F MATHEMATICAL O K O FORMULAS O B D N AMD INTEGRALS A H ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto This book is printed on acid-free paper. ® Copyright © 1995 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press, Inc. A Division of Harcourt Brace & Company 525 Β Street, Suite 1900, San Diego, California 92101-4495 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Jeffrey, Alan. Handbook of mathematical formulas and integrals / by Alan Jeffrey. p. cm. Includes index. ISBN 0-12-382580-6 (pbk.) 1. Mathematics—Tables. 2. Mathematics—Formulas. I. Title. QA47.J38 1995 510*.212—dc20 95-2344 CIP PRINTED IN THE UNITED STATES OF AMERICA 95 96 97 98 99 00 MM 9 8 7 6 5 4 3 2 1 Preface This book contains a collection of general mathematical results, formulas, and inte- grals that occur throughout applications of mathematics. Many of the entries are based on the updated fifth edition of Gradshteyn and Ryzhik's "Tables of Integrals, Series, and Products," though during the preparation of the book, results were also taken from various other reference works. The material has been arranged in a straightforward manner, and for the convenience of the user a quick reference list of the simplest and most frequently used results is to be found in Chapter 0 at the front of the book. Tab marks have been added to pages to identify the twelve main subject areas into which the entries have been divided and also to indicate the main interconnections that exist between them. Keys to the tab marks are to be found inside the front and back covers. The Table of Contents at the front of the book is sufficiently detailed to enable rapid location of the section in which a specific entry is to be found, and this information is supplemented by a detailed index at the end of the book. In the chapters listing integrals, instead of displaying them in their canonical form, as is customary in reference works, in order to make the tables more convenient to use, the integrands are presented in the more general form in which they are likely to arise. It is hoped that this will save the user the necessity of reducing a result to a canonical form before consulting the tables. Wherever it might be helpful, material has been added explaining the idea underlying a section or describing simple techniques that are often useful in the application of its results. Standard notations have been used for functions, and a list of these together with their names anova reference to the section in which they occur or are defined is to be xix XX Preface found at the front of the book. As is customary with tables of indefinite integrals, the additive arbitrary constant of integration has always been omitted. The result of an integration may take more than one form, often depending on the method used for its evaluation, so only the most common forms are listed. A user requiring more extensive tables, or results involving the less familiar special functions, is referred to the short classified reference list at the end of the book. The list contains works the author found to be most useful and which a user is likely to find readily accessible in a library, but it is in no sense a comprehensive bibliography. Further specialist references are to be found in the bibliographies contained in these reference works. Every effort has been made to ensure the accuracy of these tables and, whenever possible, results have been checked by means of computer symbolic algebra and integration programs, but the final responsibility for errors must rest with the author. Alan Jeffrey Index of Special Functions and Notations Section or formula Notation Name containing its definition \a I Absolute value of the real number 1.1.2.1 am u Amplitude of an elliptic function 12.2.1.1.2 ~ Asymptotic relationship 1.14.2.1 a Modular angle of an elliptic integral 12.1.2 arg ζ Argument of complex number ζ 2.1.1.1 A(x) A(x) = 2P(x) - 1; probability function 13.1.1.1.7 A Matrix A~1 Multiplicative inverse of a square matrix A 1.5.1.1.9 AT Transpose of matrix A 1.5.1.1.7 IAI Determinant associated with a square matrix A 1.4.1.1 Β Bernoulli number 1.3.1.1 n B* Alternative Bernoulli number 1.3.1.1.6 B(JC) Bernoulli polynomial 1.3.2.1.1 n B{x,y) Beta function 11.1.7.1 Binomial coefficient 1.2.1.1 n\ n\ 0- kJ k\(n-k)\ K C (JC) Fresnel cosine integral 14.1.1.1.1 xxi xxii Index of Special Functions Section or formula Notation Name containing its definition Cofactor of element <ζ in a square matrix A 1.4.2 /7 nC or Γ Combination symbol nC = ( n ) 1.6.2.1 m en w Jacobian elliptic function 12.2.1.1.4 en"1 w Inverse Jacobian elliptic function 12.4.1.1.4 curl F = V x F Curl of vector F 23.8.1.1.6 SM Dirac delta function 19.1.3 D{x) Dirichlet kernel 1.13.1.10.3 n dn u Jacobian elliptic function 12.2.1.1.5 dn-1 M Inverse Jacobian elliptic function 12.4.1.1.5 divF= VF Divergence of vector F 23.8.1.1.4 Euler formula; el° = cos θ + / sin θ 2.1.1.2.1 e Euler's constant 0.3 Ei(x) Exponential integral 5.1.2.2 Ε(φΛ) Incomplete elliptic integral of the second kind 12.1.1.1.5 E(k),E'(k) Complete elliptic integrals of the second kind 13.1.1.1.8, 13.1.1.1.10 Matrix exponential 1.5.4.1 erf x Error function 13.2.1.1 erfc x Complementary error function 13.2.1.1.4 E„ Euler number 1.3.1.1 Κ Alternative Euler number 1.3.1.1.6 E„(x) Euler polynomial 1.3.2.3.1 f(x) A function of χ fix) First derivative df/dx 1.15.1.1.6 f(n\x) nth derivative dn f/dxn 1.12.1.1 f{n()xo) nth derivative dn f/dx" at xo 1.12.1.1 F(<p,k) Incomplete elliptic integral of the first kind 12.1.1.1.4 II*» Il Norm of Φ„(χ) 18.1.1.1 grad φ — V</> Gradient of the scalar function φ 23.8.1.6 Γ(χ) Gamma function 11.1.1.1 y Euler-Mascheroni constant 1.11.1.1.7 H(x) Heaviside step function 19.1.2.5 H(x) Hermite polynomial 18.5.3 n i Imaginary unit 1.1.1.1 Im{z) Imaginary part of ζ = x -h iy \ Im{z} = y 1.1.1.2 I Unit (identity) matrix 1.5.1.1.3 i" erfc x nth repeated integral of erfc χ 13.2.7.1.1 1±ΛΧ) Modified Bessel function of the first kind of order y 17.6.1.1 ff(x)dx Indefinite integral (antiderivative) of f(x) 1.15.2 fïfWdx Definite integral of f(x) from χ = a to χ = b 1.15.2.5 Index of Special Functions XXJjj Section or formula Notation Name containing its definition J±(x) Bessel function of the first kind of order ν 17.1.1.1 v k Modulus of an elliptic integral 12.1.1.1 k' Complementary modulus of an elliptic integral; 12.1.1.1 vT^F V = K(k), K'(k) Complete elliptic integrals of the first kind 12.1.1.1.7, 12.1.1.1.9 K(x) Modified Bessel function of the second kind of order ν 17.6.1.1 v £[/(*); s] Laplace transform of f(x) 19.1.1 L{x) Laguerre polynomial 18.4.1 n log χ Logarithm of χ to the base a 2.2.1.1 a In χ Natural logarithm of χ (to the base e) 2.2.1.1 Mjj Minor of element in a square matrix A 1.4.2 nl Factorials; «! = 1 23··.«; 0! = 1 1.2.1.1 (2/i)!! Double factorial; (2/i)!! = 2 · 4 · 6 · · (2/i) 15.2.1 (2/1 - 1)!! Double factorial; (2n - 1)!! = 1 · 3 · 5 · · · {In - 1) 15.2.1 Γ^Ι Integral part of n/2 18.2.4.1.1 η ! "Por P Permutation symbol; n P = 1.6.1.1.3 m n m m (n — m)\ P (JC) Legendre polynomial 18.2.1 n Ρ (χ) Normal probability distribution 13.1.1.1.5 η η ]~~[ Uf( Product symbol; ]~| u^ = u \ uj · · · u 1.9.1.1.1 n Ä:= 1 k=\ PV. f(x)dx Cauchy principal value of the integral 1.15.4.IV π Ratio of the circumference of a circle to its diameter 0.3 n(jc) pi function 11.1.1.1 Π (φ, /ζ, k) Incomplete elliptic integral of the third kind 12.1.1.1.6 ψ (ζ) psi (digamma) function 11.1.6.1 Q(x) Probability function; Q(x) = 1 - P(x) 13.1.1.1.6 Q(x) Quadratic form 1.5.2.1 Q (x) Legendre function of the second kind 18.2.7 n r Modulus of ζ = χ + iy\ r = (x2 + y2)1 2/ 2.1.1 Re{z} Real part of ζ = JC + iy\ Re{z} =x 1.1.1.2 sgn(x) sign of χ defined as JC/|JC| 1.1.21 sn u Jacobian elliptic function 12.2.1.1.3 sn~1 u Inverse Jacobian elliptic function 12.4.1.1.3 S(x) Fresnel sine integral 14.1.1.1.2 Summation symbol; 1.2.3 and if η < m we define xxiv Index of Special Functionsand Notations Section or formula Notation Name containing its definition σο Σα*(χ -χοΫ Power series expanded about xo 1.11.1.1.1 k=m Chebyshev polynomial 18.3.1.1 T„{X) Trace of a square matrix A 15.1.1.10 tr A Chebyshev polynomial 18.3.11 Unix) x = f~l(y) Function inverse to y = f(x) 1.11.1.8 Bessel function of the second kind of order ν 17.1.1.1 Υν(χ) Ζ Complex number ζ = χ + iy 1.1.1.1 |z| Modulus of ζ = χ + iy; r = \z\ = (x2 + y2)l/2 1.1.1.1 ζ Complex conjugate of ζ = χ iy; ζ = χ — iy 1.1.1.1 R E ο T P A H C Quick Reference List of Frequently Used Data 0.1 Useful Identities 0.1.1 Trigonometric identities sin2 χ + cos2 χ = 1 sec2 χ = 1 + tan2 χ sin(;c + y) — sin JC cos y + cos JC sin y esc2 JC = 1 + cot2 χ sin(jc — y) — sin JC cos y — cos JC sin y sin 2x = 2 sin JC cos JC COS(JC + y) = cos JC cos y — sinjc siny cos 2x = cos2 JC — sin2 JC COS(JC — y) = cos JC cos y H- sinjc sin y = 1-2 sin2 JC tanjc + tan y tan(jc + y) = 2 cos2 JC — 1 1 — tan JC tan y tan JC — tan y sin2 χ = -2( 1 - COS2JC) tan(jc — y) 1 + tanjc tan y cos2 X = -(1 + COS2JC) 2 1 2 Chapter 0 Quick Reference List of Frequently Used Data 0.1.2 Hyperbolic identities cosh2 χ — sinh2 χ = 1 sinh(jc + y) sinh JC cosh y + cosh JC sinh sech2 χ = 1 — tanh2 χ sinh(jc — y) sinh JC cosh y — cosh JC sinh csch2 χ = coth2 χ — 1 cosh(jc + y) = cosh JC cosh y -f sinh JC sinh sinh 2x = 2 sinh χ cosh χ cosh(jc — _y) = cosh JC cosh y — sinh JC sinh cosh 2x = cosh2 JC + sinh2 χ tanh JC + tanh y = 1+2 sinh2 JC tanh(jc + = 1 + tanh JC tanh y = 2 cosh2 χ - 1 tanh JC — tanh y ? 1 sinh χ = -(cosh2jc — 1) tanh(jc — y) = 1 — tanh JC tanh y cosh JC = -(cosh2jc + 1) jc + (a2+jc2)1/2 arcsinh — = In [—oo < x/a < oo] a jc + (jc2-a2)1/2 arccosh — = In [x/a > 1] a a + χ arctanh - = ^-ln [JC2 < a2] a 2 a — χ 0.2 Complex Relationships elx = COSJC + / sinjc sinhjc = eix _ -ixe 2 sinjc = 2i ex -h e~ coshjc = eix _|_ -iex COSJC = sinix = i sinhjc (COSJC + / sinjc)w = COSMJC + / sinrcjc COS/JC = coshjc sinnjc = Im{(COSJC + / sinjc)"} sinh/jc = / s'mx cosnjc = Re{(COSJC + i sinjc)"} cosh/jc = COSJC 0.3 Constants e = 2.7182 81828 45904 π = 3.1415 92653 58979 logjo e = 0.4342 94481 90325