Integrals, Series, Eighth Edition. Table of. and Products USA. Daniel Zwillinger, Editor Rensselaer Polytechnic Institute,

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1 Table of Integrals, Series, and Products Eighth Edition I S Gradshteyn and I M Ryzhik Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Victor Moll (Scientific Editor) Tulane University, Department of Mathematics Translated from Russian by Scripta Technica, Inc AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO ELSEVIER Academic Press is an imprint of Elsevier

2 Contents Preface to the Eighth Edition Acknowledgments The Order of Presentation of the Formulas Use of the Tables Index of Special Functions Notation Note on the Bibliographic References xvii xix xxv xxix xxxvii xli xlv 0 Introduction 1 01 Finite sums Progressions Sums of powers of natural numbers Sums of reciprocals of natural numbers Sums of products of reciprocals of natural numbers Sums of the binomial coefficients 4 02 Numerical series and infinite products The convergence of numerical series Convergence tests Examples of numerical series Infinite products Examples of infinite products Functional series Definitions and theorems Power series Fourier series Asymptotic series Certain formulas from differential calculus Differentiation of a definite integral with respect to a parameter The nth derivative of a product (Leibniz's rule) The nth derivative of a composite function Integration by substitution 24 1 Elementary Functions Power of Binomials Power series Series of rational fractions The Exponential Function 26 v

3 121 Series representation Functional relations Series of exponentials Trigonometric and Hyperbolic Functions Introduction The basic functional relations The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions Certain sums of trigonometric and hyperbolic functions Sums of powers of trigonometric functions of multiple angles Sums of products of trigonometric functions of multiple angles Sums of tangents of multiple angles Sums leading to hyperbolic tangents and cotangents The representation of cosines and sines of multiples of the angle as finite products The expansion of trigonometric and hyperbolic functions in power series 142 Expansion in series of simple fractions Representation in the form of an infinite product Trigonometric (Fourier) series Series of products of exponential and trigonometric functions Series of hyperbolic functions Lobachevskiy's "Angle of parallelism" Yl(x) The hyperbolic amplitude (the Gudermannian) gdx The Logarithm Series representation Series of logarithms (cf 1431) The Inverse Trigonometric and Hyperbolic Functions The domain of definition Functional relations Series representations 60 2 Indefinite Integrals of Elementary Functions Introduction General remarks The basic integrals General formulas Rational Functions General integration rules Forms containing the binomial a + bxk Forms containing the binomial 1 ± xn Forms containing pairs of binomials: a + bx and a + /3x Forms containing the trinomial a + bxk + cx2k Forms containing the quadratic trinomial a + bx + cx2 and powers of x Forms containing the quadratic trinomial a + bx + cx2 and the binomial a + /3x Algebraic functions Introduction Forms containing the binomial a + bxk and y/x 83

4 ' vii Forms containing ^/(a + bx)k Forms containing \/q + bx and the binomial a + fix Forms containing y/a + bx + cx Forms containing y/a + bx + cx2 and integral powers of x Forms containing \/a + cx2 and integral powers of x Forms containing y/a + bx + cx2 and first-and second-degree polynomials 229 integrals that can be reduced to elliptic or pseudo-elliptic integrals The Exponential Function Forms containing eax The exponential combined with rational functions of x Hyperbolic Functions Powers of sinha;, cosha:, tanhx, and cothx Rational functions of hyperbolic functions Algebraic functions of hyperbolic functions Combinations of hyperbolic functions and powers Combinations of hyperbolic functions, exponentials, and powers Trigonometric Functions Introduction Powers of trigonometric functions Sines and cosines of multiple angles and of linear and more complicated func tions of the argument Rational functions of the sine and cosine Integrals containing y/a±bsinx or Va ± bcosx Integrals reducible to elliptic and pseudo-elliptic integrals Products of trigonometric functions and powers Combinations of trigonometric functions and exponentials Combinations of trigonometric and hyperbolic functions Logarithms and Inverse-Hyperbolic Functions The logarithm Combinations of logarithms and algebraic functions Inverse hyperbolic functions Logarithms and exponential functions Inverse Trigonometric Functions Arcsines and arccosines The arcsecant, the arccosecant, the arctangent and the arccotangent Combinations of arcsine or arccosine and algebraic functions Combinations of the arcsecant and arccosecant with powers of x Combinations of the arctangent and arccotangent with algebraic functions Definite Integrals of Elementary Functions Introduction Theorems of a general nature Change of variable in a definite integral General formulas Improper integrals The principal values of improper integrals Power and Algebraic Functions Rational functions 255

k2 k2 viii 318 312 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials 256 313-317 Expressions that can be reduced to square

5 k2 k2 viii Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with rational functions Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions Combinations of powers of x and powers of binomials of the form (a + fix) Powers of x, of binomials of the form a + /3xp and of polynomials in x Exponential Functions Exponential functions Exponentials of more complicated arguments Combinations of exponentials and rational functions Combinations of exponentials and algebraic functions Combinations of exponentials and arbitrary powers Combinations of rational functions of powers and exponentials Combinations of powers and algebraic functions of exponentials Combinations of exponentials of more complicated arguments and powers Hyperbolic Functions Hyperbolic functions Combinations of hyperbolic functions and algebraic functions Combinations of hyperbolic functions and exponentials Combinations of hyperbolic functions, exponentials, and powers Trigonometric 361 Rational functions of sines and cosines and trigonometric functions of multiple Functions 392 angles Powers of trigonometric functions Powers of trigonometric functions and trigonometric functions of linear functions Powers and rational functions of trigonometric functions Forms containing powers of linear functions of trigonometric functions Square roots of expressions containing trigonometric functions Various forms of powers of trigonometric functions Trigonometric functions of more complicated arguments Combinations of trigonometric and rational functions Combinations of trigonometric and algebraic functions Combinations of trigonometric functions and powers Rational functions of x and of trigonometric functions Powers of trigonometric functions combined with other powers Integrals containing \/l sin2 x, y/1 cos2 x, and similar expressions Trigonometric functions of more complicated arguments combined with powers Trigonometric functions and exponentials Trigonometric functions of more complicated arguments combined with exponentials Trigonometric and exponential functions of trigonometric functions Combinations involving trigonometric functions, exponentials, and powers Combinations of trigonometric and hyperbolic functions 513

6 ix Combinations involving trigonometric and hyperbolic functions and powers Combinations of trigonometric and hyperbolic functions and exponentials Combinations of trigonometric and hyperbolic functions, exponentials, and powers Logarithmic Functions Logarithmic functions Logarithms of more complicated arguments Combinations of logarithms and rational functions Combinations of logarithms and algebraic functions Combinations of logarithms and powers Combinations involving powers of the logarithm and other powers Combinations of rational functions of In a; and powers Combinations of logarithmic functions of more complicated arguments and powers Combinations of logarithms and exponentials Combinations of logarithms, exponentials, and powers Combinations of logarithms and hyperbolic functions Logarithms and trigonometric functions Combinations of logarithms, trigonometric functions, and powers Combinations of logarithms, trigonometric functions, and exponentials Inverse Trigonometric Functions Inverse trigonometric functions Combinations of arcsines, arccosines, and powers Combinations of arctangents, arccotangents, and powers Combinations of inverse trigonometric functions and exponentials A combination of the arctangent and a hyperbolic function Combinations of inverse and direct trigonometric functions A combination involving an inverse and a direct trigonometric function and a power Combinations of inverse trigonometric functions and logarithms Multiple Integrals Change of variables in multiple integrals Change of the order of integration and change of variables Double and triple integrals with constant limits Multiple integrals Indefinite Integrals of Special Functions Elliptic Integrals and Functions Complete elliptic integrals Elliptic integrals Jacobian elliptic functions Weierstrass elliptic functions The Exponential Integral Function The exponential integral function Combinations of the exponential integral function and powers Combinations of the exponential integral and the exponential The Sine Integral and the Cosine Integral The Probability Integral and Fresnel Integrals Bessel Functions 633

7 X 56 Orthogonal Polynomials Hypergeometric Functions Definite Integrals of Special Functions Elliptic Integrals and Functions Forms containing F(x, k) Forms containing E(x,k) Integration of elliptic integrals with respect to the modulus Complete elliptic integrals The theta function Generalized elliptic integrals The Exponential Integral Function and Functions Generated by It The logarithm integral The exponential integral function The sine integral and cosine integral functions The hyperbolic sine integral and hyperbolic cosine integral functions The probability integral Fresnel integrals The Gamma Function and Functions Generated by It The gamma function Combinations of the gamma function, the exponential, and powers Combinations of the gamma function and trigonometric functions The logarithm of the gamma function* The incomplete gamma function The function tp(x) Bessel Functions Bessel functions Bessel functions combined with x and x Combinations of Bessel functions and rational functions Combinations of Bessel functions and algebraic functions Combinations of Bessel functions and powers Combinations of powers and Bessel functions of more complicated arguments Combinations of Bessel functions and exponentials Combinations of Bessel functions, exponentials, and powers Combinations of Bessel functions of more complicated arguments, exponentials, and powers Combinations of Bessel and exponential functions of more complicated arguments and powers Combinations of Bessel, hyperbolic, and exponential functions Combinations of Bessel and trigonometric functions Combinations of Bessel and trigonometric functions and powers Combinations of Bessel, trigonometric, and exponential functions and powers Combinations of Bessel, trigonometric, and hyperbolic functions Combinations of Bessel functions and the logarithm, or arctangent Combinations of Bessel and other special functions Integration of Bessel functions with respect to the order Functions Generated by Bessel Functions Struve functions 760

8 xi 682 Combinations of Struve functions, exponentials, and powers Combinations of Struve and trigonometric functions Combinations of Struve and Bessel functions Lommel functions Thomson functions Mathieu Functions Mathieu functions Combinations of Mathieu, hyperbolic, and trigonometric functions Combinations of Mathieu and Bessel functions Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems Associated Legendre Functions Associated Legendre functions Combinations of associated Legendre functions and powers Combinations of associated Legendre functions, exponentials, and powers 715 Combinations of associated Legendre and hyperbolic functions Combinations of associated Legendre functions, powers, and trigonometric 783 functions A combination of an associated Legendre function and the probability integral Combinations of associated Legendre and Bessel functions Combinations of associated Legendre functions and functions generated by Bessel functions Integration of associated Legendre functions with respect to the order Combinations of Legendre polynomials, rational functions, and algebraic functions Combinations of Legendre polynomials and powers Combinations of Legendre polynomials and other elementary functions Combinations of Legendre polynomials and Bessel functions Orthogonal Polynomials Combinations of Gegenbauer polynomials C (x) and powers Combinations of Gegenbauer polynomials C%(x) and elementary functions * Complete System of Orthogonal Step Functions Combinations of the polynomials C^{x) and Bessel functions Integration of Gegenbauer functions with respect to the index Combinations of Chebyshev polynomials and powers Combinations of Chebyshev polynomials and elementary functions Combinations of Chebyshev polynomials and Bessel functions Hermite polynomials Jacobi polynomials Laguerre polynomials Hypergeometric Functions Combinations of hypergeometric functions and powers Combinations of hypergeometric functions and exponentials Hypergeometric and trigonometric functions Combinations of hypergeometric and Bessel functions Confluent Hypergeometric Functions Combinations of confluent hypergeometric functions and powers 828

9 xii Combinations of confluent hypergeometric functions and exponentials Combinations of confluent hypergeometric and trigonometric functions Combinations of confluent hypergeometric functions and Bessel functions 766 Combinations of confluent hypergeometric functions, Bessel functions, 838 and powers Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers Combinations of confluent hypergeometric functions and other special 769 Integration of confluent hypergeometric functions with respect to the index functions Parabolic Cylinder Functions Parabolic cylinder functions Combinations of parabolic cylinder functions, powers, and exponentials Combinations of parabolic cylinder and hyperbolic functions Combinations of parabolic cylinder and trigonometric functions Combinations of parabolic cylinder and Bessel functions Combinations of parabolic cylinder functions and confluent hypergeometric functions Integration of a parabolic cylinder function with respect to the index Meijer's and MacRobert's Functions (G and E) Combinations of the functions G and E and the elementary functions Combinations of the functions G and E and Bessel functions Combinations of the functions G and E and other special functions Special Functions Elliptic Integrals and Functions Elliptic integrals Functional relations between elliptic integrals Elliptic functions Jacobian elliptic functions Properties of Jacobian elliptic functions and functional relationships between them The Weierstrass function p(u) The functions (u) and <r(u) Theta functions The Exponential Integral Function and Functions Generated by It The exponential integral function Ei(x) The hyperbolic sine integral sbi x and the hyperbolic cosine integral chi x The sine integral and the cosine integral: six and cix The logarithm integral li(x) The probability integral $(x), the Fresnel integrals S(x), C{x), the error func tion erf(x), and the complementary error function erfc(x) Lobachevskiy's function L(x) Euler's Integrals of the First and Second Kinds and Functions Generated by Them The gamma function (Euler's integral of the second kind): T(z) Representation of the gamma function as series and products Functional relations involving the gamma function The logarithm of the gamma function The incomplete gamma function The psi function ip(x) 911

10 xiii 837 The function 0(x) The beta function (Euler's integral of the first kind): B(x,y) The incomplete beta function Bx(p, q) Bessel Functions and Functions Associated with Them Definitions Integral representations of the functions J (z) and Nv(z) Integral representations of the functions H^\z) and H^\z) Integral representations of the functions Iv{z) and Kv{z) Series representation Asymptotic expansions of Bessel functions Bessel functions of order equal to an integer plus one-half Functional relations Differential equations leading to Bessel functions Series of Bessel functions Expansion in products of Bessel functions The zeros of Bessel functions Struve functions Thomson functions and their generalizations Lommel functions Anger and Weber functions Jv(z) and E (z) Neumann's and Schlafli's polynomials: On(z) and Sn(z) Mathieu Functions Mathieu's equation Periodic Mathieu functions Recursion relations for the coefficients A%^\ B^ff*' Bg 2) Mathieu functions with a purely imaginary argument Non-periodic solutions of Mathieu's equation Mathieu functions for negative q Representation of Mathieu functions as series of Bessel functions The general theory Associated Legendre Functions Introduction Integral representations Asymptotic series for large values of \v\ Functional relations Special cases and particular values Derivatives with respect to the order Series representation The zeros of associated Legendre functions Series of associated Legendre functions Associated Legendre functions with integer indices Legendre functions Conical functions Toroidal functions Orthogonal Polynomials Introduction Legendre polynomials 992

11 xiv 8919 Series of products of Legendre and Chebyshev polynomials Series of Legendre polynomials Gegenbauer polynomials C*(i) The Chebyshev polynomials Tn(x) and Un(x) The Hermite polynomials Hn(x) Jacobi's polynomials The Laguerre polynomials Hypergeometric Functions Definition Integral representations Representation of elementary functions in terms of a hypergeometric functions Transformation formulas and the analytic continuation of functions defined by hypergeometric series A generalized hypergeometric series The hypergeometric differential equation Riemann's differential equation Representing the solutions to certain second-order differential equations using a Riemann scheme Hypergeometric functions of two variables A hypergeometric function of several variables Confluent Hypergeometric Functions Introduction The functions ${a,r,z) and V(a,r,z) The Whittaker functions MA,M(z) and W\,p(z) Parabolic cylinder functions Dp{z) Confluent hypergeometric series of two variables Meijer's G-Function Definition Functional relations A differential equation for the G-function Series of G-functions Connections with other special functions MacRobert's ^-Function Representation by means of multiple integrals Functional relations Riemann's Zeta Functions (,{z,q), and C,{z), and the Functions $(z,s,v) and Definition and integral representations Representation as a series or as an infinite product Functional relations Singular points and zeros The Lerch function $(z,s,v) The function (s) Bernoulli Numbers and Polynomials, Euler Numbers, the Functions v{x), u(x, a), (i(x,/3), fj,(x,0,a), X(x,y) and Euler Polynomials Bernoulli numbers Bernoulli polynomials Euler numbers 1054

12 _xv 964 The functions u(x), u{x,a), fj(x,0), (i(x,f3,a), X(x,y) Euler polynomials Constants Bernoulli numbers Euler numbers Euler's and Catalan's constants Stirling numbers Vector Field Theory Vectors, Vector Operators, and Integral Theorems Products of vectors Properties of scalar product Properties of vector product Differentiation of vectors Operators grad, div, and curl Properties of the operator V Solenoidal fields Orthogonal curvilinear coordinates Vector integral theorems Integral rate of change theorems Integral Inequalities Mean Value Theorems First mean value theorem Second mean value theorem First mean value theorem for infinite integrals Second mean value theorem for infinite integrals Differentiation of Definite Integral Containing a Parameter Differentiation when limits are finite Differentiation when a limit is infinite Integral Inequalities Cauchy-Schwarz-Buniakowsky inequality for integrals Holder's inequality for integrals Minkowski's inequality for integrals Chebyshev's inequality for integrals Young's inequality for integrals Steffensen's inequality for integrals Gram's inequality for integrals Ostrowski's inequality for integrals Convexity and Jensen's Inequality Jensen's inequality Carleman's inequality for integrals Fourier Series and Related Inequalities Riemann-Lebesgue lemma Dirichlet lemma Parseval's theorem for trigonometric Fourier series Integral representation of the nth partial sum Generalized Fourier series 1075

13 xvi Bessel's inequality for generalized Fourier series Parseval's theorem for generalized Fourier series Fourier, Laplace, and Mellin Transforms Integral Transforms Laplace transform Basic properties of the Laplace transform Table of Laplace transform pairs Fourier transform Basic properties of the Fourier transform Table of Fourier transform pairs Table of Fourier transform pairs for spherically symmetric functions Fourier sine and cosine transforms Basic properties of the Fourier sine and cosine transforms Table of Fourier sine transforms Table of Fourier cosine transforms Relationships between transforms Mellin transform Basic properties of the Mellin transform Table of Mellin transforms 1101 Bibliographic References 1105 Supplementary References 1109 Index of Functions and Constants 1115 Index of Concepts 1125

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