# Table of Integrals, Series, and Products

## 1st Edition

**Authors:**I. S. Gradshteyn I. M. Ryzhik

**eBook ISBN:**9781483265643

**Imprint:**Academic Press

**Published Date:**28th May 1980

**Page Count:**1206

## Description

Table of Integrals, Series, and Products provides information pertinent to the fundamental aspects of integrals, series, and products. This book provides a comprehensive table of integrals.

Organized into 17 chapters, this book begins with an overview of elementary functions and discusses the power of binomials, the exponential function, the logarithm, the hyperbolic function, and the inverse trigonometric function. This text then presents some basic results on vector operators and coordinate systems that are likely to be useful during the formulation of many problems. Other chapters consider inequalities that range from basic algebraic and functional inequalities to integral inequalities and fundamental oscillation and comparison theorems for ordinary differential equations. This book discusses as well the important part played by integral transforms. The final chapter deals with Fourier and Laplace transforms that provides so much information about other integrals.

This book is a valuable resource for mathematicians, engineers, scientists, and research workers.

## Table of Contents

Preface to the Corrected and Enlarged Edition

Preface to the First Edition

Preface to the Third Edition

Preface to the Fourth Edition

Editor's Foreword

Acknowledgment

The Order of Presentation of the Formulas

Use of the Tables

Index of Special Functions and Notations

Notations

Note on the Bibliographic References

0. Introduction

0.1 FInite Sums

0.11 Progressions

0.12 Sums of Powers of Natural Numbers

0.13 Sums of Reciprocals of Natural Numbers

0.14 Sums of Products of Reciprocals of Natural Numbers

0.15 Sums of the BInomial Coefficients

0.2 Numerical Series and Infinite Products

0.21 The Convergence of Numerical Series

0.22 Convergence Tests

0.23-0.24 Examples of Numerical Series

0.25 InfInite Products

0.26 Examples of Infinite Products

0.3 Functional Series

0.30 DefInitions and Theorems

0.31 Power Series

0.32 Fourier Series

0.33 Asymptotic Series

0.4 CertaIn Formulas From Differential Calculus

0.41 Differentiation of a Definite Integral with Respect to a Parameter

0.42 The nth Derivative of a Product

0.43 The nth Derivative of a Composite Function

0.44 Integration by Substitution

1. Elementary Functions

1.1 Power of Binomials

1.11 Power Series

1.12 Series of Rational Fractions

1.2 The Exponential Function

1.21 Series Representations

1.22 Functional Relations

1.23 Series of Exponentials

1.3-1.4 Trigonometric and Hyperbolic Functions

1.30 Introduction

1.31 The Basic Functional Relations

1.32 The Representation of Powers of Trigonometric and Hyperbolic Functions in Terms of Functions of Multiples of the Argument (Angle)

1.33 The Representation of Trigonometric and Hyperbolic Functions of Multiples of the Argument (angle) in Terms of Powers of these Functions

1.34 Certain Sums of Trigonometric and Hyperbolic Functions

1.35 Sums of Powers of Trigonometric Functions of Multiple Angles

1.36 Sums of Products of Trigonometric Functions of Multiple Angles

1.37 Sums of Tangents of Multiple Angles

1.38 Sums Leading To Hyperbolic Tangents and Cotangents

1.39 The Representation of Cosines and Sines of Multiples of the Angle as Finite Products

1.41 The Expansion of Trigonometric and Hyperbolic Functions in Power Series

1.42 Expansion in Series of Simple Fractions

1.43 Representation in the Form of an Infinite Product

1.44-1.45 Trigonometric (Fourier) Series

1.46 Series of Products of Exponential and Trigonometric Functions

1.47 Series of Hyperbolic Functions

1.48 Lobachevskiy's "Angle of Parallelism" Π (x)

1.49 The Hyperbolic Amplitude (The Gudermannian) gd x

1.5 The Logarithm

1.51 Series Representation

1.52 Series of Logarithms

1.6 The Inverse Trigonometric and Hyperbolic Functions

1.61 The Domain of Definition

1.62-1.63 Functional Relations

2. Indefinite Integals of Elementary Functions

2.0 Introduction

2.00 General Remarks

2.01 The Basic Integrals

2.02 General Formulas

2.1 Rational Functions

2.10 General Integration Rules

2.11-2.13 Forms Containing the Binomial a + bxk

2.14 Forms Containing the Binomial 1 ± xn

2.15 Forms Containing Pairs of Binomials: a + bx and α + βx

2.16 Forms Containing the Trinomial a + bxk + cx2k

2.17 Forms Containing the Quadratic Trinomial a + bx + Cx2 and Powers of x

2.18 Forms Containing the Quadratic Trinomial a + bx + Cx2 and the Binomial α + βx

2.2 Algebraic Functions

2.20 Introduction

2.21 Forms Containing the Binomial a + bxk and √x

2.22-2.23 Forms Containing x√(a + bx)k

2.24 Forms Containing √a + bx and he Binomial α + βx

2.25 Forms Containing √a + bx + cx2

2.26 Forms Containing √a + bx + cx2 and Integral Powers of x

2.27 Forms Containing √a + cx2 and Integral Powers of x

2.28 Forms Containing √a + bx + cx2 and First- and Second-Degree Polynomials

2.29 Integrals that can be Reduced o Elliptic Or Pseudo-Elliptic Integrals

2.3 The Exponential Function

2.31 Forms Containing eax

2.32 The Exponential Combined with Rational Functions of x

2.4 Hyperbolic Functions

2.41-2.43 Powers of Sh x, ch x, th x and cth x

2.44-2.45 Rational Functions of Hyperbolic Functions

2.46 Algebraic Functions of Hyperbolic Functions

2.47 Combinations of Hyperbolic Functions and Powers

2.48 Combinations of Hyperbolic Functions, Exponentials, and Powers

2.5-2.6 Trigonometric Functions

2.50 Introduction

2.51-2.52 Powers of Trigonometric Functions

2.53-2.54 Sines and Cosines of Multiple Angles and of Linear and More Complicated Functions of the Argument

2.55-2.56 Rational Functions of the Sine and Cosine

2.57 Forms Containing √a ± b sin x, Or √a ± b cos x and Forms Reducible to Such Expressions

2.58-2.62 Integrals Reducible to Elliptic and Pseudo-Elliptic Integrals

2.63-2.65 Products of Trigonometric Functions and Powers

2.66 Combinations of Trigonometric Functions and Exponentials

2.67 Combinations of Trigonometric and Hyperbolic Functions

2.7 Logarithms and Inverse-Hyperbolic Functions

2.71 The Logarithm

2.72-2.73 Combinations of Logarithms and Algebraic Functions

2.74 Inverse Hyperbolic Functions

2.8 Inverse Trigonometric Functions

2.81 Arcsines and Arccosines

2.82 The Arcsecant, the Arccosecant, the Arctangent, and the Arccotangent

2.83 Combinations of Arcsine or Arccosine and Algebraic Functions

2.84 Combinations of the Arcsecant and Arccosecant with Powers of x

2.85 Combinations of the Arctangent and Arccotangent with Algebraic Functions

3.-4. Definite Integrals of Elementary Functions

3.0 Introduction

3.01 Theorems of A General Nature

3.02 Change of Variable in A Definite Integral

3.03 General Formulas

3.04 Improper Integrals

3.05 The Principal Values of Improper Integrals

3.1-3.2 Powerand Algebraic Functions

3.11 Rational Functions

3.12 Products of Rational Functions and Expressions that can be Reduced to Square Roots of First- and Second-Degree Polynomials

3.13-3.17 Expressions that can be Reduced to Square Roots of Third- and Fourth-Degree Polynomials and their Products with Rational Functions

3.18 Expressions that can be Reduced to Fourth Roots of Second-Degree Polynomials and their Products with Rational Functions

3.19-3.23 Combinations of Powers of x and Powers of Binomials of the Form (a + βx)

3.24-3.27 Powers of x, of Binomials of he Form a + βxp and of Polynomials in x

3.3-3.4 Exponential Functions

3.31 Exponential Functions

3.32-3.34 Exponentials of More Complicated Arguments

3.35 Combinations of Exponentials and Rational Functions

3.36-3.37 Combinations of Exponentials and Algebraic Functions

3.38-3.39 Combinations of Exponentials and Arbitrary Powers

3.41-3.44 Combinations of Rational Functions of Powers and Exponentials

3.45 Combinations of Powers and Algebraic Functions of Exponentials

3.46-3.48 Combinations of Exponentials of More Complicated Arguments and Powers

3.5 Hyperbolic Functions

3.51 Hyperbolic Functions

3.52-3.53 Combinations of Hyperbolic Functions and Algebraic Functions

3.54 Combinations of Hyperbolic Functions and Exponentials

3.55-3.56 Combinations of Hyperbolic Functions, Exponentials, and Powers

3.6-4.1 Trigonometric Functions

3.61 Rational Functions of Sines and Cosines and Trigonometric Functions of Multiple Angles

3.62 Powers of Trigonometric Functions

3.63 Powers of Trigonometric Functions and Trigonometric Functions of Linear Functions

3.64-3.66 Powers and Rational Functions of Trigonometric Functions

3.67 Square Roots of Expressions Containing Trigonometric Functions

3.68 Various Forms of Powers of Trigonometric Functions

3.69-3.71 Trigonometric Functions of More Complicated Arguments

3.72-3.74 Combinations of Trigonometric and Rational Functions

3.75 Combinations of Trigonometric and Algebraic Functions

3.76-3.77 Combinations of Trigonometric Functions and Powers

3.78-3.81 Rational Functions of x and of Trigonometric Functions

3.82-3.83 Powers of Trigonometric Functions Combined With Other Powers

3.84 Integrals Containing the Expressions √1 - k2 sin2 x, √1 - k2 cos2 x, and Similar Expressions

3.85-3.88 Trigonometric Functions of More Complicated Arguments Combined With Powers

3.89-3.91 Trigonometric Functions and Exponentials

3.92 Trigonometric Functions of More Complicated Arguments CombIned with Exponentials

3.93 Trigonometric and Exponential Functions of Trigonometric Functions

3.94-3.97 Combinations Involving Trigonometric Functions, Exponentials, and Powers

3.98-3.99 Combinations of Trigonometric and Hyperbolic Functions

4.11-4.12 Combinations Involving Trigonometric and Hyperbolic Functions and Powers

4.13 Combinations of Trigonometric and Hyperbolic Functions and Exponentials

4.14 Combinations of Trigonometric and Hyperbolic Functions, Exponentials, and Powers

4.2-4.4 Logarithmic Functions

4.21 Logarithmic Functions

4.22 Logarithms of More Complicated Arguments

4.23 Combinations of Logarithms and Rational Functions

4.24 Combinations of Logarithms and Algebraic Functions

4.25 Combinations of Logarithms and Powers

4.26-4.27 Combinations Involving Powers of the Logarithm and Other Powers

4.28 Combinations of Rational Functions of in Jc and Powers

4.29-4.32 Combinations of Logarithmic Functions of More Complicated Arguments and Powers

4.33-4.34 Combinations of Logarithms and Exponentials

4.35-4.36 Combinations of Logarithms, Exponentials, and Powers

4.37 Combinations of Logarithms and Hyperbolic Functions

4.38-4.41 Logarithms and Trigonometric Functions

4.42-4.43 Combinations of Logarithms, Trigonometric Functions, and Powers

4.44 Combinations of Logarithms, Trigonometric Functions, and Exponentials

4.5 Inverse Trigonometric Functions

4.51 Inverse Trigonometric Functions

4.52 Combinations of Arcsines, Arccosines, and Powers

4.53-4.54 Combinations of Arctangents, Arccotangents, and Powers

4.55 Combinations of Inverse Trigonometric Functions and Exponentials

4.56 A Combination of The Arctangent and A Hyperbolic Function

4.57 Combinations of Inverse and Direct Trigonometric Functions

4.58 A Combination InvolvIng an Inverse and A Direct Trigonometric Function and a Power

4.59 Combinations of Inverse Trigonometric Functions and Logarithms

4.6 Multiple Integrals

4.60 Change of Variables in Multiple Integrals

4.61 Change of he Order of Integration and Change of Variables

4.62 Double and Triple Integrals With Constant Limits

4.63-4.64 Multiple Integrals

5. Indefinite Integrals of Special Functions

5.1 Elliptic Integrals and Functions

5.11 Complete Elliptic Integrals

5.12 Elliptic Integrals

5.13 Jacobian Elliptic Functions

5.14 Weierstrass Elliptic Functions

5.2 The Exponential-Integral Function

5.21 The Exponential-Integral Function

5.22 Combinations of the Exponential-Integral Function and Powers

5.23 Combinations of the Exponential-Integral and the Exponential

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

6.11 Forms Containing F (x, k)

6.12 Forms Containing E (x, k)

6.13 Integration of Elliptic Integrals With Respect to the Modulus

6.14-6.15 Complete Elliptic Integrals

6.16 The Theta Function

6.2-6.3 The Exponential-Integral Function and Functions Generated By It

6.21 The Logarithm-Integral

6.22-6.23 The Exponential-Integral Function

6.24-6.26 The Sine- and Cosine-Integral Functions

6.27 The Hyperbolic-Sine- and Cosine-Integral Functions

6.28-6.31 The Probability Integral

6.32 Fresnel Integrals

6.4 The Gamma Function and Functions Generated By It

6.41 The Gamma Function

6.42 Combinations of the Gamma Function, the Exponential, and Powers

6.43 Combinations of the Gamma Function and Trigonometric Functions

6.44 The Logarithm of the Gamma Function

6.45 The Incomplete Gamma Function

6.46-6.47 The Function ψ (x)

6.5-6.7 Bessel Functions

6.51 Bessel Functions

6.52 Bessel Functions Combined with x and x2

6.53-6.54 Combinations of Bessel Functions and Rational Functions

6.55 Combinations of Bessel Functions and Algebraic Functions

6.56-6.58 Combinations of Bessel Functions and Powers

6.59 Combinations of Powers and Bessel Functions of More Complicated Arguments

6.61 Combinations of Bessel Functions and Exponentials

6.62-6.63 Combinations of Bessel Functions, Exponentials, and Powers

6.64 Combinations of Bessel Functions of More Complicated Arguments, Exponentials, and Powers

6.65 Combinations of Bessel and Exponential Functions of More Complicated Arguments and Powers

6.66 Combinations of Bessel, Hyperbolic, and Exponential Functions

6.67-6.68 Combinations of Bessel and Trigonometric Functions

6.69-6.74 Combinations of Bessel and Trigonometric Functions and Powers

6.75 Combinations of Bessel, Trigonometric, and Exponential Functions and Powers

6.76 Combinations of Bessel, Trigonometric, and Hyperbolic Functions

6.77 Combinations of Bessel Functions and the Logarithm, or Arctangent

6.78 Combinations of Bessel and Other Special Functions

6.79 Integration of Bessel Functions with Respect to the Order

6.8 Functions Generated By Bessel Functions

6.81 Struve Functions

6.82 Combinations of Struve Functions, Exponentials, and Powers

6.83 Combinations of Struve and Trigonometric Functions

6.84-6.85 Combinations of Struve and Bessel Functions

6.86 Lommel Functions

6.87 Thomson Functions

6.9 Mathieu Functions

6.91 Mathieu Functions

6.92 Combinations of Mathieu, Hyperbolic, and Trigonometric Functions

6.93 Combinations of Mathieu and Bessel Functions

7.1-7.2 Associated Legendre Functions

7.11 Associated Legendre Functions

7.12-7.13 Combinations of Associated Legendre Functions and Powers

7.14 Combinations of Associated Legendre Functions, Exponentials, and Powers

7.15 Combinations of Associated Legendre and Hyperbolic Functions

7.16 Combinations of Associated Legendre Functions, Powers, and Trigonometric Functions

7.17 A Combination of an Associated Legendre Function and the Probability Integral

7.18 Combinations of Associated Legendre and Bessel Functions

7.19 Combinations of Associated Legendre Functions and Functions Generated By Bessel Functions

7.21 Integration of Associated Legendre Functions with Respect to the Order

7.22 Combinations of Legendre Polynomials, Rational Functions, and Algebraic Functions

7.23 Combinations of Legendre Polynomials and Powers

7.24 Combinations of Legendre Polynomials and Other Elementary Functions

7.25 Combinations of Legendre Polynomials and Bessel Functions

7.3-7.4 Orthogonal Polynomials

7.31 Combinations of Gegenbauer Polynomials Cvn (x) and Powers

7.32 Combinations of the Polynomials Cvn (x) and Some Elementary Functions

7.33 Combinations of the Polynomials Cvn (x) and Bessel Functions. Integration of Gegenbauer Functions with Respect to the Index

7.34 Combinations of Chebyshev Polynomials and Powers

7.35 Combinations of Chebyshev Polynomials and Some Elementary Functions

7.36 Combinations of Chebyshev Polynomials and Bessel Functions

7.37-7.38 Hermite Polynomials

7.39 Jacobi Polynomials

7.41-7.42 Laguerre Polynomials

7.5 Hypergeometric Functions

7.51 Combinations of Hypergeometric Functions and Powers

7.52 Combinations of Hypergeometric Functions and Exponentials

7.53 Hypergeometric and Trigonometric Functions

7.54 Combinations of Hypergeometric and Bessel Functions

7.6 Degenerate Hypergeometric Functions

7.61 Combinations of Degenerate Hypergeometric Functions and Powers

7.62-7.63 Combinations of Degenerate Hypergeometric Functions and Exponentials

7.64 Combinations of Degenerate Hypergeometric and Trigonometric Functions

7.65 Combinations of Degenerate Hypergeometric Functions and Bessel Functions

7.66 Combinations of Degenerate Hypergeometric Functions, Bessel Functions, and Powers

7.67 Combinations of Degenerate Hypergeometric Functions, Bessel Functions, Exponentials, and Powers

7.68 Combinations of Degenerate Hypergeometric Functions and Other Special Functions

7.69 Integration of Degenerate Hypergeometric Functions With Respect to the Index

7.7 Parabolic-Cylinder Functions

7.71 Parabolic-Cylinder Functions

7.72 Combinations of Parabolic-Cylinder Functions, Powers, and Exponentials

7.73 Combinations of Parabolic-Cylinder and Hyperbolic Functions

7.74 Combinations of Parabolic-Cylinder and Trigonometric Functions

7.75 Combinations of Parabolic-Cylinder and Bessel Functions

7.76 Combinations of Parabolic-Cylinder Functions and Degenerate Hypergeometric Functions

7.77 Integration of A Parabolic-Cylinder Function with Respect to the Index

7.8 Meijer's and Macrobert's Functions (G and E)

7.81 Combinations of the Functions G and E and the Elementary Functions

7.82 Combinations of the Functions G and E and Bessel Functions

7.83 Combinations of the Functions G and E and Other Special Functions

8.-9. Special Functions

8.1 Elliptic Integrals and Functions

8.11 Elliptic Integrals

8.12 Functional Relations Between Elliptic Integrals

8.13 Elliptic Functions

8.14 Jacobian Elliptic Functions

8.15 Properties of Jacobian Elliptic Functions and Functional Relationships Between Them

8.16 The Weierstrass Function P (u)

8.17 The Functions ζ (u) and σ (u)

8.18-8.19 Theta Functions

8.2 The Exponential-Integral Function and Functions Generated By It

8.21 The Exponential-Integral Function Ei (x)

8.22 The Hyperbolic-Sine-Integral shi x and the Hyperbolic-Cosine-Integral chi x

8.23 The Sine Integral and the Cosine Integral: si (x) and ci (x)

8.24 The Logarithm-Integral li (x)

8.25 The Probability Integral and Fresnel Integrals φ (x), S (x), and C (x)

8.26 Lobachevskiy's Function L (x)

8.3 Euler's Integrals of the First and Second Kinds and Functions Generated By Them

8.31 The Gamma Function (Euler's Integral of the Second Kind): Γ (z)

8.32 Representation of the Gamma Function as Series and Products

8.33 Functional Relations Involving the Gamma Function

8.34 The Logarithm of the Gamma Function

8.35 The Incomplete Gamma Function

8.36 The Psi Function ψ (x)

8.37 The Function β(x)

8.38 The Beta Function (Euler's Integral of the First Kind): B (x, y)

8.4-8.5 Bessel Functions and Functions Associated with them

8.40 Definitions

8.41 Integral Representations of the Functions Jv (z) and Nv (z)

8.42 Integral Representations of the Functions H(1)v(z) and H (2)v (z)

8.43 Integral Representations of the Functions Iv (z) and Kv (z)

8.44 Series Representation

8.45 Asymptotic Expansions of Bessel Functions

8.46 Bessel Functions of Order Equal to an Integer Plus One-Half

8.47-8.48 Functional Relations

8.49 Differential Equations LeadIng to Bessel Functions

8.51-8.52 Series of Bessel Functions

8.53 Expansion in Products of Bessel Functions

8.54 The Zeros of Bessel Functions

8.55 Struve Functions

8.56 Thomson Functions and their Generalizations: ber, (z), bei, (z), her, (z), hei, (z), ker (z), kei (z)

8.57 Lommel Functions

8.58 Anger and Weber Functions Jv (z) and Ev (z)

8.59 Neumann's and Schläfiys Polynomials: On (z) and Sn (z)

8.6 Mathieu Functions

8.60 Mathieu's Equation

8.61 Periodic Mathieu Functions

8.62 Recursion Relations for the Coefficients A(2n)2r, A (2n+1)2r+1, B(2n+1)2r+1, B(2n+2)2r+2

8.63 Mathieu Functions with a Purely Imaginary Argument

8.64 Nonperiodic Solutions of Mathieu's Equation

8.65 Mathieu Functions For Negative q

8.66 Representation of Mathieu Functions as Series of Bessel Functions

8.67 The General Theory

8.7-8.8 Associated Legendre Functions

8.70 Introduction

8.72 Asymptotic Series for Large Values of |v|

8.73-8.74 Functional Relations

8.75 Special Cases and Particular Values

8.76 Derivatives with Respect to the Order

8.77 Series Representation

8.78 The Zeros of Associated Legendre Functions

8.79 Series of Associated Legendre Functions

8.81 Associated Legendre Functions with Integral Indices

8.82-8.83 Legendre Functions

8.84 Conical Functions

8.85 Toroidal Functions

8.9 Orthogonal Polynomials

8.90 Introduction

8.91 Legendre Polynomials

8.92 Series of Legendre Polynomials

8.93 Gegenbauer Polynomials Cλn (t)

8.94 The Chebyshev Polynomials Tn (x) and Un (x)

8.95 The Hermite Polynomials Hn (x)

8.96 Jacobi's Polynomials

8.97 The Laguerre Polynomials

9.1 Hypergeometric Functions

9.10 Definition

9.11 Integral Representations

9.12 The Representation of Elementary Functions in Terms of a Hypergeometric Function

9.13 Transformation Formulas and the Analytic Continuation of Functions Defined By Hypergeometric Series

9.14 A Generalized Hypergeometric Series

9.15 The Hypergeometric Differential Equation

9.16 Riemann's Differential Equation

9.17 Representation of Certain Second-Order Differential Equations By Means of a Riemann Scheme

9.18 Hypergeometric Functions of two Variables

9.19 A Hypergeometric Function of Several Variables

9.2 A Degenerate Hypergeometric Function

9.20 Introduction

9.21 The Functions φ (α, y; z) and ψ (α, y; z)

9.22-9.23 The Whittaker Functions Mλμ (z) and Wλμ (z)

9.24-9.25 Parabolic Cylinder Functions Dp (z)

9.26 Degenerate Hypergeometric Series of Two Variables

9.3 Meijer's G-Function

9.30 Definition

9.31 Functional Relations

9.32 A Differential Equation For the G-Function

9.33 Series of G-Functions

9.34 Connections with Other Special Functions

9.4 Macrobert's E-Function

9.41 Representation By Means of Multiple Integrals

9.42 Functional Relations

9.5 Riemann's Zeta Functions ζ (z,q), and ζ (z), and the Functions Φ (z, s, v) and ξ (s)

9.51 Definition and Integral Representations

9.52 Representation as a Series Or as an InfInite Product

9.53 Functional Relations

9.54 Singular Points and Zeros

9.55 The Function Φ (z, s, v)

9.56 The Function ξ (s)

9.6 Bernoulli Numbers and Polynomirials, Euler Numbers, The Functions v (x), v (x, α), μ (x, β), μ (x, β, α), λ (x, y)

9.61 Bernoulli Numbers

9.62 Bernoulli Polynomials

9.63 Euler Numbers

9.64 The Functions v (x), v (x, α), μ (x, β), μ (x, β, α), λ (x, y)

9.7 Constants

9.71 Bernoulli Numbers

9.72 Euler Numbers

9.73 Euler's and Catalan's Constants

10. Vector Field Theory

10.1 Vectors, Vector Operators, and Integral Theorems

10.11 Products of Vectors

10.12 Properties of Scalar Product

10.13 Properties of Vector Product

10.14 Differentiation of Vectors

10.21 Operators Grad, Div, and Curl

10.31 Properties of The Operator ∇

10.41 Solenoidal Fields

10.51-10.61 Orthogonal Curvilinear Coordinates

10.71-10.72 Vector Integral Theorems

10.81 Integral Rate of Change Theorems

11. Algebraic Inequalities

11.1-11.3 General Algebraic Inequalities

11.11 Algebraic Inequalities InvolvIng Real Numbers

11.21 Algebraic Inequalities InvolvIng Complex Numbers

11.31 Inequalities for Sets of Complex Numbers

12. Integral Inequalities

12.1-12.5 Properties of Integrals and 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

13.1-13.4 Matrices and Quadratic Forms

13.11-13.12 Special Matrices

13.21 Quadratic Forms

13.31 Differentiation of Matrices

13.41 The Matrix Exponential

14. DetermInants

14.1-14.3 Properties of 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

16.11 First-Order Equations

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 Nonoscillation Theorems for Second Order Equations

16.71 Two Related Comparison Theorems

16.81-16.82 Nonoscillatory Solutions

16.91 Some Growth Estimates For Solutions of Second-Order Equations

16.92 Boundedness Theorems

17. Fourier and Laplace Transforms

17.1-17.3 Integral Transforms

17.11 Laplace Transform

17.12 Basic Properties of the Laplace Transform

17.13 Table of Laplace Transform Pairs

17.21 Fourier Transform

17.22 Basic Properties of the Fourier Transform

17.23 Table of Fourier Transform Pairs

17.31 Fourier Sine and Cosine Transforms

17.32 Basic Properties of the Fourier Sine and Cosine Transforms

17.33 Table of Fourier Sine Transforms

17.34 Table of Fourier Cosine Transforms

17.35 Relationships Between Transforms

Bibliographic References Used in Preparation of Text

Classified Supplementary References

## Details

- No. of pages:
- 1206

- Language:
- English

- Copyright:
- © Academic Press 1980

- Published:
- 28th May 1980

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483265643