Handbook of Mathematical Formulas and Integrals FOURTH EDITION PDF

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Handbook of Mathematical Formulas and Integrals FOURTH EDITION Handbook of Mathematical Formulas and Integrals FOURTH EDITION Alan Jeffrey Hui-Hui Dai Professor of Engineering Mathematics Associate Professor of Mathematics University of Newcastle upon Tyne City University of Hong Kong Newcastle upon...

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Handbook of Mathematical Formulas and Integrals FOURTH EDITION

Handbook of Mathematical Formulas and Integrals FOURTH EDITION

Alan Jeffrey

Hui-Hui Dai

Professor of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom

Associate Professor of Mathematics City University of Hong Kong Kowloon, China

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier

Acquisitions Editor: Lauren Schultz Yuhasz Developmental Editor: Mara Vos-Sarmiento Marketing Manager: Leah Ackerson Cover Design: Alisa Andreola Cover Illustration: Dick Hannus Production Project Manager: Sarah M. Hajduk Compositor: diacriTech Cover Printer: Phoenix Color Printer: Sheridan Books Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK This book is printed on acid-free paper. c 2008, Elsevier Inc. All rights reserved. Copyright
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. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected]. You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Application Submitted British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN: 978-0-12-374288-9 For information on all Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America 08 09 10

9 8 7 6 5 4 3 2 1

Contents

Preface

xix

Preface to the Fourth Edition

xxi

Notes for Handbook Users

xxiii

Index of Special Functions and Notations

xliii

0

Quick Reference List of Frequently Used Data 0.1. Useful Identities 0.1.1. Trigonometric Identities 0.1.2. Hyperbolic Identities 0.2. Complex Relationships 0.3. Constants, Binomial Coefficients and the Pochhammer Symbol 0.4. Derivatives of Elementary Functions 0.5. Rules of Differentiation and Integration 0.6. Standard Integrals 0.7. Standard Series 0.8. Geometry

1 1 1 2 2 3 3 4 4 10 12

1

Numerical, Algebraic, and Analytical Results for Series and Calculus 1.1. Algebraic Results Involving Real and Complex Numbers 1.1.1. Complex Numbers 1.1.2. Algebraic Inequalities Involving Real and Complex Numbers 1.2. Finite Sums 1.2.1. The Binomial Theorem for Positive Integral Exponents 1.2.2. Arithmetic, Geometric, and Arithmetic–Geometric Series 1.2.3. Sums of Powers of Integers 1.2.4. Proof by Mathematical Induction 1.3. Bernoulli and Euler Numbers and Polynomials 1.3.1. Bernoulli and Euler Numbers 1.3.2. Bernoulli and Euler Polynomials 1.3.3. The Euler–Maclaurin Summation Formula 1.3.4. Accelerating the Convergence of Alternating Series 1.4. Determinants 1.4.1. Expansion of Second- and Third-Order Determinants 1.4.2. Minors, Cofactors, and the Laplace Expansion 1.4.3. Basic Properties of Determinants

27 27 27 28 32 32 36 36 38 40 40 46 48 49 50 50 51 53

v

vi

Contents

1.5.

1.6.

1.7.

1.8.

1.9.

1.10. 1.11. 1.12.

1.13. 1.14.

1.15.

1.4.4. Jacobi’s Theorem 1.4.5. Hadamard’s Theorem 1.4.6. Hadamard’s Inequality 1.4.7. Cramer’s Rule 1.4.8. Some Special Determinants 1.4.9. Routh–Hurwitz Theorem Matrices 1.5.1. Special Matrices 1.5.2. Quadratic Forms 1.5.3. Differentiation and Integration of Matrices 1.5.4. The Matrix Exponential 1.5.5. The Gerschgorin Circle Theorem Permutations and Combinations 1.6.1. Permutations 1.6.2. Combinations Partial Fraction Decomposition 1.7.1. Rational Functions 1.7.2. Method of Undetermined Coefficients Convergence of Series 1.8.1. Types of Convergence of Numerical Series 1.8.2. Convergence Tests 1.8.3. Examples of Infinite Numerical Series Infinite Products 1.9.1. Convergence of Infinite Products 1.9.2. Examples of Infinite Products Functional Series 1.10.1. Uniform Convergence Power Series 1.11.1. Definition Taylor Series 1.12.1. Definition and Forms of Remainder Term 1.12.2. Order Notation (Big O and Little o) Fourier Series 1.13.1. Definitions Asymptotic Expansions 1.14.1. Introduction 1.14.2. Definition and Properties of Asymptotic Series Basic Results from the Calculus 1.15.1. Rules for Differentiation 1.15.2. Integration 1.15.3. Reduction Formulas 1.15.4. Improper Integrals 1.15.5. Integration of Rational Functions 1.15.6. Elementary Applications of Definite Integrals

53 54 54 55 55 57 58 58 62 64 65 67 67 67 68 68 68 69 72 72 72 74 77 77 78 79 79 82 82 86 86 88 89 89 93 93 94 95 95 96 99 101 103 104

Contents

vii

2

Functions and Identities 2.1. Complex Numbers and Trigonometric and Hyperbolic Functions 2.1.1. Basic Results 2.2. Logorithms and Exponentials 2.2.1. Basic Functional Relationships 2.2.2. The Number e 2.3. The Exponential Function 2.3.1. Series Representations 2.4. Trigonometric Identities 2.4.1. Trigonometric Functions 2.5. Hyperbolic Identities 2.5.1. Hyperbolic Functions 2.6. The Logarithm 2.6.1. Series Representations 2.7. Inverse Trigonometric and Hyperbolic Functions 2.7.1. Domains of Definition and Principal Values 2.7.2. Functional Relations 2.8. Series Representations of Trigonometric and Hyperbolic Functions 2.8.1. Trigonometric Functions 2.8.2. Hyperbolic Functions 2.8.3. Inverse Trigonometric Functions 2.8.4. Inverse Hyperbolic Functions 2.9. Useful Limiting Values and Inequalities Involving Elementary Functions 2.9.1. Logarithmic Functions 2.9.2. Exponential Functions 2.9.3. Trigonometric and Hyperbolic Functions

109 109 109 121 121 123 123 123 124 124 132 132 137 137 139 139 139 144 144 145 146 146 147 147 147 148

3

Derivatives of Elementary Functions 3.1. Derivatives of Algebraic, Logarithmic, and Exponential Functions 3.2. Derivatives of Trigonometric Functions 3.3. Derivatives of Inverse Trigonometric Functions 3.4. Derivatives of Hyperbolic Functions 3.5. Derivatives of Inverse Hyperbolic Functions

149 149 150 150 151 152

4

Indefinite Integrals of Algebraic Functions 4.1. Algebraic and Transcendental Functions 4.1.1. Definitions 4.2. Indefinite Integrals of Rational Functions 4.2.1. Integrands Involving xn 4.2.2. Integrands Involving a + bx 4.2.3. Integrands Involving Linear Factors 4.2.4. Integrands Involving a2 ± b2 x2 4.2.5. Integrands Involving a + bx + cx2

153 153 153 154 154 154 157 158 162

viii

Contents

4.3.

5

6

7

4.2.6. Integrands Involving a + bx3 4.2.7. Integrands Involving a + bx4 Nonrational Algebraic Functions √ 4.3.1. Integrands Containing a + bxk and x 1/2 4.3.2. Integrands Containing (a + bx) 1/2 4.3.3. Integrands Containing (a + cx2 ) 1/2
4.3.4. Integrands Containing a + bx + cx2

164 165 166 166 168 170 172

Indefinite Integrals of Exponential Functions 5.1. Basic Results 5.1.1. Indefinite Integrals Involving eax 5.1.2. Integrals Involving the Exponential Functions Combined with Rational Functions of x 5.1.3. Integrands Involving the Exponential Functions Combined with Trigonometric Functions

175 175 175

Indefinite Integrals of Logarithmic Functions 6.1. Combinations of Logarithms and Polynomials 6.1.1. The Logarithm 6.1.2. Integrands Involving Combinations of ln(ax) and Powers of x 6.1.3. Integrands Involving (a + bx)m lnn x 6.1.4. Integrands Involving ln(x2 ± a2 ) 1/2 
6.1.5. Integrands Involving xm ln x + x2 ± a2

181 181 181

Indefinite Integrals of Hyperbolic Functions 7.1. Basic Results 7.1.1. Integrands Involving sinh(a + bx) and cosh(a + bx) 7.2. Integrands Involving Powers of sinh(bx) or cosh(bx) 7.2.1. Integrands Involving Powers of sinh(bx) 7.2.2. Integrands Involving Powers of cosh(bx) 7.3. Integrands Involving (a + bx)m sinh(cx) or (a + bx)m cosh(cx) 7.3.1. General Results 7.4. Integrands Involving xm sinhn x or xm coshn x 7.4.1. Integrands Involving xm sinhn x 7.4.2. Integrands Involving xm coshn x 7.5. Integrands Involving xm sinhn x or xm coshn x 7.5.1. Integrands Involving xm sinhn x 7.5.2. Integrands Involving xm coshn x 7.6. Integrands Involving (1 ± cosh x)−m 7.6.1. Integrands Involving (1 ± cosh x)−1 7.6.2. Integrands Involving (1 ± cosh x)−2

189 189 189 190 190 190 191 191 193 193 193 193 193 194 195 195 195

175 177

182 183 185 186

Contents

8

9

ix

7.7.

Integrands Involving sinh(ax) cosh−n x or cosh(ax) sinh−n x 7.7.1. Integrands Involving sinh(ax) coshn x 7.7.2. Integrands Involving cosh(ax) sinhn x 7.8. Integrands Involving sinh(ax + b) and cosh(cx + d) 7.8.1. General Case 7.8.2. Special Case a = c 7.8.3. Integrands Involving sinhp x coshq x 7.9. Integrands Involving tanh kx and coth kx 7.9.1. Integrands Involving tanh kx 7.9.2. Integrands Involving coth kx 7.10. Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 7.10.1. Integrands Involving (a + bx)m sinh kx 7.10.2. Integrands Involving (a + bx)m cosh kx

195 195 196 196 196 197 197 198 198 198 199 199 199

Indefinite Integrals Involving Inverse Hyperbolic Functions 8.1. Basic Results 8.1.1. Integrands Involving Products of xn and arcsinh(x/a) or arc(x/c) 8.2. Integrands Involving x−n arcsinh(x/a) or x−n arccosh(x/a) 8.2.1. Integrands Involving x−n arcsinh(x/a) 8.2.2. Integrands Involving x−n arccosh(x/a) 8.3. Integrands Involving xn arctanh(x/a) or xn arccoth(x/a) 8.3.1. Integrands Involving xn arctanh(x/a) 8.3.2. Integrands Involving xn arccoth(x/a) 8.4. Integrands Involving x−n arctanh(x/a) or x−n arccoth(x/a) 8.4.1. Integrands Involving x−n arctanh(x/a) 8.4.2. Integrands Involving x−n arccoth(x/a)

201 201

Indefinite Integrals of Trigonometric Functions 9.1. Basic Results 9.1.1. Simplification by Means of Substitutions 9.2. Integrands Involving Powers of x and Powers of sin x or cos x 9.2.1. Integrands Involving xn sinm x 9.2.2. Integrands Involving x−n sinm x 9.2.3. Integrands Involving xn sin−m x 9.2.4. Integrands Involving xn cosm x 9.2.5. Integrands Involving x−n cosm x 9.2.6. Integrands Involving xn cos−m x m 9.2.7. Integrands Involving xn sin x/(a + b cos x) m or xn cos x/(a + b sin x) 9.3. Integrands Involving tan x and/or cot x 9.3.1. Integrands Involving tann x or tann x/(tan x ± 1) 9.3.2. Integrands Involving cotn x or tan x and cot x

207 207 207 209 209 210 211 212 213 213

201 202 202 203 204 204 204 205 205 205

214 215 215 216

x

Contents

9.4.

9.5.

Integrands Involving sin x and cos x 9.4.1. Integrands Involving sinm x cosn x 9.4.2. Integrands Involving sin−n x 9.4.3. Integrands Involving cos−n x 9.4.4. Integrands Involving sinm x/ cosn x cosm x/ sinn x 9.4.5. Integrands Involving sin−m x cos−n x Integrands Involving Sines and Cosines with Linear Arguments and Powers of x 9.5.1. Integrands Involving Products of (ax + b)n , sin(cx + d), and/or cos(px + q) 9.5.2. Integrands Involving xn sinm x or xn cosm x

10 Indefinite Integrals of Inverse Trigonometric Functions 10.1. Integrands Involving Powers of x and Powers of Inverse Trigonometric Functions 10.1.1. Integrands Involving xn arcsinm (x/a) 10.1.2. Integrands Involving x−n arcsin(x/a) 10.1.3. Integrands Involving xn arccosm (x/a) 10.1.4. Integrands Involving x−n arccos(x/a) 10.1.5. Integrands Involving xn arctan(x/a) 10.1.6. Integrands Involving x−n arctan(x/a) 10.1.7. Integrands Involving xn arccot(x/a) 10.1.8. Integrands Involving x−n arccot(x/a) 10.1.9. Integrands Involving Products of Rational Functions and arccot(x/a) 11 The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions 11.1. The Euler Integral Limit and Infinite Product Representations for the Gamma Function
(x). The Incomplete Gamma Functions
(α, x) and γ(α, x) 11.1.1. Definitions and Notation 11.1.2. Special Properties of
(x) 11.1.3. Asymptotic Representations of
(x) and n! 11.1.4. Special Values of
(x) 11.1.5. The Gamma Function in the Complex Plane 11.1.6. The Psi (Digamma) Function 11.1.7. The Beta Function 11.1.8. Graph of
(x) and Tabular Values of
(x) and ln
(x) 11.1.9. The Incomplete Gamma Function 12 Elliptic Integrals and Functions 12.1. Elliptic Integrals 12.1.1. Legendre Normal Forms

217 217 217 218 218 220 221 221 222 225 225 225 226 226 227 227 227 228 228 229

231

231 231 232 233 233 233 234 235 235 236 241 241 241

Contents

xi

12.1.2. Tabulations and Trigonometric Series Representations of Complete Elliptic Integrals 12.1.3. Tabulations and Trigonometric Series for E(ϕ, k) and F (ϕ, k) 12.2. Jacobian Elliptic Functions 12.2.1. The Functions sn u, cn u, and dn u 12.2.2. Basic Results 12.3. Derivatives and Integrals 12.3.1. Derivatives of sn u, cn u, and dn u 12.3.2. Integrals Involving sn u, cn u, and dn u 12.4. Inverse Jacobian Elliptic Functions 12.4.1. Definitions

243 245 247 247 247 249 249 249 250 250

13 Probability Distributions and Integrals, and the Error Function 13.1. Distributions 13.1.1. Definitions 13.1.2. Power Series Representations (x ≥ 0) 13.1.3. Asymptotic Expansions (x  0) 13.2. The Error Function 13.2.1. Definitions 13.2.2. Power Series Representation 13.2.3. Asymptotic Expansion (x  0) 13.2.4. Connection Between P (x) and erf x 13.2.5. Integrals Expressible in Terms of erf x 13.2.6. Derivatives of erf x 13.2.7. Integrals of erfc x 13.2.8. Integral and Power Series Representation of in erfc x 13.2.9. Value of in erfc x at zero

253 253 253 256 256 257 257 257 257 258 258 258 258 259 259

14 Fresnel Integrals, Sine and Cosine Integrals 14.1. Definitions, Series Representations, and Values at Infinity 14.1.1. The Fresnel Integrals 14.1.2. Series Representations 14.1.3. Limiting Values as x → ∞ 14.2. Definitions, Series Representations, and Values at Infinity 14.2.1. Sine and Cosine Integrals 14.2.2. Series Representations 14.2.3. Limiting Values as x → ∞

261 261 261 261 263 263 263 263 264

15 Definite Integrals 15.1. Integrands Involving 15.2. Integrands Involving 15.3. Integrands Involving 15.4. Integrands Involving

265 265 267 270 273

Powers of x Trigonometric Functions the Exponential Function the Hyperbolic Function

xii

Contents

15.5. Integrands Involving the Logarithmic Function 15.6. Integrands Involving the Exponential Integral Ei(x)

273 274

16 Different Forms of Fourier Series 16.1. Fourier Series for f (x) on −π ≤ x ≤ π 16.1.1. The Fourier Series 16.2. Fourier Series for f (x) on −L ≤ x ≤ L 16.2.1. The Fourier Series 16.3. Fourier Series for f (x) on a ≤ x ≤ b 16.3.1. The Fourier Series 16.4. Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ π 16.4.1. The Fourier Series 16.5. Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ L 16.5.1. The Fourier Series 16.6. Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ π 16.6.1. The Fourier Series 16.7. Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ L 16.7.1. The Fourier Series 16.8. Complex (Exponential) Fourier Series for f (x) on −π ≤ x ≤ π 16.8.1. The Fourier Series 16.9. Complex (Exponential) Fourier Series for f (x) on −L ≤ x ≤ L 16.9.1. The Fourier Series 16.10. Representative Examples of Fourier Series 16.11. Fourier Series and Discontinuous Functions 16.11.1. Periodic Extensions and Convergence of Fourier Series 16.11.2. Applications to Closed-Form Summations of Numerical Series

275 275 275 276 276 276 276 277 277 277 277 278 278 278 278 279 279 279 279 280 285 285

17 Bessel Functions 17.1. Bessel’s Differential Equation 17.1.1. Different Forms of Bessel’s Equation 17.2. Series Expansions for Jν (x) and Yν (x) 17.2.1. Series Expansions for Jn (x) and Jν (x) 17.2.2. Series Expansions for Yn (x) and Yν (x) 17.2.3. Expansion of sin(x sin θ) and cos(x sin θ) in Terms of Bessel Functions 17.3. Bessel Functions of Fractional Order 17.3.1. Bessel Functions J±(n+1/2) (x) 17.3.2. Bessel Functions Y±(n+1/2) (x) 17.4. Asymptotic Representations for Bessel Functions 17.4.1. Asymptotic Representations for Large Arguments 17.4.2. Asymptotic Representation for Large Orders 17.5. Zeros of Bessel Functions 17.5.1. Zeros of Jn (x) and Yn (x)

289 289 289 290 290 291

285

292 292 292 293 294 294 294 294 294

Contents

17.6. 17.7.

17.8.

17.9. 17.10.

17.11. 17.12. 17.13. 17.14.

17.15.

xiii

Bessel’s Modified Equation 17.6.1. Different Forms of Bessel’s Modified Equation Series Expansions for Iν (x) and Kν (x) 17.7.1. Series Expansions for In (x) and Iν (x) 17.7.2. Series Expansions for K0 (x) and Kn (x) Modified Bessel Functions of Fractional Order 17.8.1. Modified Bessel Functions I±(n+1/2) (x) 17.8.2. Modified Bessel Functions K±(n+1/2) (x) Asymptotic Representations of Modified Bessel Functions 17.9.1. Asymptotic Representations for Large Arguments Relationships Between Bessel Functions 17.10.1. Relationships Involving Jν (x) and Yν (x) 17.10.2. Relationships Involving Iν (x) and Kν (x) Integral Representations of Jn (x), In (x), and Kn (x) 17.11.1. Integral Representations of Jn (x) Indefinite Integrals of Bessel Functions 17.12.1. Integrals of Jn (x), In (x), and Kn (x) Definite Integrals Involving Bessel Functions 17.13.1. Definite Integrals Involving Jn (x) and Elementary Functions Spherical Bessel Functions 17.14.1. The Differential Equation 17.14.2. The Spherical Bessel Function jn (x) and yn (x) 17.14.3. Recurrence Relations 17.14.4. Series Representations 17.14.5. Limiting Values as x→ 0 17.14.6. Asymptotic Expansions of jn (x) and yn (x) When the Order n Is Large Fourier-Bessel Expansions

18 Orthogonal Polynomials 18.1. Introduction 18.1.1. Definition of a System of Orthogonal Polynomials 18.2. Legendre Polynomials Pn (x) 18.2.1. Differential Equation Satisfied by Pn (x) 18.2.2. Rodrigues’ Formula for Pn (x) 18.2.3. Orthogonality Relation for Pn (x) 18.2.4. Explicit Expressions for Pn (x) 18.2.5. Recurrence Relations Satisfied by Pn (x) 18.2.6. Generating Function for Pn (x) 18.2.7. Legendre Functions of the Second Kind Qn (x) 18.2.8. Definite Integrals Involving Pn (x) 18.2.9. Special Values

294 294 297 297 298 298 298 299 299 299 299 299 301 302 302 302 302 303 303 304 304 305 306 306 306 307 307 309 309 309 310 310 310 310 310 312 313 313 315 315

xiv

Contents

18.3.

18.4.

18.5.

18.6.

18.2.10. Associated Legendre Functions 18.2.11. Spherical Harmonics Chebyshev Polynomials Tn (x) and Un (x) 18.3.1. Differential Equation Satisfied by Tn (x) and Un (x) 18.3.2. Rodrigues’ Formulas for Tn (x) and Un (x) 18.3.3. Orthogonality Relations for Tn (x) and Un (x) 18.3.4. Explicit Expressions for Tn (x) and Un (x) 18.3.5. Recurrence Relations Satisfied by Tn (x) and Un (x) 18.3.6. Generating Functions for Tn (x) and Un (x) Laguerre Polynomials Ln (x) 18.4.1. Differential Equation Satisfied by Ln (x) 18.4.2. Rodrigues’ Formula for Ln (x) 18.4.3. Orthogonality Relation for Ln (x) 18.4.4. Explicit Expressions for Ln (x) and xn in Terms of Ln (x) 18.4.5. Recurrence Relations Satisfied by Ln (x) 18.4...