Table of integrals, series, and products / I.S.Gradshteyn and I.M.Ryzhik ; Alan Jeffrey, editor ; Daniel Zwillinger, associate editor ; translated from the Russian by Scripta Technica, Inc.

Format:

Book

Author/Creator:

Gradshtyn, I. S., author.

Ryzhik, I. M. (Iosif Moiseevich), author.

Contributor:

Zwillinger, Daniel, 1957- editor.

Moll, Victor, 1956- editor.

Scripta Technica, inc., translator.

Standardized Title:

Tablitsy integralov, summ, riadov i proizvedenii. English

Language:

English

Subjects (All):

Mathematics--Tables.

Mathematics.

Physical Description:

1 online resource (1184 pages)

Edition:

8th edition.

Place of Publication:

Amsterdam : Academic Press 2014.

Language Note:

Translated from the Russian.

System Details:

text file

Summary:

The eighth edition of the classic Gradshteyn and Ryzhik is an updated completely revised edition of what is acknowledged universally by mathematical and applied science users as the key reference work concerning the integrals and special functions. The book is valued by users of previous editions of the work both for its comprehensive coverage of integrals and special functions, and also for its accuracy and valuable updates. Since the first edition, published in 1965, the mathematical content of this book has significantly increased due to the addition of new material, though the size of the book has remained almost unchanged. The new 8th edition contains entirely new results and amendments to the auxiliary conditions that accompany integrals and wherever possible most entries contain valuable references to their source. Over 10, 000 mathematical entries Most up to date listing of integrals, series and products (special functions) Provides accuracy and efficiency in industry work 25% of new material not including changes to the restrictions on results that revise the range of validity of results, which lend to approximately 35% of new updates

Contents:

Front Cover

Table of Integrals, Series, and Products

Copyright

Contents

Preface to the Eighth Edition

Acknowledgments

The Order of Presentation of the Formulas

Use of the Tables

Bernoulli and Euler Polynomials and Numbers

Elliptic Functions and Elliptic Integrals

The Jacobi Zeta Function and Theta Functions

Exponential and Related Integrals

Hermite and Chebyshev Orthogonal Polynomials

Bessel Functions

Parabolic Cylinder Functions and Whittaker Functions

Mathieu Functions

Index of Special Functions

Notation

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 (Leibniz's rule)

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 representation

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 argu ...

1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers...

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 (cf. 1.431)

1.6 The Inverse Trigonometric and Hyperbolic Functions

1.61 The domain of definition

1.62-1.63 Functional relations

1.64 Series representations

2 Indefinite Integrals 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+c x2k

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 n(a + bx)k

The square root

Cube root

2.24 Forms containing a+bx and the binomial α+βx

2.25 Forms containing a+bx+cx2

Integration techniques

2.26 Forms containing a+bx+cx2 and integral powers of x

2.2712 Forms containing a+c x2 and integral powers of x

2.28 Forms containing a+bx+c x2 and first-and second-degree polynomials

2.29 Integrals that can be reduced to 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 sinh x, cosh x, tanh x, and coth x

Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument

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 Integrals containing a ± b sin x or a ± b cos x

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

2.75 Logarithms and exponential 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 Power and 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 ration

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 (α+βx)

3.24-3.27 Powers of x, of binomials of the form α+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

3.64-3.65 Powers and rational functions of trigonometric functions

3.66 Forms containing powers of linear 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 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.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references and index.

OCLC:

1055555849