Table of Integrals, Series, and Products, 7/Ed > 수학/통계학

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 Introduction 1
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 Elementary Functions 25
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 Indefinite Integrals of Elementary Functions 61
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
3-4 Definite Integrals of Elementary Functions 243
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 Indefinite Integrals of Special Functions 615
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
6-7 Definite Integrals of Special Functions 625
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
8-9 Special Functions 851
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 Vector Field Theory 1039
10.1-10.8 Vectors, Vector Operators, and Integral Theorems 1039
11 Algebraic Inequalities 1049
11.1-11.3 General Algebraic Inequalities 1049
12 Integral Inequalities 1053
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 Matrices and related results 1059
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 Determinants 1065
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 Norms 1071
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 Ordinary differential equations 1083
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 Fourier, Laplace, and Mellin Transforms 1099
17.1-17.4 Integral Transforms 1099
18 The z-transform 1127
18.1-18.3 Definition, Bilateral, and Unilateral z-Transforms 1127
References 1133
Supplemental references 1137
Function and constant index 1143
General index 1153