, 936, 940941, 944950, 953, 960, 963, 989990, 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: 6460036
 II
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 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 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.
 III
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 100,000th 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. 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
 IIIa
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 2day 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 conclusions and recommendations, which were set forth in the 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 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.
 V
 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.
 VI
 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 selfcontained summary of the mathematical functions that arise in physical and engineering problems. The wellknown 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 halfcentury. 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 13, by A. Erdélyi, 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.
^{1} The most recent, the sixth, with F. Loesch added as coauthor, 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.
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, fouror fivefigure accuracy, which suffices in most physical applications. Users requiring higher 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 6/10 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.
 IX
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.
 M. ABRAMOWITZ.
 XIV