Milton Abramowitz (editor) - Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Books on Mathematics)

HANDBOOK OF MATHEMATICAL FUNCTIONS WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES Edited by Milton Abramowitz and Irene A. Stegun DOVER PUBLICATIONS, INC., NEW YORK The text relating to physical constants and conversion factors (page 6) has been modified to take into account the newly adopted Systeme International dUnites (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. This Dover edition, first published in 1965, is an unabridged and unaltered republication of the work originally published by the National Bureau of Standards in 1964.

This ninth Dover printing conforms to the tenth (December 1972) printing by the Government Printing Office, except that additional corrections have been made on pages 18, 79, 80, 82, 408, 450, 786, 825 and 934. Library of Congress Catalog Card Number: 65-12253International Standard Book NumberISBN-13: 978-0-486-61272-0ISBN-10: 0-486-61272-4 Manufactured in the United States by Courier Corporation 61272423 www.doverpublications.com 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 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. Erdlyi, M. C. Gray, N. 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. 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. 6-8. 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. 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, 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 1516, 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.