Table of Integrals, Series, and Products (eBook)

Front Cover 1
Table of Integrals, Series, and Products 4
Copyright Page 5
Contents 6
Preface to the Sixth Edition 22
Acknowledgments 24
The order of presentation of the formulas 28
Use of the tables 32
Special functions 40
Notation 44
Note on the bibliographic references 48
Chapter 0. Introduction 50
0.1 Finite sums 50
0.2 Numerical series and infinite products 55
0.3 Functional series 64
0.4 Certain formulas from differential calculus 70
Chapter 1. Elementary Functions 74
1.1 Power of Binomials 74
1.2 The Exponential Function 75
1.3–1.4 Trigonometric and Hyperbolic Functions 77
1.5 The Logarithm 100
1.6 The Inverse Trigonometric and Hyperbolic Functions 103
Chapter 2. Indefinite Integrals of Elementary Functions 110
2.0 Introduction 110
2.1 Rational functions 113
2.2 Algebraic functions 129
2.3 The Exponential Function 153
2.4 Hyperbolic Functions 154
2.5–2.6 Trigonometric Functions 196
2.7 Logarithms and Inverse-Hyperbolic Functions 282
2.8 Inverse Trigonometric Functions 286
Chapter 3–4. Definite Integrals of Elementary Functions 292
3.0 Introduction 292
3.1–3.2 Power and Algebraic Functions 298
3.3–3.4 Exponential Functions 380
3.5 Hyperbolic Functions 415
3.6–4.1 Trigonometric Functions 434
4.2–4.4 Logarithmic Functions 571
4.5 Inverse Trigonometric Functions 645
4.6 Multiple Integrals 653
Chapter 5. Indefinite Integrals of Special Functions 664
5.1 Elliptic Integrals and Functions 664
5.2 The Exponential Integral Function 671
5.3 The Sine Integral and the Cosine Integral 672
5.4 The Probability Integral and Fresnel Integrals 673
5.5 Bessel Functions 673
Chapter 6–7. Definite Integrals of Special Functions 674
6.1 Elliptic Integrals and Functions 674
6.2–6.3 The Exponential Integral Function and Functions Generated by It 679
6.4 The Gamma Function and Functions Generated by It 693
6.5–6.7 Bessel Functions 702
6.8 Functions Generated by Bessel Functions 794
6.9 Mathieu Functions 804
7.1–7.2 Associated Legendre Functions 811
7.3–7.4 Orthogonal Polynomials 837
7.5 Hypergeometric Functions 855
7.6 Confluent Hypergeometric Functions 863
7.7 Parabolic Cylinder Functions 884
7.8 Meijer’s and MacRobert’s Functions (G and E) 892
Chapter 8–9. Special Functions 900
8.1 Elliptic integrals and functions 900
8.2 The Exponential Integral Function and Functions Generated by It 924
8.3 Euler’s Integrals of the First and Second Kinds 932
8.4–8.5 Bessel Functions and Functions Associated with Them 949
8.6 Mathieu Functions 989
8.7–8.8 Associated Legendre Functions 997
8.9 Orthogonal Polynomials 1021
9.1 Hypergeometric Functions 1044
9.2 Confluent Hypergeometric Functions 1061
9.3 Meijer’s G-Function 1071
9.4 MacRobert’s E-Function 1074
9.5 Riemann’s Zeta Functions (z, q), and (z), and the Functions F (z s
9.6 Bernoulli numbers and polynomials, Euler numbers 1079
9.7 Constants 1084
Chapter 10. Vector Field Theory 1088
10.1–10.8 Vectors, Vector Operators, and Integral Theorems 1088
Chapter 11. Algebraic Inequalities 1098
11.1–11.3 General Algebraic Inequalities 1098
Chapter 12. Integral Inequalities 1102
12.11 Mean value theorems 1102
12.21 Differentiation of definite integral containing a parameter 1103
12.31 Integral inequalities 1103
12.41 Convexity and Jensen’s inequality 1105
12.51 Fourier series and related inequalities 1105
Chapter 13. Matrices and related results 1108
13.11-13.12 Special matrices 1108
13.21 Quadratic forms 1110
13.31 Differentiation of matrices 1112
13.41 The matrix exponential 1113
Chapter 14. Determinants 1114
14.11 Expansion of second- and third-order determinants 1114
14.12 Basic properties 1114
14.13 Minors and cofactors of a determinant 1114
14.14 Principal minors 1115
14.15* Laplace expansion of a determinant 1115
14.16 Jacobi’s theorem 1115
14.17 Hadamard’s theorem 1116
14.18 Hadamard’s inequality 1116
14.21 Cramer’s rule 1116
14.31 Some special determinants 1117
Chapter 15. Norms 1120
15.1–15.9 Vector Norms 1120
15.11 General properties 1120
15.21 Principal vector norms 1120
15.31 Matrix norms 1121
15.41 Principal natural norms 1121
15.51 Spectral radius of a square matrix 1122
15.61 Inequalities involving eigenvalues of matrices 1123
15.71 Inequalities for the characteristic polynomial 1123
15.81–15.82 Named theorems on eigenvalues 1126
15.91 Variational principles 1130
Chapter 16. Ordinary differential equations 1132
16.1–16.9 Results relating to the solution of ordinary differential equations 1132
16.11 First-order equations 1132
16.21 Fundamental inequalities and related results 1133
16.31 First-order systems 1134
16.41 Some special types of elementary differential equations 1136
16.51 Second-order equations 1137
16.61–16.62 Oscillation and non-oscillation theorems for second-order equations 1139
16.71 Two related comparison theorems 1142
16.81–16.82 Non-oscillatory solutions 1142
16.91 Some growth estimates for solutions of second-order equations 1143
16.92 Boundedness theorems 1145
Chapter 17. Fourier, Laplace, and Mellin Transforms 1148
17.1– 17.4 Integral Transforms 1148
Chapter 18. The z-transform 1176
18.1–18.3 Definition, Bilateral, and Unilateral z-Transforms 1176
References 1182
Supplemental references 1186
Function and constant index 1192
General index 1202