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 Seventh Edition 22
Acknowledgments 24
The Order of Presentation of the Formulas 28
Use of the Tables 32
Index of 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 102
1.6 The Inverse Trigonometric and Hyperbolic Functions 105
Chapter 2 Indefinite Integrals of Elementary Functions 112
2.0 Introduction 112
2.1 Rational Functions 115
2.2 Algebraic Functions 131
2.3 The Exponential Function 155
2.4 Hyperbolic Functions 159
2.5-2.6 Trigonometric Functions 200
2.7 Logarithms and Inverse-Hyperbolic Functions 286
2.8 Inverse Trigonometric Functions 290
Chapter 3-4 Definite Integrals of Elementary Functions 296
3.0 Introduction 296
3.1-3.2 Power and Algebraic Functions 302
3.3-3.4 Exponential Functions 383
3.5 Hyperbolic Functions 420
3.6-4.1 Trigonometric Functions 439
4.2-4.4 Logarithmic Functions 576
4.5 Inverse Trigonometric Functions 648
4.6 Multiple Integrals 656
Chapter 5 Indefinite Integrals of Special Functions 668
5.1 Elliptic Integrals and Functions 668
5.2 The Exponential Integral Function 676
5.3 The Sine Integral and the Cosine Integral 677
5.4 The Probability Integral and Fresnel Integrals 678
5.5 Bessel Functions 678
Chapter 6-7 Definite Integrals of Special Functions 680
6.1 Elliptic Integrals and Functions 680
6.2-6.3 The Exponential Integral Function and Functions Generated by It 685
6.4 The Gamma Function and Functions Generated by It 699
6.5-6.7 Bessel Functions 708
6.8 Functions Generated by Bessel Functions 802
6.9 Mathieu Functions 812
7.1-7.2 Associated Legendre Functions 818
7.3-7.4 Orthogonal Polynomials 844
7.5 Hypergeometric Functions 861
7.6 Confluent Hypergeometric Functions 869
7.7 Parabolic Cylinder Functions 890
7.8 Meijer's and MacRobert's Functions (G and E) 899
Chapter 8-9 Special Functions 908
8.1 Elliptic Integrals and Functions 908
8.2 The Exponential Integral Function and Functions Generated by It 932
8.3 Euler's Integrals of the First and Second Kinds 941
8.4-8.5 Bessel Functions and Functions Associated with Them 959
8.6 Mathieu Functions 999
8.7-8.8 Associated Legendre Functions 1007
8.9 Orthogonal Polynomials 1031
9.1 Hypergeometric Functions 1054
9.2 Confluent Hypergeometric Functions 1071
9.3 Meijer's G-Function 1081
9.4 MacRobert's E-Function 1084
9.5 Riemann's Zeta Functions zeta(z,q) and zeta(z), and the Functions Phi(z,s,v) and xi(s) 1085
9.6 Bernoulli Numbers and Polynomials, Euler Numbers 1089
9.7 Constants 1094
Chapter 10 Vector Field Theory 1098
10.1-10.8 Vectors, Vector Operators, and Integral Theorems 1098
Chapter 11 Algebraic Inequalities 1108
11.1-11.3 General Algebraic Inequalities 1108
Chapter 12 Integral Inequalities 1112
12.11 Mean Value Theorems 1112
12.21 Differentiation of Definite Integral Containing a Parameter 1113
12.31 Integral Inequalities 1113
12.41 Convexity and Jensen's Inequality 1115
12.51 Fourier Series and Related Inequalities 1115
Chapter 13 Matrices and Related Results 1118
13.11-13.12 Special Matrices 1118
13.21 Quadratic Forms 1120
13.31 Differentiation of Matrices 1122
13.41 The Matrix Exponential 1123
Chapter 14 Determinants 1124
14.11 Expansion of Second- and Third-Order Determinants 1124
14.12 Basic Properties 1124
14.13 Minors and Cofactors of a Determinant 1124
14.14 Principal Minors 1125
14.15* Laplace Expansion of a Determinant 1125
14.16 Jacobi's Theorem 1125
14.17 Hadamard's Theorem 1126
14.18 Hadamard's Inequality 1126
14.21 Cramer's Rule 1126
14.31 Some Special Determinants 1127
Chapter 15 Norms 1130
15.1-15.9 Vector Norms 1130
15.11 General Properties 1130
15.21 Principal Vector Norms 1130
15.31 Matrix Norms 1131
15.41 Principal Natural Norms 1131
15.51 Spectral Radius of a Square Matrix 1132
15.61 Inequalities Involving Eigenvalues of Matrices 1133
15.71 Inequalities for the Characteristic Polynomial 1133
15.81-15.82 Named Theorems on Eigenvalues 1136
15.91 Variational Principles 1140
Chapter 16 Ordinary Differential Equations 1142
16.1-16.9 Results Relating to the Solution of Ordinary Differential Equations 1142
16.11 First-Order Equations 1142
16.21 Fundamental Inequalities and Related Results 1143
16.31 First-Order Systems 1143
16.41 Some Special Types of Elementary Differential Equations 1146
16.51 Second-Order Equations 1147
16.61-16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations 1149
16.71 Two Related Comparison Theorems 1152
16.81-16.82 Non-Oscillatory Solutions 1152
16.91 Some Growth Estimates for Solutions of Second-Order Equations 1153
16.92 Boundedness Theorems 1155
Chapter 17 Fourier, Laplace, and Mellin Transforms 1156
17.1-17.4 Integral Transforms 1156
Chapter 18 The z-Transform 1184
18.1-18.3 Definition, Bilateral, and Unilateral z-Transforms 1184
References 1190
Supplemental references 1194
Index of Functions and Constants 1200
General Index of Concepts 1210