"积分与序列手册：第七版完整目录与特殊函数索引"

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The "Table of Integrals, Series, and Products" is a comprehensive mathematical handbook that provides a thorough collection of formulas, functions, and notation for use in various mathematical calculations and analyses. This seventh edition includes a preface, acknowledgments, and an index of special functions, in addition to the content of the tables themselves. The handbook begins with an introduction that covers finite sums, progressions, and sums of powers of natural numbers. It also includes a section on the use of the tables and a note on bibliographic references, providing guidance on how to effectively utilize the information provided. The tables themselves cover a wide range of mathematical functions and formulas, including integrals, series, and products. The handbook is organized in a logical order to facilitate ease of use, with the order of presentation of the formulas carefully considered to provide a comprehensive resource for mathematicians, engineers, and other professionals who rely on mathematical analysis in their work. The seventh edition of the "Table of Integrals, Series, and Products" is edited by I.S. Gradshteyn and I.M. Ryzhik, with Alan Jeffrey and Daniel Zwillinger serving as editors. Translated from Russian by Scripta Technica, Inc., the handbook is a valuable resource for anyone working in the fields of mathematics, engineering, and numerical analysis. Overall, the "Table of Integrals, Series, and Products" is a comprehensive and indispensable resource for anyone working with mathematical functions and formulas. The seventh edition, in particular, provides a valuable update to this essential handbook, ensuring its continued relevance and importance in the field of mathematics and related disciplines.

CONTENTS xv

9.73 Euler’s and Catalan’s constants . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

9.74 Stirling numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046

10 Vector Field Theory 1049

10.1–10.8 Vectors, Vector Operators, and Integral Theorems . . . . . . . . . . . . . . . . 1049

10.11 Products of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

10.12 Properties of scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

10.13 Properties of vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1049

10.14 Diﬀerentiation of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

10.21 Operators grad, div, and curl . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050

10.31 Properties of the operator ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051

10.41 Solenoidal ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1052

10.51–10.61 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1052

10.71–10.72 Vector integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055

10.81 Integral rate of change theorems . . . . . . . . . . . . . . . . . . . . . . . . . 1057

11 Algebraic Inequalities 1059

11.1–11.3 General Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059

11.11 Algebraic inequalities involving real numbers . . . . . . . . . . . . . . . . . . . 1059

11.21 Algebraic inequalities involving complex numbers . . . . . . . . . . . . . . . . . 1060

11.31 Inequalities for sets of complex numbers . . . . . . . . . . . . . . . . . . . . . 1061

12 Integral Inequalities 1063

12.11 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

12.111 First mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

12.112 Second mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063

12.113 First mean value theorem for inﬁnite integrals . . . . . . . . . . . . . . . . . . 1063

12.114 Second mean value theorem for inﬁnite integrals . . . . . . . . . . . . . . . . . 1064

12.21 Diﬀerentiation of Deﬁnite Integral Containing a Parameter . . . . . . . . . . . 1064

12.211 Diﬀerentiation when limits are ﬁnite . . . . . . . . . . . . . . . . . . . . . . . . 1064

12.212 Diﬀerentiation when a limit is inﬁnite . . . . . . . . . . . . . . . . . . . . . . . 1064

12.31 Integral Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064

12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals . . . . . . . . . . . . . . 1064

12.312 H¨older’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1064

12.313 Minkowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065

12.314 Chebyshev’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065

12.315 Young’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 1065

12.316 Steﬀensen’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1065

12.317 Gram’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065

12.318 Ostrowski’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . 1066

12.41 Convexity and Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 1066

12.411 Jensen’s inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066

12.412 Carleman’s inequality for integrals . . . . . . . . . . . . . . . . . . . . . . . . . 1066

12.51 Fourier Series and Related Inequalities . . . . . . . . . . . . . . . . . . . . . . 1066

12.511 Riemann-Lebesgue lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067

12.512 Dirichlet lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067

12.513 Parseval’s theorem for trigonometric Fourier series . . . . . . . . . . . . . . . . 1067

12.514 Integral representation of the n

partial sum . . . . . . . . . . . . . . . . . . . 1067

xvi CONTENTS

12.515 Generalized Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067

12.516 Bessel’s inequality for generalized Fourier series . . . . . . . . . . . . . . . . . 1068

12.517 Parseval’s theorem for generalized Fourier series . . . . . . . . . . . . . . . . . 1068

13 Matrices and Related Results 1069

13.11–13.12 Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.111 Diagonal matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.112 Identity matrix and null matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.113 Reducible and irreducible matrices . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.114 Equivalent matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.115 Transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069

13.116 Adjoint matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.117 Inverse matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.118 Trace of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.119 Symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.120 Skew-symmetric matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.121 Triangular matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.122 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.123 Hermitian transpose of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.124 Hermitian matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1070

13.125 Unitary matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.126 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.127 Nilpotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.128 Idempotent matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.129 Positive deﬁnite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.130 Non-negative deﬁnite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.131 Diagonally dominant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.21 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071

13.211 Sylvester’s law of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

13.212 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

13.213 Signature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072

13.214 Positive deﬁnite and semideﬁnite quadratic form . . . . . . . . . . . . . . . . . 1072

13.215 Basic theorems on quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . 1072

13.31 Diﬀerentiation of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073

13.41 The Matrix Exponential

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074

3.411 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1074

14 Determinants 1075

14.11 Expansion of Second- and Third-Order Determinants . . . . . . . . . . . . . . 1075

14.12 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075

14.13 Minors and Cofactors of a Determinant . . . . . . . . . . . . . . . . . . . . . . 1075

14.14 Principal Minors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076

14.15

Laplace Expansion of a Determinant . . . . . . . . . . . . . . . . . . . . . . . 1076

14.16 Jacobi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076

14.17 Hadamard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077

14.18 Hadamard’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077

14.21 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077

14.31 Some Special Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

CONTENTS xvii

14.311 Vandermonde’s determinant (alternant) . . . . . . . . . . . . . . . . . . . . . . 1078

14.312 Circulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

14.313 Jacobian determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078

14.314 Hessian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079

14.315 Wronskian determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079

14.316 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1079

14.317 Gram-Kowalewski theorem on linear dependence . . . . . . . . . . . . . . . . . 1080

15 Norms 1081

15.1–15.9 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

15.11 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

15.21 Principal Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

15.211 The norm ||x||

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

15.212 The norm ||x||

(Euclidean or L

norm) . . . . . . . . . . . . . . . . . . . . . 1081

15.213 The norm ||x||

∞

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081

15.31 Matrix Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.311 General properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.312 Induced norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.313 Natural norm of unit matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.41 Principal Natural Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.411 Maximum absolute column sum norm . . . . . . . . . . . . . . . . . . . . . . . 1082

15.412 Spectral norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1082

15.413 Maximum absolute row sum norm . . . . . . . . . . . . . . . . . . . . . . . . . 1083

15.51 Spectral Radius of a Square Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1083

15.511 Inequalities concerning matrix norms and the spectral radius . . . . . . . . . . 1083

15.512 Deductions from Gerschgorin’s theorem (see 15.814) . . . . . . . . . . . . . . 1083

15.61 Inequalities Involving Eigenvalues of Matrices . . . . . . . . . . . . . . . . . . . 1084

15.611 Cayley-Hamilton theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084

15.612 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084

15.71 Inequalities for the Characteristic Polynomial . . . . . . . . . . . . . . . . . . . 1084

15.711 Named and unnamed inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 1085

15.712 Parodi’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086

15.713 Corollary of Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086

15.714 Ballieu’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086

15.715 Routh-Hurwitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086

15.81–15.82 Named Theorems on Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . 1087

15.811 Schur’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087

15.812 Sturmian separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087

15.813 Poincare’s separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087

15.814 Gerschgorin’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.815 Brauer’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.816 Perron’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.817 Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.818 Perron–Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.819 Wielandt’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088

15.820 Ostrowski’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

15.821 First theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . 1089

15.822 Second theorem due to Lyapunov

. . . . . . . . . . . . . . . . . . . . . . . . . 1089

xviii CONTENTS

15.823 Hermitian matrices and diophantine relations involving circular functions of

rational angles due to Calogero and Perelomov . . . . . . . . . . . . . . . . . . 1089

15.91 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091

15.911 Rayleigh quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091

15.912 Basic theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091

16 Ordinary Diﬀerential Equations 1093

16.1–16.9 Results Relating to the Solution of Ordinary Diﬀerential Equations . . . . . . . 1093

16.11 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

16.111 Solution of a ﬁrst-order equation . . . . . . . . . . . . . . . . . . . . . . . . . 1093

16.112 Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093

16.113 Approximate solution to an equation . . . . . . . . . . . . . . . . . . . . . . . 1093

16.114 Lipschitz continuity of a function . . . . . . . . . . . . . . . . . . . . . . . . . 1094

16.21 Fundamental Inequalities and Related Results . . . . . . . . . . . . . . . . . . 1094

16.211 Gronwall’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

16.212 Comparison of approximate solutions of a diﬀerential equation . . . . . . . . . 1094

16.31 First-Order Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094

16.311 Solution of a system of equations . . . . . . . . . . . . . . . . . . . . . . . . . 1094

16.312 Cauchy problem for a system . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095

16.313 Approximate solution to a system . . . . . . . . . . . . . . . . . . . . . . . . . 1095

16.314 Lipschitz continuity of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1095

16.315 Comparison of approximate solutions of a system . . . . . . . . . . . . . . . . 1096

16.316 First-order linear diﬀerential equation . . . . . . . . . . . . . . . . . . . . . . . 1096

16.317 Linear systems of diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . 1096

16.41 Some Special Types of Elementary Diﬀerential Equations . . . . . . . . . . . . 1097

16.411 Variables separable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

16.412 Exact diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1097

16.413 Conditions for an exact equation . . . . . . . . . . . . . . . . . . . . . . . . . 1097

16.414 Homogeneous diﬀerential equations . . . . . . . . . . . . . . . . . . . . . . . . 1097

16.51 Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

16.511 Adjoint and self-adjoint equations . . . . . . . . . . . . . . . . . . . . . . . . . 1098

16.512 Abel’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

16.513 Lagrange identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099

16.514 The Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099

16.515 Solutions of the Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . 1099

16.516

Solution of a second-order linear diﬀerential equation . . . . . . . . . . . . . . 1100

16.61–16.62 Oscillation and Non-Oscillation Theorems for Second-Order Equations . . . . . 1100

16.611 First basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1100

16.622 Second basic comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1101

16.623 Interlacing of zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101

16.624 Sturm separation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101

16.625 Sturm comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101

16.626 Szeg¨o’s comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1101

16.627 Picone’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

16.628 Sturm-Picone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

16.629 Oscillation on the half line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102

16.71 Two Related Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1103

16.711 Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

CONTENTS xix

16.712 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

16.81–16.82 Non-Oscillatory Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

16.811 Kneser’s non-oscillation theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1103

16.822 Comparison theorem for non-oscillation . . . . . . . . . . . . . . . . . . . . . . 1104

16.823 Necessary and suﬃcient conditions for non-oscillation . . . . . . . . . . . . . . 1104

16.91 Some Growth Estimates for Solutions of Second-Order Equations . . . . . . . . 1104

16.911 Strictly increasing and decreasing solutions . . . . . . . . . . . . . . . . . . . . 1104

16.912 General result on dominant and subdominant solutions . . . . . . . . . . . . . 1104

16.913 Estimate of dominant solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105

16.914 A theorem due to Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105

16.92 Boundedness Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

16.921 All solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

16.922 If all solutions of the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

16.923 If a(x) →∞monotonically as x →∞, then all solutions of . . . . . . . . . . . 1106

16.924 Consider the equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

16.93 Growth of maxima of |y| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106

17 Fourier, Laplace, and Mellin Transforms 1107

17.1–17.4 Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

17.11 Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107

17.12 Basic properties of the Laplace transform . . . . . . . . . . . . . . . . . . . . . 1107

17.13 Table of Laplace transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . 1108

17.21 Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117

17.22 Basic properties of the Fourier transform . . . . . . . . . . . . . . . . . . . . . 1118

17.23 Table of Fourier transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 1118

17.24 Table of Fourier transform pairs for spherically symmetric functions . . . . . . . 1120

17.31 Fourier sine and cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1121

17.32 Basic properties of the Fourier sine and cosine transforms . . . . . . . . . . . . 1121

17.33 Table of Fourier sine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . 1122

17.34 Table of Fourier cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . 1126

17.35 Relationships between transforms

. . . . . . . . . . . . . . . . . . . . . . . . . 1129

17.41 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1129

17.42 Basic properties of the Mellin transform . . . . . . . . . . . . . . . . . . . . . 1130

17.43 Table of Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1131

18 The z-Transform 1135

18.1–18.3 Deﬁnition, Bilateral, and Unilateral z-Transforms . . . . . . . . . . . . . . . . . 1135

18.1 Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135

18.2 Bilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136

18.3 Unilateral z-transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1138

References 1141

Supplemental references 1145

Index of Functions and Constants 1151

General Index of Concepts 1161

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