Table of integrals, series, and products

Table of Integrals, Series, and Products
Copyright Page
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
Introduction
	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
	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
Elementary Functions
	Power of Binomials
		Power series
		Series of rational fractions
	The Exponential Function
		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  ...
		The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of th ...
		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
		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\'\' Π(x)
		The hyperbolic amplitude (the Gudermannian) gd x
	The Logarithm
		Series representation
		Series of logarithms (cf. 1.431)
	The Inverse Trigonometric and Hyperbolic Functions
		The domain of definition
		Functional relations
		Series representations
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 α+βx
		Forms containing the trinomial a+bxk+c  x2k
		Forms containing the quadratic trinomial a+bx+cx2 and powers of x
		Forms containing the quadratic trinomial a+bx+cx2 and the binomial α+βx
	Algebraic functions
		Introduction
		Forms containing the binomial a+bxk and √x
		Forms containing n√(a+bx)k
		Forms containing √a+bx and the binomial α+βx
		Forms containing √a+bx+cx2
		Forms containing √a+bx+cx2 and integral powers of x
		Forms containing √a+cx2 and integral powers of x
		Forms containing √a+bx+cx2 and first-and second-degree polynomials
		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 sinh x, cosh x, tanh x, and coth x
		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 functions of the argument
		Rational functions of the sine and cosine
		Integrals containing √a ± b sin x or √a ± b cos x
		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
		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 ration func ...
		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 (α+βx)
		Powers of x, of binomials of the form α+βxp 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 Functions
		Rational functions of sines and cosines and trigonometric functions of multiple 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 √1 − k2 sin2 x, √1 − k2 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
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 ln x 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
	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 ψ(x)
	Bessel Functions
		Bessel functions
		Bessel functions combined with x and  x2
		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
		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
	Combinations of associated Legendre and hyperbolic functions
	Combinations of associated Legendre functions, powers, and trigonometric 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 Cnν(x) and powers
	Combinations of Gegenbauer polynomials Cnν(x) and elementary functions
Complete System of Orthogonal Step Functions
	Combinations of the polynomials Cνn(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
	Combinations of confluent hypergeometric functions and exponentials
	Combinations of confluent hypergeometric and trigonometric functions
	Combinations of confluent hypergeometric functions and Bessel functions
	Combinations of confluent hypergeometric functions, Bessel functions, and powers
	Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers
	Combinations of confluent hypergeometric functions and other special functions
	Integration of confluent hypergeometric functions with respect to the index
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 ℘(u)
		The functions ζ(u) and σ(u)
		Theta functions
	The Exponential Integral Function and Functions Generated by It
		The exponential integral function Ei(x)
		The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x
		The sine integral and the cosine integral: si x and ci x
		The logarithm integral li(x)
		The probability integral Φ(x), the Fresnel integrals S(x), C(x), the error function erf(x), and the complementary error fu ...
		Lobachevskiy\'s function L(x)
	Euler\'s Integrals of the First and Second Kinds
		The gamma function (Euler\'s integral of the second kind): Γ(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 ψ(x)
		The function β(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 Jν(z) and Nν(z)
		Integral representations of the functions H(1)ν(z) and H(2)ν(z)
		Integral representations of the functions Iν(z) and Kν(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 Jν(z) and Eν(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(2n)2r, A(2n+1)2r+1, B(2n+1)2r+1, B(2n+2)2r+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 |ν|
		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
		Series of products of Legendre and Chebyshev polynomials
		Series of Legendre polynomials
		Gegenbauer polynomials Cnλ(t)
		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 Φ(α,γ;z) and Ψ(α,γ;z)
	The Whittaker functions Mλ,μ ( z )  and Wλ,μ ( 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 E-Function
	Representation by means of multiple integrals
	Functional relations
Riemann\'s Zeta Functions ζ(z,q), and ζ(z), and the Functions Φ(z,s,v) and ξ(s)
	Definition and integral representations
	Representation as a series or as an infinite product
	Functional relations
	Singular points and zeros
	The Lerch function Φ(z, s, v)
	The function ξ ( s )
Bernoulli Numbers and Polynomials, Euler Numbers, the Functions ν(x), ν(x,α), μ(x,β), μ(x,β,α), λ(x,y) and Euler Polynomial
	Bernoulli numbers
	Bernoulli polynomials
	Euler numbers
	The functions ν(x), ν(x,α), μ(x,β), μ(x,β,α), λ(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
		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
		Hölder\'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
		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
Bibliographic References
Supplementary References
Index of Functions and Constants
	Symbols
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	W
	X
	Y
	Z
Index of Concepts
	A
	B
	C
	D
	E
	F
	G
	H
	I
	J
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	Y
	Z