San Francisco Public Library

Table of integrals, series, and products, Daniel Zwillinger, Victor Moll, I.S. Gradshteyn, I.M. Ryzhik

eng

Includes bibliographical references and index

illustrations

index present

non fiction

Table of integrals, series, and products

dictionariesbibliography

1055555849

Daniel Zwillinger, Victor Moll, I.S. Gradshteyn, I.M. Ryzhik

The Table of Integrals, Series, and Products is the essential reference for integrals in the English language. Mathematicians, scientists, and engineers, rely on it when identifying and subsequently solving extremely complex problems. Since publication of the first English-language edition in 1965, it has been thoroughly revised and enlarged on a regular basis, with substantial additions and, where necessary, existing entries corrected or revised. The seventh edition includes a fully searchable CD-Rom.- Fully searchable CD that puts information at your fingertips included with text- Most up to date listing of integrals, series andproducts - Provides accuracy and efficiency in work

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 relations3.46-3.48 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 functions505-00/(S, 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 logarithm505-00/(S, 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 exponentials505-00/(S, 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