Table of Integrals, Series and Products

Contents

• Preface to the Sixth Edition xxi
• Acknowledgments xxiii
• The order of presentation of the formulas xxvii
• Use of the tables xxxi
• Special functions xxxix
• Notation xliii
• Note on the bibliographic references xlvii

• 0.1 Finite sums 1
• 0.2 Numerical series and infinite products 6
• 0.3 Functional series 15
• 0.4 Certain formulas from differential calculus 21

• 1.1 Power of Binomials 25
• 1.2 The Exponential Function 26
• 1.3-1.4 Trigonometric and Hyperbolic Functions 27
• 1.5 The Logarithm 51
• 1.6 The Inverse Trigonometric and Hyperbolic Functions 54

• 2.0 Introduction 61
• 2.1 Rational functions 64
• 2.2 Algebraic functions 80
• 2.3 The Exponential Function 104
• 2.4 Hyperbolic Functions 105
• 2.5-2.6 Trigonometric Functions 147
• 2.7 Logarithms and Inverse-Hyperbolic Functions 233
• 2.8 Inverse Trigonometric Functions 237

• 3.0 Introduction 243
• 3.1-3.2 Power and Algebraic Functions 248
• 3.3-3.4 Exponential Functions 331
• 3.5 Hyperbolic Functions 365
• 3.6-4.1 Trigonometric Functions 384
• 4.2-4.4 Logarithmic Functions 522
• 4.5 Inverse Trigonometric Functions 596
• 4.6 Multiple Integrals 604

• 5.1 Elliptic Integrals and Functions 615
• 5.2 The Exponential Integral Function 622
• 5.3 The Sine Integral and the Cosine Integral 623
• 5.4 The Probability Integral and Fresnel Integrals 623
• 5.5 Bessel Functions 624

• 6.1 Elliptic Integrals and Functions 625
• 6.2-6.3 The Exponential Integral Function and Functions Generated by It 630
• 6.4 The Gamma Function and Functions Generated by It 644
• 6.5-6.7 Bessel Functions 652
• 6.8 Functions Generated by Bessel Functions 745
• 6.9 Mathieu Functions 755
• 7.1-7.2 Associated Legendre Functions 762
• 7.3-7.4 Orthogonal Polynomials 788
• 7.5 Hypergeometric Functions 806
• 7.6 Confluent Hypergeometric Functions 814
• 7.7 Parabolic Cylinder Functions 835
• 7.8 Meijer's and MacRobert's Functions (G and E) 843

• 8.1 Elliptic integrals and functions 851
• 8.2 The Exponential Integral Function and Functions Generated by It 875
• 8.3 Euler's Integrals of the First and Second Kinds 883
• 8.4-8.5 Bessel Functions and Functions Associated with Them 900
• 8.6 Mathieu Functions 940
• 8.7-8.8 Associated Legendre Functions 948
• 8.9 Orthogonal Polynomials 972
• 9.1 Hypergeometric Functions 995
• 9.2 Confluent Hypergeometric Functions 1012
• 9.3 Meijer's G-Function 1022
• 9.4 MacRobert's E-Function 1025
• 9.5 Riemann's Zeta Functions [zeta] (z, q), and [zeta] (z), and the Functions [Phi] (z, s, v) and [xi] (s) 1026
• 9.6 Bernoulli numbers and polynomials, Euler numbers 1030
• 9.7 Constants 1035

• 10.1-10.8 Vectors, Vector Operators, and Integral Theorems 1039

• 11.1-11.3 General Algebraic Inequalities 1049

• 12.11 Mean value theorems 1053
• 12.21 Differentiation of definite integral containing a parameter 1054
• 12.31 Integral inequalities 1054
• 12.41 Convexity and Jensen's inequality 1056
• 12.51 Fourier series and related inequalities 1056

• 13.11-13.12 Special matrices 1059
• 13.21 Quadratic forms 1061
• 13.31 Differentiation of matrices 1063
• 13.41 The matrix exponential 1064

• 14.11 Expansion of second- and third-order determinants 1065
• 14.12 Basic properties 1065
• 14.13 Minors and cofactors of a determinant 1065
• 14.14 Principal minors 1066
• 14.15 Laplace expansion of a determinant 1066
• 14.16 Jacobi's theorem 1066
• 14.17 Hadamard's theorem 1066
• 14.18 Hadamard's inequality 1067
• 14.21 Cramer's rule 1067
• 14.31 Some special determinants 1068

• 15.1-15.9 Vector Norms 1071
• 15.11 General properties 1071
• 15.21 Principal vector norms 1071
• 15.31 Matrix norms 1072
• 15.41 Principal natural norms 1072
• 15.51 Spectral radius of a square matrix 1073
• 15.61 Inequalities involving eigenvalues of matrices 1074
• 15.71 Inequalities for the characteristic polynomial 1074
• 15.81-15.82 Named theorems on eigenvalues 1076
• 15.91 Variational principles 1081

• 16.1-16.9 Results relating to the solution of ordinary differential equations 1083
• 16.11 First-order equations 1083
• 16.21 Fundamental inequalities and related results 1084
• 16.31 First-order systems 1085
• 16.41 Some special types of elementary differential equations 1087
• 16.51 Second-order equations 1088
• 16.61-16.62 Oscillation and non-oscillation theorems for second-order equations 1090
• 16.71 Two related comparison theorems 1093
• 16.81-16.82 Non-oscillatory solutions 1093
• 16.91 Some growth estimates for solutions of second-order equations 1094
• 16.92 Boundedness theorems 1096

• 17.1-17.4 Integral Transforms 1099

• 18.1-18.3 Definition, Bilateral, and Unilateral z-Transforms 1127

• References 1133
• Supplemental references 1137
• Function and constant index 1143
• General index 1153