hand book of mathematical formulas and integrals second edition pdf

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HANDBOOK OF

MATHEMATICAL
FORMULAS
AND INTEGRALS
Second Edition

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HANDBOOK OF

.

MATHEMATICAL
FORMULAS
AND INTEGRALS
Second Edition
ALAN JEFFREY
Department of Engineering Mathematics
University of Newcastle upon Tyne
Newcastle upon Tyne
United Kingdom

San Diego San Francisco New York Boston London Sydney Tokyo

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This book is printed on acid-free paper. ∞
Copyright

C

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 or
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
525 B Street, Suite 1900, San Diego, CA 92101-4495 USA

Academic Press
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
00 01 02 03 04 COB 9 8 7 6 5 4 3 2 1

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Contents

Preface xix
Preface to the Second Edition xxi
Index of Special Functions and Notations

0

xxiii

Quick Reference List of Frequently Used Data
0.1 Useful Identities 1
0.1.1 Trigonometric identities 1
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
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1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10
1.11

Finite Sums 29
1.2.1 The binomial theorem for positive integral

exponents 29
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
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
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 51
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
Matrices 55
1.5.1 Special matrices 55
1.5.2 Quadratic forms 58
1.5.3 Differentiation and integration of matrices 60
1.5.4 The matrix exponential 61
1.5.5 The Gerschgorin circle theorem 61
Permutations and Combinations 62
1.6.1 Permutations 62
1.6.2 Combinations 62
Partial Fraction Decomposition 63

1.7.1 Rational functions 63
1.7.2 Method of undetermined coefficients 63
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
Infinite Products 71
1.9.1 Convergence of infinite products 71
1.9.2 Examples of infinite products 71
Functional Series 73
1.10.1 Uniform convergence 73
Power Series 74
1.11.1 Definition 74

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1.12 Taylor Series 79
1.12.1 Definition and forms of remainder term 79
1.12.4 Order notation (Big O and little o) 80
1.13 Fourier Series 81
1.13.1 Definitions 81
1.14 Asymptotic Expansions 85
1.14.1 Introduction 85
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
1.15.4 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 101
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
2.3.1 Series representations 114
2.4 Trigonometric Identities 115
2.4.1 Trigonometric functions 115
2.5 Hyperbolic Identities 121
2.5.1 Hyperbolic functions 121
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

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Contents

3

Derivatives of Elementary Functions
3.1
3.2
3.3
3.4
3.5

4

Derivatives of Algebraic, Logarithmic, and Exponential Functions
Derivatives of Trigonometric Functions 140
Derivatives of Inverse Trigonometric Functions 140
Derivatives of Hyperbolic Functions 141
Derivatives of Inverse Hyperbolic Functions 142

139

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 n 146
4.2.2 Integrands involving a + bx 146
4.2.3 Integrands involving linear factors 149
4.2.4 Integrands involving a 2 ± b2 x 2 150
4.2.5 Integrands involving a + bx + cx 2 153
4.2.6 Integrands involving a + bx 3 155
4.2.7 Integrands involving a + bx 4 156
4.3 Nonrational Algebraic Functions 158 √
4.3.1 Integrands containing a + bx k and x 158
4.3.2 Integrands containing (a + bx)1/2 160
4.3.3 Integrands containing (a + cx 2 )1/2 161
4.3.4 Integrands containing (a + bx + cx 2 )1/2 164

5

Indefinite Integrals of Exponential Functions
5.1 Basic Results 167
5.1.1 Indefinite integrals involving eax 167
5.1.2 Integrands involving the exponential functions combined with rational
functions of x 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 ln(ax) and powers of x
6.1.3 Integrands involving (a + bx)m lnn x 175

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6.1.4 Integrands involving ln(x 2 ± a 2 ) 177
6.1.5 Integrands involving x m ln[x + (x 2 ± a 2 )1/2 ]

7

Indefinite Integrals of Hyperbolic Functions
7.1
7.2

7.3
7.4

7.5

7.6

7.7

7.8

7.9

7.10

8

178

Basic Results 179
7.1.1 Integrands involving sinh(a + bx) and cosh(a + bx) 179
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
Integrands Involving (a ± bx)m sinh(cx) or (a + bx)m cosh(cx) 181
7.3.1 General results 181
Integrands Involving x m sinhn x or x m coshn x 183
7.4.1 Integrands involving x m sinhn x 183
7.4.2 Integrals involving x m coshn x 183
Integrands Involving x m sinh−n x or x m cosh−n x 183
7.5.1 Integrands involving x m sinh−n x 183
7.5.2 Integrands involving x m cosh−n x 184
Integrands Involving (1 ± cosh x)−m 185
7.6.1 Integrands involving (1 ± cosh x)−1 185

7.6.2 Integrands involving (1 ± cosh x)−2 185
Integrands Involving sinh(ax)cosh−n x or cosh(ax)sinh−n x 185
7.7.1 Integrands involving sinh(ax) cosh−n x 185
7.7.2 Integrands involving cosh(ax) sinh−n x 186
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 p xcoshq x 187
Integrands Involving tanh kx and coth kx 188
7.9.1 Integrands involving tanh kx 188
7.9.2 Integrands involving coth kx 188
Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 189
7.10.1 Integrands involving (a + bx)m sinh kx 189
7.10.2 Integrands involving (a + bx)m cosh kx 189

Indefinite Integrals Involving Inverse Hyperbolic Functions
8.1 Basic Results 191
8.1.1 Integrands involving products of x n and arcsinh(x/a) or
arccosh(x/a) 191
8.2 Integrands Involving x −n arcsinh(x/a) or x −n arccosh(x/a) 193
8.2.1 Integrands involving x −n arcsinh(x/a) 193
8.2.2 Integrands involving x −n arccosh(x/a) 193

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8.3 Integrands Involving x n arctanh(x/a) or x n arccoth(x/a) 194
8.3.1 Integrands involving x n arctanh(x/a) 194
8.3.2 Integrands involving x n arccoth(x/a) 194
8.4 Integrands Involving x −n arctanh(x/a) or x −n arccoth(x/a) 195
8.4.1 Integrands involving x −n arctanh(x/a) 195
8.4.2 Integrands involving x −n arccoth(x/a) 195

9

Indefinite Integrals of Trigonometric Functions
9.1 Basic Results 197
9.1.1 Simplification by means of substitutions 197
9.2 Integrands Involving Powers of x and Powers of sin x or cos x 197
9.2.1 Integrands involving x n sinm x 199
9.2.2 Integrands involving x −n sinm x 200
9.2.3 Integrands involving x n sin−m x 201
9.2.4 Integrands involving x n cosm x 201
9.2.5 Integrands involving x −n cosm x 203
9.2.6 Integrands involving x n cos−m x 203
9.2.7 Integrands involving x n sin x/(a + b cos x)m or
x n cos x/(a + b sin x)m 204
9.3 Integrands Involving tan x and/or cot x 205
9.3.1 Integrands involving tann x or tann x/(tan x ± 1) 205
9.3.2 Integrands involving cotn x or tan x and cot x 206
9.4 Integrands Involving sin x and cos x 207
9.4.1 Integrands involving sinm x cosn x 207
9.4.2 Integrands involving sin−n x 207
9.4.3 Integrands involving cos−n x 208
9.4.4 Integrands involving sinm x/cosn x or cosm x/sinn x 208
9.4.5 Integrands involving sin−m x cos−n x 210

9.5 Integrands Involving Sines and Cosines with Linear Arguments and Powers
of x 211
9.5.1 Integrands involving products of (ax + b)n , sin(cx + d), and/or
cos( px + q) 211
9.5.2 Integrands involving x n sinm x or x n cosm x 211

10

Indefinite Integrals of Inverse Trigonometric Functions
10.1 Integrands Involving Powers of x and Powers of Inverse Trigonometric
Functions 215
10.1.1 Integrands involving x n arcsinm (x/a) 215
10.1.2 Integrands involving x −n arcsin(x/a) 216
10.1.3 Integrands involving x n arccosm (x/a) 216
10.1.4 Integrands involving x −n arccos(x/a) 217

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10.1.5
10.1.6
10.1.7
10.1.8
10.1.9

11

Integrands involving x n arctan(x/a) 217
Integrands involving x −n arctan(x/a) 218
Integrands involving x n arccot(x/a) 218
Integrands involving x −n arccot(x/a) 219
Integrands involving products of rational functions and
arccot(x/a) 219

The Gamma, Beta, Pi, and Psi Functions
11.1 The Euler Integral and Limit and Infinite Product Representations
for Ŵ(x) 221
11.1.1 Definitions and notation 221
11.1.2 Special properties of Ŵ(x) 222
11.1.3 Asymptotic representations of Ŵ(x) and n! 223
11.1.4 Special values of Ŵ(x) 223
11.1.5 The gamma function in the complex plane 223
11.1.6 The psi (digamma) function 224
11.1.7 The beta function 224
11.1.8 Graph of Ŵ(x) and tabular values of Ŵ(x) and ln Ŵ(x) 225

12

Elliptic Integrals and Functions
12.1 Elliptic Integrals 229
12.1.1 Legendre normal forms 229
12.1.2 Tabulations and trigonometric series representations of complete
elliptic integrals 231
12.1.3 Tabulations and trigonometric series for E(ϕ, k) and F(ϕ, k) 233
12.2 Jacobian Elliptic Functions 235
12.2.1 The functions sn u, cn u, and dn u 235

12.2.2 Basic results 235
12.3 Derivatives and Integrals 237
12.3.1 Derivatives of sn u, cn u, and dn u 237
12.3.2 Integrals involving sn u, cn u, and dn u 237
12.4 Inverse Jacobian Elliptic Functions 237
12.4.1 Definitions 237

13

Probability Integrals and the Error Function
13.1 Normal Distribution 239
13.1.1 Definitions 239
13.1.2 Power series representations (x ≥ 0) 240
13.1.3 Asymptotic expansions (x ≫ 0) 241

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13.2 The Error Function 242
13.2.1 Definitions 242
13.2.2 Power series representation 242
13.2.3 Asymptotic expansion (x ≫ 0) 243
13.2.4 Connection between P(x) and erf x 243
13.2.5 Integrals expressible in terms of erf x 243
13.2.6 Derivatives of erf x 243
13.2.7 Integrals of erfc x 243

13.2.8 Integral and power series representation of in erfc x
13.2.9 Value of in erfc x at zero 244

14

244

Fresnel Integrals, Sine and Cosine Integrals
14.1 Definitions, Series Representations, and Values at Infinity 245
14.1.1 The Fresnel integrals 245
14.1.2 Series representations 245
14.1.3 Limiting values as x → ∞ 247
14.2 Definitions, Series Representations, and Values at Infinity 247
14.2.1 Sine and cosine integrals 247
14.2.2 Series representations 247
14.2.3 Limiting values as x → ∞ 248

15

Definite Integrals
15.1
15.2
15.3
15.4
15.5

16

Integrands Involving Powers of x 249
Integrands Involving Trigonometric Functions 251

Integrands Involving the Exponential Function 254
Integrands Involving the Hyperbolic Function 256
Integrands Involving the Logarithmic Function 256

Different Forms of Fourier Series
16.1 Fourier Series for f (x) on −π ≤ x ≤ π 257
16.1.1 The Fourier series 257
16.2 Fourier Series for f (x) on −L ≤ x ≤ L 258
16.2.1 The Fourier series 258
16.3 Fourier Series for f (x) on a ≤ x ≤ b 258
16.3.1 The Fourier series 258
16.4 Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ π
16.4.1 The Fourier series 259
16.5 Half-Range Fourier Cosine Series for f (x) on 0 ≤ x ≤ L
16.5.1 The Fourier series 259

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16.6
16.7
16.8
16.9

16.10
16.11

17

Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ π 260
16.6.1 The Fourier series 260
Half-Range Fourier Sine Series for f (x) on 0 ≤ x ≤ L 260
16.7.1 The Fourier series 260
Complex (Exponential) Fourier Series for f (x) on −π ≤ x ≤ π 260
16.8.1 The Fourier series 260
Complex (Exponential) Fourier Series for f (x) on −L ≤ x ≤ L 261
16.9.1 The Fourier series 261
Representative Examples of Fourier Series 261
Fourier Series and Discontinuous Functions 265
16.11.1 Periodic extensions and convergence of Fourier series 265
16.11.2 Applications to closed-form summations of numerical
series 266

Bessel Functions
17.1
17.2

17.3

17.4

17.5
17.6
17.7

17.8

17.9
17.10

17.11

Bessel’s Differential Equation 269
17.1.1 Different forms of Bessel’s equation 269
Series Expansions for Jν (x) and Yν (x) 270
17.2.1 Series expansions for Jn (x) and Jν (x) 270
17.2.2 Series expansions for Yn (x) and Yν (x) 271
Bessel Functions of Fractional Order 272
17.3.1 Bessel functions J±(n+1/2) (x) 272
17.3.2 Bessel functions Y±(n+1/2) (x) 272
Asymptotic Representations for Bessel Functions 273
17.4.1 Asymptotic representations for large arguments 273
17.4.2 Asymptotic representation for large orders 273
Zeros of Bessel Functions 273
17.5.1 Zeros of Jn (x) and Yn (x) 273
Bessel’s Modified Equation 274
17.6.1 Different forms of Bessel’s modified equation 274
Series Expansions for Iν (x) and K ν (x) 276
17.7.1 Series expansions for In (x) and Iν (x) 276
17.7.2 Series expansions for K 0 (x) and K n (x) 276
Modified Bessel Functions of Fractional Order 277
17.8.1 Modified Bessel functions I±(n+1/2) (x) 277
17.8.2 Modified Bessel functions K ±(n+1/2) (x) 278
Asymptotic Representations of Modified Bessel Functions 278

17.9.1 Asymptotic representations for large arguments 278
Relationships between Bessel Functions 278
17.10.1 Relationships involving Jν (x) and Yν (x) 278
17.10.2 Relationships involving Iν (x) and K ν (x) 280
Integral Representations of Jn (x), In (x), and K n (x) 281
17.11.1 Integral representations of Jn (x) 281

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17.12 Indefinite Integrals of Bessel Functions 281
17.12.1 Integrals of Jn (x), In (x), and K n (x) 281
17.13 Definite Integrals Involving Bessel Functions 282
17.13.1 Definite integrals involving Jn (x) and elementary
functions 282
17.14 Spherical Bessel Functions 283
17.14.1 The differential equation 283
17.14.2 The spherical Bessel functions jn (x) and yn (x) 284
17.14.3 Recurrence relations 284
17.14.4 Series representations 284

18

Orthogonal Polynomials
18.1 Introduction 285
18.1.1 Definition of a system of orthogonal polynomials 285

18.2 Legendre Polynomials Pn (x) 286
18.2.1 Differential equation satisfied by Pn (x) 286
18.2.2 Rodrigues’ formula for Pn (x) 286
18.2.3 Orthogonality relation for Pn (x) 286
18.2.4 Explicit expressions for Pn (x) 286
18.2.5 Recurrence relations satisfied by Pn (x) 288
18.2.6 Generating function for Pn (x) 289
18.2.7 Legendre functions of the second kind Q n (x) 289
18.3 Chebyshev Polynomials Tn (x) and Un (x) 290
18.3.1 Differential equation satisfied by Tn (x) and Un (x) 290
18.3.2 Rodrigues’ formulas for Tn (x) and Un (x) 290
18.3.3 Orthogonality relations for Tn (x) and Un (x) 290
18.3.4 Explicit expressions for Tn (x) and Un (x) 291
18.3.5 Recurrence relations satisfied by Tn (x) and Un (x) 294
18.3.6 Generating functions for Tn (x) and Un (x) 294
18.4 Laguerre Polynomials L n (x) 294
18.4.1 Differential equation satisfied by L n (x) 294
18.4.2 Rodrigues’ formula for L n (x) 295
18.4.3 Orthogonality relation for L n (x) 295
18.4.4 Explicit expressions for L n (x) 295
18.4.5 Recurrence relations satisfied by L n (x) 295
18.4.6 Generating function for L n (x) 296
18.5 Hermite Polynomials Hn (x) 296
18.5.1 Differential equation satisfied by Hn (x) 296
18.5.2 Rodrigues’ formula for Hn (x) 296
18.5.3 Orthogonality relation for Hn (x) 296
18.5.4 Explicit expressions for Hn (x) 296
18.5.5 Recurrence relations satisfied by Hn (x) 297
18.5.6 Generating function for Hn (x) 297

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19

Laplace Transformation
19.1 Introduction 299
19.1.1 Definition of the Laplace transform 299
19.1.2 Basic properties of the Laplace transform 300
19.1.3 The Dirac delta function δ(x) 301
19.1.4 Laplace transform pairs 301

20

Fourier Transforms
20.1 Introduction 307
20.1.1 Fourier exponential transform 307
20.1.2 Basic properties of the Fourier transforms 308
20.1.3 Fourier transform pairs 309
20.1.4 Fourier cosine and sine transforms 309
20.1.5 Basic properties of the Fourier cosine and sine transforms 312
20.1.6 Fourier cosine and sine transform pairs 312

21

Numerical Integration

21.1 Classical Methods 315
21.1.1 Open- and closed-type formulas 315
21.1.2 Composite midpoint rule (open type) 316
21.1.3 Composite trapezoidal rule (closed type) 316
21.1.4 Composite Simpson’s rule (closed type) 316
21.1.5 Newton–Cotes formulas 317
21.1.6 Gaussian quadrature (open-type) 318
21.1.7 Romberg integration (closed-type) 318

22

Solutions of Standard Ordinary Differential Equations
22.1 Introduction 321
22.1.1 Basic definitions 321
22.1.2 Linear dependence and independence 322
22.2 Separation of Variables 323
22.3 Linear First-Order Equations 323
22.4 Bernoulli’s Equation 324
22.5 Exact Equations 325
22.6 Homogeneous Equations 325
22.7 Linear Differential Equations 326
22.8 Constant Coefficient Linear Differential Equations—Homogeneous
Case 327

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22.9 Linear Homogeneous Second-Order Equation 330
22.10 Constant Coefficient Linear Differential Equations—Inhomogeneous
Case 331
22.11 Linear Inhomogeneous Second-Order Equation 333
22.12 Determination of Particular Integrals by the Method of Undetermined
Coefficients 334
22.13 The Cauchy–Euler Equation 336
22.14 Legendre’s Equation 337
22.15 Bessel’s Equations 337
22.16 Power Series and Frobenius Methods 339
22.17 The Hypergeometric Equation 344
22.18 Numerical Methods 345

23

Vector Analysis
23.1

23.2
23.3
23.4
23.5
23.6
23.7
23.8
23.9
23.10
23.11
23.12

24

Scalars and Vectors 353
23.1.1 Basic definitions 353
23.1.2 Vector addition and subtraction 355
23.1.3 Scaling vectors 356
23.1.4 Vectors in component form 357
Scalar Products 358
Vector Products 359
Triple Products 360
Products of Four Vectors 361
Derivatives of Vector Functions of a Scalar t 361
Derivatives of Vector Functions of Several Scalar Variables
Integrals of Vector Functions of a Scalar Variable t 363
Line Integrals 364
Vector Integral Theorems 366
A Vector Rate of Change Theorem 368
Useful Vector Identities and Results 368

Systems of Orthogonal Coordinates
24.1 Curvilinear Coordinates 369
24.1.1 Basic definitions 369
24.2 Vector Operators in Orthogonal Coordinates 371
24.3 Systems of Orthogonal Coordinates 371

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25

Partial Differential Equations and Special Functions
25.1 Fundamental Ideas 381
25.1.1 Classification of equations 381
25.2 Method of Separation of Variables 385
25.2.1 Application to a hyperbolic problem 385
25.3 The Sturm–Liouville Problem and Special Functions 387
25.4 A First-Order System and the Wave Equation 390
25.5 Conservation Equations (Laws) 391
25.6 The Method of Characteristics 392
25.7 Discontinuous Solutions (Shocks) 396
25.8 Similarity Solutions 398
25.9 Burgers’s Equation, the KdV Equation, and the KdVB Equation

26

The z -Transform
26.1 The z -Transform and Transform Pairs

27

400

403

Numerical Approximation
27.1 Introduction 409
27.1.1 Linear interpolation 410
27.1.2 Lagrange polynomial interpolation 410
27.1.3 Spline interpolation 410
27.2 Economization of Series 411
27.3 Pad´e Approximation 413
27.4 Finite Difference Approximations to Ordinary and Partial Derivatives
Short Classified Reference List
Index

419

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Preface

This book contains a collection of general mathematical results, formulas, and integrals 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 and a reference to the section in which they occur or are defined is to be
xix
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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.

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Preface
to the
Second Edition

The publication of a second edition of my Handbook has provided me with the
opportunity to correct a few typographical errors that were present in the first edition,
and to enhance it by the addition of small amounts of new material throughout the
book together with two new chapters. The first of these chapters provides basic information on the z-transform together with tables of transform pairs, and the second
is on numerical approximation. Included in this chapter is information about interpolation, economization of series, Pad´e approximation and the basic finite difference
approximations to ordinary and partial derivatives.
Alan Jeffrey

xxi
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Index of Special Functions
and Notations

Notation
|a|
am u
∼
α
arg z
A(x)
A
A−1
AT
|A|
Bn
Bn∗
Bn (x)
B(x, y)
n
k

C(x)

Section or formula
containing its definition

Name
Absolute value of the real number
Amplitude of an elliptic function
Asymptotic relationship
Modular angle of an elliptic integral
Argument of complex number z
A(x) = 2P(x) − 1; probability function
Matrix
Multiplicative inverse of a square matrix A
Transpose of matrix A
Determinant associated with a square matrix A
Bernoulli number
Alternative Bernoulli number
Bernoulli polynomial
Beta function
Binomial coefficient
n
n!
,
=
k!(n − k)!
k
Fresnel cosine integral

1.1.2.1
12.2.1.1.2
1.14.2.1
12.1.2
2.1.1.1

13.1.1.1.7
1.5.1.1.9
1.5.1.1.7
1.4.1.1
1.3.1.1
1.3.1.1.6
1.3.2.1.1
11.1.7.1
1.2.1.1

n
=1
0
14.1.1.1.1
xxiii

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xxiv
Notation
Ci j
n

Cm or n Cm

cn u
cn−1 u
curl F = ∇ × F
δ(x)

Dn (x)
dn u
dn−1 u
div F = ∇ · F
eiθ
e
Ei(x)
E(ϕ, k)
E(k), E′ (k)
e Az
erf x
erfc x
En
E n∗
E n (x)
f (x)
f ′ (x)
f (n) (x)
f (n) (x0 )
F(ϕ, k)
n

grad φ = ∇φ
Ŵ(x)
γ
H (x)
Hn (x)
i
Im{z}
I

in erfc x
I±ν (x)
f (x) d x
b
a f (x) d x

Index of Special Functions

Name
Cofactor of element ai j in a square matrix A
n
Combination symbol n Cm =
m
Jacobian elliptic function
Inverse Jacobian elliptic function
Curl of vector F
Dirac delta function
Dirichlet kernel
Jacobian elliptic function
Inverse Jacobian elliptic function
Divergence of vector F
Euler formula; eiθ = cos θ + i sin θ
Euler’s constant
Exponential integral
Incomplete elliptic integral of the second kind
Complete elliptic integrals of the second kind
Matrix exponential
Error function
Complementary error function
Euler number

Alternative Euler number
Euler polynomial
A function of x
First derivative d f/d x
nth derivative d n f/d x n
nth derivative d n f/d x n at x0
Incomplete elliptic integral of the first kind
Norm of n (x)
Gradient of the scalar function φ
Gamma function
Euler–Mascheroni constant
Heaviside step function
Hermite polynomial
Imaginary unit
Imaginary part of z = x + i y; Im{z} = y
Unit (identity) matrix
nth repeated integral of erfc x
Modified Bessel function of the first kind of order ν
Indefinite integral (antiderivative) of f (x)
Definite integral of f (x) from x = a to x = b

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Section or formula
containing its definition
1.4.2
1.6.2.1
12.2.1.1.4
12.4.1.1.4
23.8.1.1.6

19.1.3
1.13.1.10.3
12.2.1.1.5
12.4.1.1.5
23.8.1.1.4
2.1.1.2.1
0.3
5.1.2.2
12.1.1.1.5
13.1.1.1.8,
13.1.1.1.10
1.5.4.1
13.2.1.1
13.2.1.1.4
1.3.1.1
1.3.1.1.6
1.3.2.3.1
1.15.1.1.6
1.12.1.1
1.12.1.1
12.1.1.1.4
18.1.1.1
23.8.1.6
11.1.1.1
1.11.1.1.7
19.1.2.5
18.5.3
1.1.1.1
1.1.1.2
1.5.1.1.3

13.2.7.1.1
17.6.1.1
1.15.2
1.15.2.5