Handbook of Mathematical Formulas and Integrals, Second Edition

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Handbook of Mathematical Formulas and Integrals, Second Edition
This Page Intentionally Left Blank HANDBOOK OF MATHEMATICAL EODMIIM AND INTEGRALS Second edition ALAN JEFFREY Department of Engineering Mathematics University of newcastle upon tyne Newcastle upon Tyne United Kingdon ACADEMIC PRESS A Harcourt Science and Technology company San Diego San Francisco New York Boston London Sydney Tokyo This book is printed on acid-free paper. Copyright 2000, 1995 by Academic Press All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Requests for permission to make copies of any part of the work should be mailed to the following address Permissions Department, Harcourt, Inc, 6277 Sea Harbor Drive, Orlando, Florida 32887-6777 ACADEMIC PRESS a Harcourt Science and Technology Company 525 B Street, Suite 1900. San Diego, CA 92101-4495 USA http://www.academicpress.com Academic pre Harcourt Place. 32 Jamestown Road. London nW1 7BY UK Library of Congress Catalog Number: 95-2344 International Standard Book number: 0-12-382251-3 Printed in the United States of america 0001020304COB987654321 Contents Preface xix Preface to the second edition xxi Index of Special Functions and Notations xxiii Quick Reference List of Frequently Used Data 0.1 Useful Identities 1 0. 1. I Trigonometric identities I 0.1.2 Hyperbolic identities 2 0.2 Complex relationships 2 0. 3 Constants 2 0.4 Derivatives of Elementary Functions 3 0.5 Rules of Differentiation and Integration 3 0.6 Standard Integrals 4 0.7 Standard Series 11 0.8 Geometry 13 1 Numerical, Algebraic, and Analytical Results for Series and calculus 1. 1 Algebraic Results Involving Real and Complex Numbers 25 1.1.1 Complex numbers 25 1.1. 2 Algebraic inequalities involving real and complex numbers 26 Contents 2 Finite Sums 29 1.2. 1 The binomial theorem for positive integral exponents 1.2.2 Arithmetic, geometric, and arithmetic-geometric series 33 1. 2. 3 Sums of powers of integers 34 1. 2. 4 Proof by mathematical induction 36 3 Bernoulli and euler numbers and polynomials 37 1.3.1 Bernoulli and Euler numbers 37 1.3.2 Bernoulli and euler polynomials 43 1. 3. 3 The Euler Maclaurin summation formula 45 1.3.4 Accelerating the convergence of alternating series 46 1. 4 Determinants 47 1.4.1 Expansion of second-and third-order determinants 47 1.4.2 Minors, cofactors, and the Laplace expansion 48 1.4.3 Basic properties of determinants 50 1.4.4 Jacobi's theorem 50 1. 4.5 Hadamard's theorem 5 1. 4.6 Hadamard's inequality 51 1.4.7 Cramer's rule 52 1.4.8 Some special determinants 52 1. 4.9 Routh-Hurwitz theorem 54 1.5 Matrices 55 1.5.1 Special matrices 55 5.2 Quadratic forms 58 .5.3 Differentiation and integration of matrices 60 1.5. 4 The matrix exponential 61 1.5.5 The Gerschgorin circle theorem 61 1. 6 Permutations and Combinations 62 1. 6.1 Permutations 62 1. 6.2 Combinations 62 1.7 Partial fraction Decomposition 63 1.7.1 Rational functions 63 1.7.2 Method of undetermined coefficients 63 1.8 Convergence of Series 66 1.8.1 Types of convergence of numerical series 66 1.8.2 Convergence tests 66 1.8.3 Examples of infinite numerical series 68 1.9 Infinite Products 71 1.9.1 Convergence of infinite products 71 1.9.2 Examples of infinite products 7 1.10 Functional Series 73 1.10.1 Uniform convergence 73 11 Power Series 74 1.11.1 Definition 74 Contents aylor Series 1. 12. 1 Definition and forms of remainder term 79 1. 12.4 Order notation(BigO and little o)80 1. 13 Fourier series 81 1. 13.1 Definitions 81 1.14 Asymptotic Expansions 85 1. 14.1 Introduct 1. 14.2 Definition and properties of asymptotic series 86 1.15 Basic Results from the calculus 86 1. 15.1 Rules for differentiation 86 1. 15.2 Integration 88 1. 15.3 Reduction formulas 91 15.4 Improper integrals 92 1. 15.5 Integration of rational functions 94 1. 15.6 Elementary applications of definite integrals 96 2 Functions and Identities 2. 1 Complex Numbers and Trigonometric and Hyperbolic Functions 101 2.1.1 Basic results 10 2.2 Logarithms and Exponentials 112 2.2.1 Basic functional relationships 112 2.2.2 The number e 113 2.3 The Exponential Function 114 2.3.1S ns114 2.4 Trigonometric Identities 115 2.4.1 Trigonometric functions 115 2.5 Hyperbolic Identities 121 2.5.1 Hyperbolic functi 2.6 The Logarithm 126 2.6. 1 Series representations 126 2.7 Inverse Trigonometric and hyperbolic functions 128 2.7.1 Domains of definition and principal values 128 2.7.2 Functional relations 128 2. 8 Series Representations of Trigonometric and Hyperbolic Functions 133 2.8.1 Trigonometric functions 133 2.8.2 Hyperbolic functions 134 2.8.3 Inverse trigonometric functions 134 2.8.4 Inverse hyperbolic functions 135 2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 136 2.9.1 Logarithmic functions 136 2.9.2 Exponential functions 136 2.9.3 Trigonometric and hyperbolic functions 137 Contents 3 Derivatives of Elementary Functions 3. 1 Derivatives of Algebraic, Logarithmic, and Exponential Functions 139 3.2 Derivatives of Trigonometric Functions 140 3.3 Derivatives of Inverse Trigonometric Functions 140 3.4 Derivatives of Hyperbolic Functions 141 3.5 Derivatives of Inverse Hyperbolic Functions 142 4 Indefinite Integrals of Algebraic Functions 4.1 Algebraic and Transcendental Functions 145 4.1.1 Definitions 145 4.2 Indefinite Integrals of Rational Functions 146 4.2. 1 Integrands involving x" 146 4.2.2 Integrands involving a bx 146 4.2.3 Integrands involving linear factors 149 4.2.4 Integrands involving a+bx 150 4.2.5 Integrands involving a bx +cx 153 4.2.6 Integrands involving a + bx 155 4.2.7 Integrands involving a+bx 156 4.3 Nonrational Algebraic Functions 158 4.3.1 Integrands containing a+bxk and vx 158 4.3.2 Integrands containing(a+ bx)1/2 160 4.3.3 Integrands containing (a+cr2)1/2 161 4.3.4 Integ ing(a+bx+cx2)/2 164 5 Indefinite Integrals of Exponential Functions 5.1 Basic Results 167 5.1.1 Indefinite integrals involving ex 167 5.1.2 Integrands involving the exponential functions combined with rational functions ofx 168 5.1.3 Integrands involving the exponential functions combined with trigonometric functions 169 6 Indefinite Integrals of Logarithmic Functions 6. 1 Combinations of logarithms and polynomials 173 6. 1. 1 The logarithm 173 6. 1.2 Integrands involving combinations of In(ax) and powers ofx 174 6. 1.3 Integrands involving(a+bx )mIn 175 Contents 6.1.4 Integrands involving In(x+a) 177 6.1.5 Integrands involving xmIn[x +(x2+a2)/2]178 7 Indefinite Integrals of Hyperbolic Functions 7.1 Basic results 179 7. 1. 1 Integrands involving sinh(a+ bx)and cosh(a+ bx) 179 7.2 Integrands Involving Powers of sinh(bx )or cosh(bx) 180 7. 2. 1 Integrands involving powers of sinh(bx) 180 7.2.2 Integrands involving powers of cosh(bx) 180 7.3 Integrands Involving(a+ bx )"sinh(cx)or(a+bx)"mcosh(cx) 181 7.3. 1 General results 181 7.4 Integrands Involving x sinh"x or cosh"x 183 7.4.1 Integrands involving xm"x 18 7.4.2 Integrals involving xm cosh"x 183 7.5 Integrands invol or xm cosh x 183 7.5. 1 Integrands involving xmsinh x 183 7.5.2 Integrands involving xmcosh x 184 7.6 Integrands Involving(1 +t cosh x)m 185 7.6. 1 Integrands involving(1 +t cosh x) 185 7.6.2 Integrands involving(1 coshx) 185 7.7 Integrands Involving sinh(ax )cosh x or cosh(ax)sinh x 185 7.7. 1 Integrands involving sinh(ax )cosh x 185 7.7.2 Integrands involving cosh(ax)sinh x 186 7.8 Integrands Involving sinh(ax +b)and cosh(cx +d) 186 7. 8.1 General case 186 7.8.2 Special case a =C 187 7.8.3 Integrands involving sinh"xcosh'x 187 7.9 Integrands Involving tanh kx and coth kx 188 7.9.1 Integrands involving tanh kx 188 7.9.2 Integrands involving coth kx 188 7. 10 Integrands Involving(a+ bx)"sinh kx or(a+bx)"mcosh kx 189 7. 10. 1 Integrands involving(a+ bx)"sinh kx 189 7. 10.2 Integrands involving(a+bx)m cosh kx 189 8 Indefinite Integrals Involving Inverse Hyperbolic Functions 8. 1 Basic results 191 8. 1. 1 Integrands involving products of x"and arcsinh(x/a)or arccos(x/ a) 191 8.2 Integrands Involving x arcsinh(x a)orxarccosh(/ a) 193 8. 2. 1 Integrands involving xn arcsinh(x/a) 193 8.2.2 Integrands involving xnarccosh(x/a) 193

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