内容简介
This completely reset sixth edition of Gradshteyn and Ryzhik is a corrected and expanded version of the previous edition. The book was completely reset in order to add more material and to enhance the visual appearance of the material. To preserve compatibility with the previous edition, the original numbering system for entries has been retained. New entries and sections have been inserted in a manner consistent with the original scheme. Whenever possible, new entries and corrections have been checked by means of symbolic computation.
The diverse ways in which corrections have been contributed have made it impossible to attribute them to reference sources that are accessible to users of this book. However, as in previous editions, our indebtedness to these contributors is shown in the form of an acknowledgment list on page xxiii. This list gives the names of those who have written to ns directly' sending corrections and suggestions for addenda, and added to it are the names of those who have published errata in Mathematics of Computation. Certain individuals must be singled out for special thanks due to their significant contributions: Professors H. van Haeringen and L; P. Kok of The Netherlands and Dr. K. S. Ko1big have contributed new material, corrections, and suggestions for new material.
目录
Note on the bibliographic references
Introduction
0.1 Finite sums
0.2 Numerical series and infinite products
0.3 Functional series
0.4 Certain formulas from differential calculus
1.1 Power of Binomials
1.2 The Exponential Function
1.3-1.4 Trigonometric and Hyperbolic Functions
1.5
1.6 The Inverse Trigonometric and Hyperbolic Functions
2 Indefinite Integrals of Elementary Functions
2.0 Introduction
2.1 Rational functions
2.2 Algebraic functions
2.3 The Exponential Function
2.4 Hyperbolic Functions
2.5-2.6 Trigonometric Functions
2.7 Logarithms and Inverse-Hyperbolic Functions
2.8 Inverse Trigonometric Functions
3-4 Definite Integrals of Elementary Functions
3.0 Introduction
3.1-3.2 Power and Algebraic Functions
3.3-3.4 Exponential Functions
3.5 Hyperbolic Functions
3.6-4.1 Trigonometric Functions
4.2-4.4 Logarithmic Functions
4.5 Inverse Trigonometric Functions
4.6 Multiple Integrals
5 Indefinite Integrals of Special Functions
5.1 Elliptic Integrals and Functions
5.2 The Exponential Integral Function
5.3 The Sine Integral and the Cosine Integral
5.4 The Probability Integral and Fresnel Integrals
5.5 Bessel Functions
6-7 Definite Integrals of Special Functions
6.1 Elliptic Integrals and Functions
6.2-6.3 The Exponential Integral Function and Functions Generated by It
6.4 The Gamma Function and Functions Generated by It
6.5-6.7 Bessel Functions
6.8 Functions Generated by Bessel Functions
6.9 Mathieu Functions
7.1-7.2 Associated Legendre Functions
7.3-7.4 Orthogonal Polynomials
7.5 Hypergeometric Functions
7.6 Confluent Hypergeometric Functions
7.7 Parabolic Cylinder Functions
7.8 Meijer's and MacRobert's Functions (G and E)
8-9 Special Functions
8.1 Elliptic integrals and functions
8.2 The Exponential Integral Function and Functions Generated by It
8.3 Euler's Integrals of the First and Second Kinds
8.4-8.5 Bessel Functions and Functions Associated with Them
8.6 Mathieu Functions
8.7-8.8 Associated Legendre Functions
8.9 Orthogonal Polynomials
9.1 Hypergeometric Functions
9.2 Confluent Hypergeometric Functions
9.3 Meijer's G-Function
9.4 MacRobert's E-Function
9.5 Riemann's Zeta Functions (z, q), and (z), and the Functions (z, s, v) and (s)
9.6 Bernoulli numbers and polynomials, Euler numbers
9.7 Constants
10 Vector Field Theory
10.1-10.8 Vectors, Vector Operators, and Integral Theorems
11 Algebraic Inequalities
11.1-11.:3 General Algebraic Inequalities
12 Integral Inequalities
12.11 Mean value theorems
12.21 Differentiation of definite integral containing a parameter
12.31 Integral inequalities
12.41 Convexity and Jensen's inequality
12.51 Fourier series and related inequalities
13 Matrices and related results
13.11-13.12 Special matrices
13.21 Quadratic forms
13.31 Differentiation of matrices
13.41 The matrix exponential
14 Determinants
14.11 Expansion of second- and third-order determinants
14.12 Basic properties
14.13 Minors and cofactors of a determinant
14.14 Principal minors
14.15 Laplace expansion of a determinant
14.16 Jacobi's theorem
14.17 Hadamard's theorem
14.18 Hadamard's inequality
14.21 Cramer's rule
14.31 Some special determinants
15 Norms
15.1-15.9 Vector Norms
15.11 General properties
15.21 Principal vector norms
15.31 Matrix norms
15.41 Principal natural norms
15.51 Spectral radius of a square matrix
15.61 Inequalities involving eigenvalues of matrices
15.71 Inequalities for the characteristic polynomial
15.81-15.82 Named theorems on eigenvalues
15.91 Variational principles
16 Ordinary differential equations
16.1-16.9 Results relating to the solution of ordinary differential equations
16. tl First-order eq uations
16.21 Fundamental inequalities and related results
16.31 First-order systems
16.41 Some special types of elementary differential equations
16.51 Second-order equations
16.61-16.62 Oscillation and non-oscillation theorems for second-order equations
16.71 Two related comparison theorems
16.81-16.82 Non-oscillatory solutions
16.91 Some growth estimates for solutions of second-order equations
16.92 Boundedness theorems
17 Fourier, Laplace, and Mellin Transforms
References
Supplemental references
Function and constant index
General index
