E-raamat: Table of Integrals, Series, and Products

Preface to the Seventh Edition		xxi
Acknowledgments		xxiii
The Order of Presentation of the Formulas		xxvii
Use of the Tables		xxxi
Index of Special Functions		xxxix
Notation		xliii
Note on the Bibliographic References		xlvii
Introduction		1	(24)
Finite Sums		1	(5)
Progressions		1	(1)
Sums of powers of natural numbers		1	(2)
Sums of reciprocals of natural numbers		3	(1)
Sums of products of reciprocals of natural numbers		3	(1)
Sums of the binomial coefficients		3	(3)
Numerical Series and Infinite Products		6	(9)
The convergence of numerical series		6	(1)
Convergence tests		6	(2)
Examples of numerical series		8	(6)
Infinite products		14	(1)
Examples of infinite products		14	(1)
Functional Series		15	(6)
Definitions and theorems		15	(1)
Power series		16	(3)
Fourier series		19	(2)
Asymptotic series		21	(1)
Certain Formulas from Differential Calculus		21	(4)
Differentiation of a definite integral with respect to a parameter		21	(1)
The nth derivative of a product (Leibniz's rule)		22	(1)
The nth derivative of a composite function		22	(1)
Integration by substitution		23	(2)
Elementary Functions		25	(38)
Power of Binomials		25	(1)
Power series		25	(1)
Series of rational fractions		26	(1)
The Exponential Function		26	(2)
Series representation		26	(1)
Functional relations		27	(1)
Series of exponentials		27	(1)
Trigonometric and Hyperbolic Functions		28	(25)
Introduction		28	(1)
The basic functional relations		28	(3)
The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle)		31	(2)
The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions		33	(3)
Certain sums of trigonometric and hyperbolic functions		36	(1)
Sums of powers of trigonometric functions of multiple angles		37	(1)
Sums of products of trigonometric functions of multiple angles		38	(1)
Sums of tangents of multiple angles		39	(1)
Sums leading to hyperbolic tangents and cotangents		39	(2)
The representation of cosines and sines of multiples of the angle as finite products		41	(1)
The expansion of trigonometric and hyperbolic functions in power series		42	(2)
Expansion in series of simple fractions		44	(1)
Representation in the form of an infinite product		45	(1)
Trigonometric (Fourier) series		46	(5)
Series of products of exponential and trigonometric functions		51	(1)
Series of hyperbolic functions		51	(1)
Lobachevskiy's ``Angle of Parallelism'' II (x)		51	(1)
The hyperbolic amplitude (the Gudermannian) gd x		52	(1)
The Logarithm		53	(3)
Series representation		53	(2)
Series of logarithms (cf. 1.431)		55	(1)
The Inverse Trigonometric and Hyperbolic Functions		56	(7)
The domain of definition		56	(1)
Functional relations		56	(4)
Series representations		60	(3)
Indefinite Integrals of Elementary Functions		63	(184)
Introduction		63	(3)
General remarks		63	(1)
The basic integrals		64	(1)
General formulas		65	(1)
Rational Functions		66	(16)
General integration rules		66	(2)
Forms containing the binomial a + bxk		68	(6)
Forms containing the binomial 1 ± xn		74	(4)
Forms containing pairs of binomials: a + bx and α + βx		78	(1)
Forms containing the trinomial a + bxk + cx2k		78	(1)
Forms containing the quadratic trinomial a + bx + cx2 and powers of x		79	(2)
Forms containing the quadratic trinomial a + bx + cx2 and the binomial α + βx		81	(1)
Algebraic Functions		82	(24)
Introduction		82	(1)
Forms containing the binomial a + bxk and √x		83	(1)
Forms containing n√(a + bx)k		84	(4)
Forms containing √a + bx and the binomial α + βx		88	(4)
Forms containing √a + bx + cx2		92	(2)
Forms containing √a + bx + cx2 and integral powers of x		94	(5)
Forms containing √a + cx2 and integral powers of x		99	(4)
Forms containing √a + bx + cx2 and first- and second-degree polynomials		103	(1)
Integrals that can be reduced to elliptic or pseudo-elliptic integrals		104	(2)
The Exponential Function		106	(4)
Forms containing eax		106	(1)
The exponential combined with rational functions of x		106	(4)
Hyperbolic Functions		110	(41)
Powers of sinh x, cosh x, tanh x, and coth x		110	(15)
Rational functions of hyperbolic functions		125	(7)
Algebraic functions of hyperbolic functions		132	(7)
Combinations of hyperbolic functions and powers		139	(9)
Combinations of hyperbolic functions, exponentials, and powers		148	(3)
Trigonometric Functions		151	(86)
Introduction		151	(1)
Powers of trigonometric functions		151	(10)
Sines and cosines of multiple angles and of linear and more complicated functions of the argument		161	(10)
Rational functions of the sine and cosine		171	(8)
Integrals containing √a ± b sin x or √a ± b cos x		179	(5)
Integrals reducible to elliptic and pseudo-elliptic integrals		184	(30)
Products of trigonometric functions and powers		214	(13)
Combinations of trigonometric functions and exponentials		227	(4)
Combinations of trigonometric and hyperbolic functions		231	(6)
Logarithms and Inverse-Hyperbolic Functions		237	(4)
The logarithm		237	(1)
Combinations of logarithms and algebraic functions		238	(2)
Inverse hyperbolic functions		240	(1)
Inverse Trigonometric Functions		241	(6)
Arcsines and arccosines		241	(1)
The arcsecant, the arccosecant, the arctangent, and the arccotangent		242	(1)
Combinations of arcsine or arccosine and algebraic functions		242	(2)
Combinations of the arcsecant and arccosecant with powers of x		244	(1)
Combinations of the arctangent and arccotangent with algebraic functions		244	(3)
Definite Integrals of Elementary Functions		247	(372)
Introduction		247	(6)
Theorems of a general nature		247	(1)
Change of variable in a definite integral		248	(1)
General formulas		249	(2)
Improper integrals		251	(1)
The principal values of improper integrals		252	(1)
Power and Algebraic Functions		253	(81)
Rational functions		253	(1)
Products of rational functions and expressions that can be reduced to square roots of first- and second-degree polynomials		254	(1)
Expressions that can be reduced to square roots of third- and fourth-degree polynomials and their products with rational functions		254	(59)
Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions		313	(2)
Combinations of powers of x and powers of binomials of the form (α + βx)		315	(7)
Powers of x, of binomials of the form α + βxp and of polynomials in x		322	(12)
Exponential Functions		334	(37)
Exponential functions		334	(2)
Exponentials of more complicated arguments		336	(4)
Combinations of exponentials and rational functions		340	(4)
Combinations of exponentials and algebraic functions		344	(2)
Combinations of exponentials and arbitrary powers		346	(7)
Combinations of rational functions of powers and exponentials		353	(10)
Combinations of powers and algebraic functions of exponentials		363	(1)
Combinations of exponentials of more complicated arguments and powers		364	(7)
Hyperbolic Functions		371	(19)
Hyperbolic functions		371	(4)
Combinations of hyperbolic functions and algebraic functions		375	(7)
Combinations of hyperbolic functions and exponentials		382	(4)
Combinations of hyperbolic functions, exponentials, and powers		386	(4)
Trigonometric Functions		390	(137)
Rational functions of sines and cosines and trigonometric functions of multiple angles		390	(5)
Powers of trigonometric functions		395	(2)
Powers of trigonometric functions and trigonometric functions of linear functions		397	(4)
Powers and rational functions of trigonometric functions		401	(4)
Forms containing powers of linear functions of trigonometric functions		405	(3)
Square roots of expressions containing trigonometric functions		408	(3)
Various forms of powers of trigonometric functions		411	(4)
Trigonometric functions of more complicated arguments		415	(8)
Combinations of trigonometric and rational functions		423	(11)
Combinations of trigonometric and algebraic functions		434	(2)
Combinations of trigonometric functions and powers		436	(11)
Rational functions of x and of trigonometric functions		447	(12)
Powers of trigonometric functions combined with other powers		459	(13)
Integrals containing √1 -- k2 sin2 x, √1 -- k2 cos2 x, and similar expressions		472	(3)
Trigonometric functions of more complicated arguments combined with powers		475	(10)
Trigonometric functions and exponentials		485	(8)
Trigonometric functions of more complicated arguments combined with exponentials		493	(2)
Trigonometric and exponential functions of trigonometric functions		495	(2)
Combinations involving trigonometric functions, exponentials, and powers		497	(12)
Combinations of trigonometric and hyperbolic functions		509	(7)
Combinations involving trigonometric and hyperbolic functions and powers		516	(6)
Combinations of trigonometric and hyperbolic functions and exponentials		522	(3)
Combinations of trigonometric and hyperbolic functions, exponentials, and powers		525	(2)
Logarithmic Functions		527	(72)
Logarithmic functions		527	(2)
Logarithms of more complicated arguments		529	(6)
Combinations of logarithms and rational functions		535	(3)
Combinations of logarithms and algebraic functions		538	(2)
Combinations of logarithms and powers		540	(2)
Combinations involving powers of the logarithm and other powers		542	(11)
Combinations of rational functions of In x and powers		553	(2)
Combinations of logarithmic functions of more complicated arguments and powers		555	(16)
Combinations of logarithms and exponentials		571	(2)
Combinations of logarithms, exponentials, and powers		573	(5)
Combinations of logarithms and hyperbolic functions		578	(3)
Logarithms and trigonometric functions		581	(13)
Combinations of logarithms, trigonometric functions, and powers		594	(5)
Combinations of logarithms, trigonometric functions, and exponentials		599	(1)
Inverse Trigonometric Functions		599	(8)
Inverse trigonometric functions		599	(1)
Combinations of arcsines, arccosines, and powers		600	(1)
Combinations of arctangents, arccotangents, and powers		601	(4)
Combinations of inverse trigonometric functions and exponentials		605	(1)
A combination of the arctangent and a hyperbolic function		605	(1)
Combinations of inverse and direct trigonometric functions		605	(2)
A combination involving an inverse and a direct trigonometric function and a power		607	(1)
Combinations of inverse trigonometric functions and logarithms		607	(1)
Multiple Integrals		607	(12)
Change of variables in multiple integrals		607	(1)
Change of the order of integration and change of variables		608	(2)
Double and triple integrals with constant limits		610	(2)
Multiple integrals		612	(7)
Indefinite Integrals of Special Functions		619	(12)
Elliptic Integrals and Functions		619	(8)
Complete elliptic integrals		619	(2)
Elliptic integrals		621	(2)
Jacobian elliptic functions		623	(3)
Weierstrass elliptic functions		626	(1)
The Exponential Integral Function		627	(1)
The exponential integral function		627	(1)
Combinations of the exponential integral function and powers		627	(1)
Combinations of the exponential integral and the exponential		628	(1)
The Sine Integral and the Cosine Integral		628	(1)
The Probability Integral and Fresnel Integrals		629	(1)
Bessel Functions		629	(2)
Definite Integrals of Special Functions		631	(228)
Elliptic Integrals and Functions		631	(5)
Forms containing F(x, k)		631	(1)
Forms containing E(x, k)		632	(1)
Integration of elliptic integrals with respect to the modulus		632	(1)
Complete elliptic integrals		632	(1)
The theta function		633	(2)
Generalized elliptic integrals		635	(1)
The Exponential Integral Function and Functions Generated by It		636	(14)
The logarithm integral		636	(2)
The exponential integral function		638	(1)
The sine integral and cosine integral functions		639	(5)
The hyperbolic sine integral and hyperbolic cosine integral functions		644	(1)
The probability integral		645	(4)
Fresnel integrals		649	(1)
The Gamma Function and Functions Generated by It		650	(9)
The gamma function		650	(2)
Combinations of the gamma function, the exponential, and powers		652	(3)
Combinations of the gamma function and trigonometric functions		655	(1)
The logarithm of the gamma function		656	(1)
The incomplete gamma function		657	(1)
The function ψ(x)		658	(1)
Bessel Functions		659	(94)
Bessel functions		659	(5)
Bessel functions combined with x and x2		664	(6)
Combinations of Bessel functions and rational functions		670	(4)
Combinations of Bessel functions and algebraic functions		674	(1)
Combinations of Bessel functions and powers		675	(14)
Combinations of powers and Bessel functions of more complicated arguments		689	(5)
Combinations of Bessel functions and exponentials		694	(5)
Combinations of Bessel functions, exponentials, and powers		699	(9)
Combinations of Bessel functions of more complicated arguments, exponentials, and powers		708	(3)
Combinations of Bessel and exponential functions of more complicated arguments and powers		711	(2)
Combinations of Bessel, hyperbolic, and exponential functions		713	(4)
Combinations of Bessel and trigonometric functions		717	(10)
Combinations of Bessel and trigonometric functions and powers		727	(15)
Combinations of Bessel, trigonometric, and exponential functions and powers		742	(5)
Combinations of Bessel, trigonometric, and hyperbolic functions		747	(1)
Combinations of Bessel functions and the logarithm, or arctangent		747	(1)
Combinations of Bessel and other special functions		748	(1)
Integration of Bessel functions with respect to the order		749	(4)
Functions Generated by Bessel Functions		753	(10)
Struve functions		753	(1)
Combinations of Struve functions, exponentials, and powers		754	(1)
Combinations of Struve and trigonometric functions		755	(1)
Combinations of Struve and Bessel functions		756	(4)
Lommel functions		760	(1)
Thomson functions		761	(2)
Mathieu Functions		763	(6)
Mathieu functions		763	(1)
Combinations of Mathieu, hyperbolic, and trigonometric functions		763	(4)
Combinations of Mathieu and Bessel functions		767	(1)
Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems		767	(2)
Associated Legendre Functions		769	(26)
Associated Legendre functions		769	(1)
Combinations of associated Legendre functions and powers		770	(6)
Combinations of associated Legendre functions, exponentials, and powers		776	(2)
Combinations of associated Legendre and hyperbolic functions		778	(1)
Combinations of associated Legendre functions, powers, and trigonometric functions		779	(2)
A combination of an associated Legendre function and the probability integral		781	(1)
Combinations of associated Legendre and Bessel functions		782	(5)
Combinations of associated Legendre functions and functions generated by Bessel functions		787	(1)
Integration of associated Legendre functions with respect to the order		788	(1)
Combinations of Legendre polynomials, rational functions, and algebraic functions		789	(2)
Combinations of Legendre polynomials and powers		791	(1)
Combinations of Legendre polynomials and other elementary functions		792	(2)
Combinations of Legendre polynomials and Bessel functions		794	(1)
Orthogonal Polynomials		795	(17)
Combinations of Gegenbauer polynomials Cvn(x) and powers		795	(2)
Combinations of Gegenbauer polynomials Cvn(x) and elementary functions		797	(1)
Complete System of Orthogonal Step Functions		798	(1)
Combinations of the polynomials Cvn(x) and Bessel functions; Integration of Gegenbauer functions with respect to the index		798	(2)
Combinations of Chebyshev polynomials and powers		800	(2)
Combinations of Chebyshev polynomials and elementary functions		802	(1)
Combinations of Chebyshev polynomials and Bessel functions		803	(1)
Hermite polynomials		803	(3)
Jacobi polynomials		806	(2)
Laguerre polynomials		808	(4)
Hypergeometric Functions		812	(8)
Combinations of hypergeometric functions and powers		812	(2)
Combinations of hypergeometric functions and exponentials		814	(3)
Hypergeometric and trigonometric functions		817	(1)
Combinations of hypergeometric and Bessel functions		817	(3)
Confluent Hypergeometric Functions		820	(21)
Combinations of confluent hypergeometric functions and powers		820	(2)
Combinations of confluent hypergeometric functions and exponentials		822	(7)
Combinations of confluent hypergeometric and trigonometric functions		829	(1)
Combinations of confluent hypergeometric functions and Bessel functions		830	(1)
Combinations of confluent hypergeometric functions, Bessel functions, and powers		831	(3)
Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers		834	(5)
Combinations of confluent hypergeometric functions and other special functions		839	(2)
Integration of confluent hypergeometric functions with respect to the index		841	(1)
Parabolic Cylinder Functions		841	(9)
Parabolic cylinder functions		841	(1)
Combinations of parabolic cylinder functions, powers, and exponentials		842	(1)
Combinations of parabolic cylinder and hyperbolic functions		843	(1)
Combinations of parabolic cylinder and trigonometric functions		844	(1)
Combinations of parabolic cylinder and Bessel functions		845	(4)
Combinations of parabolic cylinder functions and confluent hypergeometric functions		849	(1)
Integration of a parabolic cylinder function with respect to the index		849	(1)
Meijer's and MacRobert's Functions (G and E)		850	(9)
Combinations of the functions G and E and the elementary functions		850	(4)
Combinations of the functions G and E and Bessel functions		854	(2)
Combinations of the functions G and E and other special functions		856	(3)
Special Functions		859	(190)
Elliptic Integrals and Functions		859	(24)
Elliptic integrals		859	(4)
Functional relations between elliptic integrals		863	(2)
Elliptic functions		865	(1)
Jacobian elliptic functions		866	(4)
Properties of Jacobian elliptic functions and functional relationships between them		870	(3)
The Weierstrass function (u)		873	(3)
The functions ζ(u) and σ(u)		876	(1)
Theta functions		877	(6)
The Exponential Integral Function and Functions Generated by It		883	(9)
The exponential integral function Ei(x)		883	(3)
The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x		886	(1)
The sine integral and the cosine integral: si x and ci x		886	(1)
The logarithm integral li(x)		887	(1)
The probability integral Φ(x), the Fresnel integrals S(x) and C(x), the error function erf(x), and the complementary error function erfc(x)		887	(4)
Lobachevskiy's function L(x)		891	(1)
Euler's Integrals of the First and Second Kinds		892	(18)
The gamma function (Euler's integral of the second kind): Γ(z)		892	(2)
Representation of the gamma function as series and products		894	(1)
Functional relations involving the gamma function		895	(3)
The logarithm of the gamma function		898	(1)
The incomplete gamma function		899	(3)
The psi function ψ(x)		902	(4)
The function β(x)		906	(2)
The beta function (Euler's integral of the first kind): B(x, y)		908	(2)
The incomplete beta function Bx(p, q)		910	(1)
Bessel Functions and Functions Associated with Them		910	(40)
Definitions		910	(2)
Integral representations of the functions Jv(z) and Nv(z)		912	(2)
Integral representations of the functions Hv(1)(z) and Hv(2)(z)		914	(2)
Integral representations of the functions Iv(z) and Kv(z)		916	(2)
Series representation		918	(2)
Asymptotic expansions of Bessel functions		920	(4)
Bessel functions of order equal to an integer plus one-half		924	(2)
Functional relations		926	(5)
Differential equations leading to Bessel functions		931	(2)
Series of Bessel functions		933	(7)
Expansion in products of Bessel functions		940	(1)
The zeros of Bessel functions		941	(1)
Struve functions		942	(2)
Thomson functions and their generalizations		944	(1)
Lommel functions		945	(3)
Anger and Weber functions Jv(z) and Ev(z)		948	(1)
Neumann's and Schlafli's polynomials: On(z) and Sn(z)		949	(1)
Mathieu Functions		950	(8)
Mathieu's equation		950	(1)
Periodic Mathieu functions		951	(1)
Recursion relations for the coefficients A(2n)2r, A(2n + 1)2r + 1, B(2n + 1)2r + 1, B(2n + 2)2r + 2		951	(1)
Mathieu functions with a purely imaginary argument		952	(1)
Non-periodic solutions of Mathieu's equation		953	(1)
Mathieu functions for negative q		953	(1)
Representation of Mathieu functions as series of Bessel functions		954	(3)
The general theory		957	(1)
Associated Legendre Functions		958	(24)
Introduction		958	(2)
Integral representations		960	(2)
Asymptotic series for large values of |v|		962	(2)
Functional relations		964	(4)
Special cases and particular values		968	(1)
Derivatives with respect to the order		969	(1)
Series representation		970	(2)
The zeros of associated Legendre functions		972	(1)
Series of associated Legendre functions		972	(2)
Associated Legendre functions with integer indices		974	(1)
Legendre functions		975	(5)
Conical functions		980	(1)
Toroidal functions		981	(1)
Orthogonal Polynomials		982	(23)
Introduction		982	(1)
Legendre polynomials		983	(5)
Series of products of Legendre and Chebyshev polynomials		988	(1)
Series of Legendre polynomials		988	(2)
Gegenbauer polynomials Cλn(t)		990	(3)
The Chebyshev polynomials Tn(x) and Un(x)		993	(3)
The Hermite polynomials Hn(x)		996	(2)
Jacobi's polynomials		998	(2)
The Laguerre polynomials		1000	(5)
Hypergeometric Functions		1005	(17)
Definition		1005	(1)
Integral representations		1005	(1)
Representation of elementary functions in terms of a hypergeometric functions		1006	(2)
Transformation formulas and the analytic continuation of functions defined by hypergeometric series		1008	(2)
A generalized hypergeometric series		1010	(1)
The hypergeometric differential equation		1010	(4)
Riemann's differential equation		1014	(3)
Representing the solutions to certain second-order differential equations using a Riemann scheme		1017	(1)
Hypergeometric functions of two variables		1018	(4)
A hypergeometric function of several variables		1022	(1)
Confluent Hypergeometric Functions		1022	(10)
Introduction		1022	(1)
The functions Φ(α, γ; z) and Ψ(α, γ; z)		1023	(1)
The Whittaker functions Mλ,μ(z) and Wλ,μ(z)		1024	(4)
Parabolic cylinder functions Dp (z)		1028	(3)
Confluent hypergeometric series of two variables		1031	(1)
Meijer's G-Function		1032	(3)
Definition		1032	(1)
Functional relations		1033	(1)
A differential equation for the G-function		1034	(1)
Series of G-functions		1034	(1)
Connections with other special functions		1034	(1)
MacRobert's E-Function		1035	(1)
Representation by means of multiple integrals		1035	(1)
Functional relations		1035	(1)
Riemann's Zeta Functions ζ(z, q) and ζ(z), and the Functions Φ(z, s, v) and ξ(s)		1036	(4)
Definition and integral representations		1036	(1)
Representation as a series or as an infinite product		1037	(1)
Functional relations		1037	(1)
Singular points and zeros		1038	(1)
The Lerch function Φ(z, s, v)		1039	(1)
The function ξ(s)		1040	(1)
Bernoulli Numbers and Polynomials, Euler Numbers		1040	(5)
Bernoulli numbers		1040	(1)
Bernoulli polynomials		1041	(2)
Euler numbers		1043	(1)
The functions v(x), v(x, α), μ(x, β), μ(x, β, α), and λ(x, y)		1043	(1)
Euler polynomials		1044	(1)
Constants		1045	(4)
Bernoulli numbers		1045	(1)
Euler numbers		1045	(1)
Euler's and Catalan's constants		1046	(1)
Stirling numbers		1046	(3)
Vector Field Theory		1049	(10)
Vectors, Vector Operators, and Integral Theorems		1049	(1)
Products of vectors		1049	(1)
Properties of scalar product		1049	(1)
Properties of vector product		1049	(1)
Differentiation of vectors		1050	(1)
Operators grad, div, and curl		1050	(1)
Properties of the operator		1051	(1)
Solenoidal fields		1052	(1)
Orthogonal curvilinear coordinates		1052	(3)
Vector integral theorems		1055	(2)
Integral rate of change theorems		1057	(2)
Algebraic Inequalities		1059	(4)
General Algebraic Inequalities		1059	(1)
Algebraic inequalities involving real numbers		1059	(1)
Algebraic inequalities involving complex numbers		1060	(1)
Inequalities for sets of complex numbers		1061	(2)
Integral Inequalities		1063	(6)
Mean Value Theorems		1063	(1)
First mean value theorem		1063	(1)
Second mean value theorem		1063	(1)
First mean value theorem for infinite integrals		1063	(1)
Second mean value theorem for infinite integrals		1064	(1)
Differentiation of Definite Integral Containing a Parameter		1064	(1)
Differentiation when limits are finite		1064	(1)
Differentiation when a limit is infinite		1064	(1)
Integral Inequalities		1064	(2)
Cauchy-Schwarz-Buniakowsky inequality for integrals		1064	(1)
Holder's inequality for integrals		1064	(1)
Minkowski's inequality for integrals		1065	(1)
Chebyshev's inequality for integrals		1065	(1)
Young's inequality for integrals		1065	(1)
Steffensen's inequality for integrals		1065	(1)
Gram's inequality for integrals		1065	(1)
Ostrowski's inequality for integrals		1066	(1)
Convexity and Jensen's Inequality		1066	(1)
Jensen's inequality		1066	(1)
Carleman's inequality for integrals		1066	(1)
Fourier Series and Related Inequalities		1066	(3)
Riemann-Lebesgue lemma		1067	(1)
Dirichlet lemma		1067	(1)
Parseval's theorem for trigonometric Fourier series		1067	(1)
Integral representation of the nth partial sum		1067	(1)
Generalized Fourier series		1067	(1)
Bessel's inequality for generalized Fourier series		1068	(1)
Parseval's theorem for generalized Fourier series		1068	(1)
Matrices and Related Results		1069	(6)
Special Matrices		1069	(2)
Diagonal matrix		1069	(1)
Identity matrix and null matrix		1069	(1)
Reducible and irreducible matrices		1069	(1)
Equivalent matrices		1069	(1)
Transpose of a matrix		1069	(1)
Adjoint matrix		1070	(1)
Inverse matrix		1070	(1)
Trace of a matrix		1070	(1)
Symmetric matrix		1070	(1)
Skew-symmetric matrix		1070	(1)
Triangular matrices		1070	(1)
Orthogonal matrices		1070	(1)
Hermitian transpose of a matrix		1070	(1)
Hermitian matrix		1070	(1)
Unitary matrix		1071	(1)
Eigenvalues and eigenvectors		1071	(1)
Nilpotent matrix		1071	(1)
Idempotent matrix		1071	(1)
Positive definite		1071	(1)
Non-negative definite		1071	(1)
Diagonally dominant		1071	(1)
Quadratic Forms		1071	(2)
Sylvester's law of inertia		1072	(1)
Rank		1072	(1)
Signature		1072	(1)
Positive definite and semidefinite quadratic form		1072	(1)
Basic theorems on quadratic forms		1072	(1)
Differentiation of Matrices		1073	(1)
The Matrix Exponential		1074	(1)
Basic properties		1074	(1)
Determinants		1075	(6)
Expansion of Second- and Third-Order Determinants		1075	(1)
Basic Properties		1075	(1)
Minors and Cofactors of a Determinant		1075	(1)
Principal Minors		1076	(1)
Laplace Expansion of a Determinant		1076	(1)
Jacobi's Theorem		1076	(1)
Hadamard's Theorem		1077	(1)
Hadamard's Inequality		1077	(1)
Cramer's Rule		1077	(1)
Some Special Determinants		1078	(3)
Vandermonde's determinant (alternant)		1078	(1)
Circulants		1078	(1)
Jacobian determinant		1078	(1)
Hessian determinants		1079	(1)
Wronskian determinants		1079	(1)
Properties		1079	(1)
Gram-Kowalewski theorem on linear dependence		1080	(1)
Norms		1081	(12)
Vector Norms		1081	(1)
General Properties		1081	(1)
Principal Vector Norms		1081	(1)
The norm ||x||1		1081	(1)
The norm ||x||2 (Euclidean or L2 norm)		1081	(1)
The norm ||x||∞		1081	(1)
Matrix Norms		1082	(1)
General properties		1082	(1)
Induced norms		1082	(1)
Natural norm of unit matrix		1082	(1)
Principal Natural Norms		1082	(1)
Maximum absolute column sum norm		1082	(1)
Spectral norm		1082	(1)
Maximum absolute row sum norm		1083	(1)
Spectral Radius of a Square Matrix		1083	(1)
Inequalities concerning matrix norms and the spectral radius		1083	(1)
Deductions from Gerschgorin's theorem (see 15.814)		1083	(1)
Inequalities Involving Eigenvalues of Matrices		1084	(1)
Cayley-Hamilton theorem		1084	(1)
Corollaries		1084	(1)
Inequalities for the Characteristic Polynomial		1084	(3)
Named and unnamed inequalities		1085	(1)
Parodi's theorem		1086	(1)
Corollary of Brauer's theorem		1086	(1)
Ballieu's theorem		1086	(1)
Routh-Hurwitz theorem		1086	(1)
Named Theorems on Eigenvalues		1087	(4)
Schur's inequalities		1087	(1)
Sturmian separation theorem		1087	(1)
Poincare's separation theorem		1087	(1)
Gerschgorin's theorem		1088	(1)
Brauer's theorem		1088	(1)
Perron's theorem		1088	(1)
Frobenius theorem		1088	(1)
Perron--Frobenius theorem		1088	(1)
Wielandt's theorem		1088	(1)
Ostrowski's theorem		1089	(1)
First theorem due to Lyapunov		1089	(1)
Second theorem due to Lyapunov		1089	(1)
Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov		1089	(2)
Variational Principles		1091	(2)
Rayleigh quotient		1091	(1)
Basic theorems		1091	(2)
Ordinary Differential Equations		1093	(14)
Results Relating to the Solution of Ordinary Differential Equations		1093	(1)
First-Order Equations		1093	(1)
Solution of a first-order equation		1093	(1)
Cauchy problem		1093	(1)
Approximate solution to an equation		1093	(1)
Lipschitz continuity of a function		1094	(1)
Fundamental Inequalities and Related Results		1094	(1)
Gronwall's lemma		1094	(1)
Comparison of approximate solutions of a differential equation		1094	(1)
First-Order Systems		1094	(3)
Solution of a system of equations		1094	(1)
Cauchy problem for a system		1095	(1)
Approximate solution to a system		1095	(1)
Lipschitz continuity of a vector		1095	(1)
Comparison of approximate solutions of a system		1096	(1)
First-order linear differential equation		1096	(1)
Linear systems of differential equations		1096	(1)
Some Special Types of Elementary Differential Equations		1097	(1)
Variables separable		1097	(1)
Exact differential equations		1097	(1)
Conditions for an exact equation		1097	(1)
Homogeneous differential equations		1097	(1)
Second-Order Equations		1098	(2)
Adjoint and self-adjoint equations		1098	(1)
Abel's identity		1098	(1)
Lagrange identity		1099	(1)
The Riccati equation		1099	(1)
Solutions of the Riccati equation		1099	(1)
Solution of a second-order linear differential equation		1100	(1)
Oscillation and Non-Oscillation Theorems for Second-Order Equations		1100	(3)
First basic comparison theorem		1100	(1)
Second basic comparison theorem		1101	(1)
Interlacing of zeros		1101	(1)
Sturm separation theorem		1101	(1)
Sturm comparison theorem		1101	(1)
Szego's comparison theorem		1101	(1)
Picone's identity		1102	(1)
Sturm-Picone theorem		1102	(1)
Oscillation on the half line		1102	(1)
Two Related Comparison Theorems		1103	(1)
Theorem 1		1103	(1)
Theorem 2		1103	(1)
Non-Oscillatory Solutions		1103	(1)
Kneser's non-oscillation theorem		1103	(1)
Comparison theorem for non-oscillation		1104	(1)
Necessary and sufficient conditions for non-oscillation		1104	(1)
Some Growth Estimates for Solutions of Second-Order Equations		1104	(2)
Strictly increasing and decreasing solutions		1104	(1)
General result on dominant and subdominant solutions		1104	(1)
Estimate of dominant solution		1105	(1)
A theorem due to Lyapunov		1105	(1)
Boundedness Theorems		1106	(1)
All solutions of the equation		1106	(1)
If all solutions of the equation		1106	(1)
If a(x) → ∞ monotonically as x → ∞, then all solutions of		1106	(1)
Consider the equation		1106	(1)
Growth of maxima of |y|		1106	(1)
Fourier, Laplace, and Mellin Transforms		1107	(28)
Integral Transforms		1107	(1)
Laplace transform		1107	(1)
Basic properties of the Laplace transform		1107	(1)
Table of Laplace transform pairs		1108	(9)
Fourier transform		1117	(1)
Basic properties of the Fourier transform		1118	(1)
Table of Fourier transform pairs		1118	(2)
Table of Fourier transform pairs for spherically symmetric functions		1120	(1)
Fourier sine and cosine transforms		1121	(1)
Basic properties of the Fourier sine and cosine transforms		1121	(1)
Table of Fourier sine transforms		1122	(4)
Table of Fourier cosine transforms		1126	(3)
Relationships between transforms		1129	(1)
Mellin transform		1129	(1)
Basic properties of the Mellin transform		1130	(1)
Table of Mellin transforms		1131	(4)
The z-Transform		1135	(6)
Definition, Bilateral, and Unilateral z-Transforms		1135	(1)
Definitions		1135	(1)
Bilateral z-transform		1136	(2)
Unilateral z-transform		1138	(3)
References		1141	(4)
Supplemental references		1145	(6)
Index of Functions and Constants		1151	(10)
General Index of Concepts		1161

Dr. Daniel Zwillinger is a Senior Principal Systems Engineer for the Raytheon Company. He was a systems requirements book boss” for the Cobra Judy Replacement (CJR) ship and was a requirements and test lead for tracking on the Ungraded Early Warning Radars (UEWR). He has improved the Zumwalt destroyers software accreditation process and he was test lead on an Active Electronically Scanned Array (AESA) radar. Dan is a subject matter expert (SME) in Design for Six Sigma (DFSS) and is a DFSS SME in Test Optimization, Critical Chain Program Management, and Voice of the Customer. He is currently leading a project creating Trust in Autonomous Systems. At Raytheon, he twice won the Presidents award for best Six Sigma project of the year: on converting planning packages to work packages for the Patriot missile, and for revising Raytheons timecard system. He has managed the Six Sigma white belt training program. Prior to Raytheon, Dan worked at Sandia Labs, JPL, Exxon, MITRE, IDA, BBN, and The Mathworks (where he developed an early version of their Statistics Toolbox).

For ten years, Zwillinger was owner and president of Aztec Corporation. As a small business, Aztec won several Small Business Innovation Research (SBIR) contracts. The company also created several software packages for publishing companies. Prior to Aztec, Zwillinger was a college professor at Rensselaer Polytechnic Institute in the department of mathematics.

Dan has written several books on mathematics on the topics of differential equations, integration, statistics, and general mathematics. He is editor-in-chief of the Chemical Rubber Companys (CRCs) Standard Mathematical Tables and Formulae”, and is on the editorial board for CRCs Handbook of Chemistry and Physics”. Zwillinger holds a bachelor's degree in mathematics from the Massachusetts Institute of Technology (MIT). He earned his doctorate in applied mathematics from the California Institute of Technology (Caltech). Zwillinger is a certified Raytheon Six Sigma Expert and an ASQ certified Six Sigma Black Belt. He also holds a pilots license.