Tables of Integrals, Series, and Products

Jeffrey and Zwillinger

Table of Contents

Preface to the Sixth Edition (page vii)
Acknowledgments (page ix)
The order of presentation of the formulas (page xiii)
Use of the tables (page xvii)
Special functions (page xxv)
Notation (page xxix)
Note on the bibliographic references (page xxxiii)

0 Introduction (page 1)
0.1 Finite sums (page 1)
0.11 Progressions (page 1)
0.12 Sums of powers of natural numbers (page 1)
0.13 Sums of reciprocals of natural numbers (page 2)
0.14 Sums of products of reciprocals of natural numbers (page 3)
0.15 Sums of the binomial coefficients (page 3)
0.2 Numerical series and infinite products (page 6)
0.21 The convergence of numerical series (page 6)
0.22 Convergence tests (page 6)
0.23-0.24 Examples of numerical series (page 8)
0.25 Infinite products (page 14)
0.26 Examples of infinite products (page 14)
0.3 Functional series (page 15)
0.30 Definitions and theorems (page 15)
0.31 Power series (page 16)
0.32 Fourier series (page 18)
0.33 Asymptotic series (page 20)
0.4 Certain formulas from differential calculus (page 21)
0.41 Differentiation of a definite integral with respect to a parameter (page 21)
0.42 The nth derivative of a product (Leibniz's rule) (page 21)
0.43 The nth derivative of a composite function (page 21)
0.44 Integration by substitution (page 23)

1 Elementary Functions (page 25)
1.1 Power of Binomials (page 25)
1.11 Power series (page 25)
1.12 Series of rational fractions (page 26)
1.2 The Exponential Function (page 26)
1.21 Series representation (page 26)
1.22 Functional relations (page 27)
1.23 Series of exponentials (page 27)
1.3-1.4 Trigonometric and Hyperbolic Functions (page 27)
1.30 Introduction (page 28)
1.31 The basic functional relations (page 28)
1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) (page 30)
1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions (page 32)
1.34 Certain sums of trigonometric and hyperbolic functions (page 35)
1.35 Sums of powers of trigonometric functions of multiple angles (page 36)
1.36 Sums of products of trigonometric functions of multiple angles (page 37)
1.37 Sums of tangents of multiple angles (page 38)
1.38 Sums leading to hyperbolic tangents and cotangents (page 38)
1.39 The representation of cosines and sines of multiples of the angle as finite products (page 39)
1.41 The expansion of trigonometric and hyperbolic functions in power series (page 41)
1.42 Expansion in series of simple fractions (page 42)
1.43 Representation in the form of an infinite product (page 43)
1.44-1.45 Trigonometric (Fourier) series (page 44)
1.46 Series of products of exponential and trigonometric functions (page 48)
1.47 Series of hyperbolic functions (page 49)
1.48 Lobachevskiy's "Angle of parallelism" ${\Pi (x)}$ (page 49)
1.49 The hyperbolic amplitude (the Gudermannian) gd(x) (page 50)
1.5 The Logarithm (page 51)
1.51 Series representation (page 51)
1.52 Series of logarithms (cf. 1.431) (page 53)
1.6 The Inverse Trigonometric and Hyperbolic Functions (page 54)
1.61 The domain of definition (page 54)
1.62-1.63 Functional relations (page 54)
1.64 Series representations (page 58)

2 Indefinite Integrals of Elementary Functions (page 61)
2.0 Introduction (page 61)
2.00 General remarks (page 61)
2.01 The basic integrals (page 61)
2.02 General formulas (page 62)
2.1 Rational functions (page 64)
2.10 General integration rules (page 64)
2.11-2.13 Forms containing the binomial a+bxk (page 66)
2.14 Forms containing the binomial $1\pm {x}^{n}$ (page 72)
2.15 Forms containing pairs of binomials: a+bx and $\alpha +\beta x$ (page 76)
2.16 Forms containing the trinomial a+bxk+c x2k (page 76)
2.17 Forms containing the quadratic trinomial a+bx+cx2 and powers of x (page 77)
2.18 Forms containing the quadratic trinomial a+bx+cx2 and the binomial $\alpha +\beta x$ (page 79)
2.2 Algebraic functions (page 80)
2.20 Introduction (page 80)
2.21 Forms containing the binomial a+bxk and $ \sqrt {x}$ (page 81)
2.22-2.23 Forms containing $\root{n}\of{(a+bx)^k}$ (page 83)
2.24 Forms containing $ \sqrt {a+bx}$ and the binomial $\alpha +\beta x$ (page 86)
2.25 Forms containing $ \sqrt {a+bx+c {x}^{2}}$ (page 90)
2.26 Forms containing $ \sqrt {a+b+c {x}^{2}}$ and integral powers of x (page 92)
2.27 Forms containing $ \sqrt {a+c {x}^{2}}$ and integral powers of x (page 97)
2.28 Forms containing $ \sqrt {a+bx+c {x}^{2}}$ and first-and second-degree polynomials (page 101)
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals (page 102)
2.3 The Exponential Function (page 104)
2.31 Forms containing eax (page 104)
2.32 The exponential combined with rational functions of x (page 104)
2.4 Hyperbolic Functions (page 105)
2.41-2.43 Powers of sinh x, cosh x, tanh x, and coth x, (page 105)
2.44-2.45 Rational functions of hyperbolic functions (page 121)
2.46 Algebraic functions of hyperbolic functions (page 128)
2.47 Combinations of hyperbolic functions and powers (page 136)
2.48 Combinations of hyperbolic functions, exponentials, and powers (page 145)
2.5-2.6 Trigonometric Functions (page 147)
2.50 Introduction (page 147)
2.51-2.52 Powers of trigonometric functions (page 147)
2.53-2.54 Sines and cosines of multiple angles and of linear and more complicated functions of the argument (page 157)
2.55-2.56 Rational functions of the sine and cosine (page 167)
2.57 Integrals containing $ \sqrt {a\pm b {\sin}x}$ or $ \sqrt {a\pm b {\cos}x}$ (page 175)
2.58-2.62 Integrals reducible to elliptic and pseudo-elliptic integrals (page 180)
2.63-2.65 Products of trigonometric functions and powers (page 210)
2.66 Combinations of trigonometric functions and exponentials (page 222)
2.67 Combinations of trigonometric and hyperbolic functions (page 227)
2.7 Logarithms and Inverse-Hyperbolic Functions (page 233)
2.71 The logarithm (page 233)
2.72-2.73 Combinations of logarithms and algebraic functions (page 233)
2.74 Inverse hyperbolic functions (page 236)
2.8 Inverse Trigonometric Functions (page 237)
2.81 Arcsines and arccosines (page 237)
2.82 The arcsecant, the arccosecant, the arctangent and the arccotangent (page 238)
2.83 Combinations of arcsine or arccosine and algebraic functions (page 238)
2.84 Combinations of the arcsecant and arccosecant with powers of x (page 240)
2.85 Combinations of the arctangent and arccotangent with algebraic functions (page 240)

3-4 Definite Integrals of Elementary Functions (page 243)
3.0 Introduction (page 243)
3.01 Theorems of a general nature (page 243)
3.02 Change of variable in a definite integral (page 244)
3.03 General formulas (page 245)
3.04 Improper integrals (page 247)
3.05 The principal values of improper integrals (page 248)
3.1-3.2 Power and Algebraic Functions (page 248)
3.11 Rational functions (page 249)
3.12 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials (page 249)
3.13-3.17 Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with rational functions (page 250)
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions (page 310)
3.19-3.23 Combinations of powers of x and powers of binomials of the form $(\alpha +\beta x)$ (page 312)
3.24-3.27 Powers of x, of binomials of the form $\alpha +\beta x^{p}$ and of polynomials in x (page 319)
3.3-3.4 Exponential Functions (page 331)
3.31 Exponential functions (page 331)
3.32-3.34 Exponentials of more complicated arguments (page 333)
3.35 Combinations of exponentials and rational functions (page 336)
3.36-3.37 Combinations of exponentials and algebraic functions (page 340)
3.38-3.39 Combinations of exponentials and arbitrary powers (page 342)
3.41-3.44 Combinations of rational functions of powers and exponentials (page 349)
3.45 Combinations of powers and algebraic functions of exponentials (page 358)
3.46-3.48 Combinations of exponentials of more complicated arguments and powers (page 360)
3.5 Hyperbolic Functions (page 365)
3.51 Hyperbolic functions (page 366)
3.52-3.53 Combinations of hyperbolic functions and algebraic functions (page 369)
3.54 Combinations of hyperbolic functions and exponentials (page 376)
3.55-3.56 Combinations of hyperbolic functions, exponentials, and powers (page 380)
3.6-4.1 Trigonometric Functions (page 384)
3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles (page 385)
3.62 Powers of trigonometric functions (page 388)
3.63 Powers of trigonometric functions and trigonometric functions of linear functions (page 390)
3.64-3.65 Powers and rational functions of trigonometric functions (page 395)
3.66 Forms containing powers of linear functions of trigonometric functions (page 399)
3.67 Square roots of expressions containing trigonometric functions (page 402)
3.68 Various forms of powers of trigonometric functions (page 404)
3.69-3.71 Trigonometric functions of more complicated arguments (page 408)
3.72-3.74 Combinations of trigonometric and rational functions (page 417)
3.75 Combinations of trigonometric and algebraic functions (page 428)
3.76-3.77 Combinations of trigonometric functions and powers (page 429)
3.78-3.81 Rational functions of x and of trigonometric functions (page 440)
3.82-3.83 Powers of trigonometric functions combined with other powers (page 453)
3.84 Integrals containing $ \sqrt {1- {k}^{2} {\sin}^2 x}$, $ \sqrt {1- {k}^{2} {\cos}^2 x}$, and similar expressions (page 466)
3.85-3.88 Trigonometric functions of more complicated arguments combined with powers (page 469)
3.89-3.91 Trigonometric functions and exponentials (page 479)
3.92 Trigonometric functions of more complicated arguments combined with exponentials (page 487)
3.93 Trigonometric and exponential functions of trigonometric functions (page 490)
3.94-3.97 Combinations involving trigonometric functions, exponentials, and powers (page 492)
3.98-3.99 Combinations of trigonometric and hyperbolic functions (page 504)
4.11-4.12 Combinations involving trigonometric and hyperbolic functions and powers (page 511)
4.13 Combinations of trigonometric and hyperbolic functions and exponentials (page 517)
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers (page 520)
4.2-4.4 Logarithmic Functions (page 522)
4.21 Logarithmic functions (page 522)
4.22 Logarithms of more complicated arguments (page 525)
4.23 Combinations of logarithms and rational functions (page 530)
4.24 Combinations of logarithms and algebraic functions (page 532)
4.25 Combinations of logarithms and powers (page 534)
4.26-4.27 Combinations involving powers of the logarithm and other powers (page 537)
4.28 Combinations of rational functions of ln x and powers (page 549)
4.29-4.32 Combinations of logarithmic functions of more complicated arguments and powers (page 551)
4.33-4.34 Combinations of logarithms and exponentials (page 567)
4.35-4.36 Combinations of logarithms, exponentials, and powers (page 568)
4.37 Combinations of logarithms and hyperbolic functions (page 574)
4.38-4.41 Logarithms and trigonometric functions (page 577)
4.42-4.43 Combinations of logarithms, trigonometric functions, and powers (page 590)
4.44 Combinations of logarithms, trigonometric functions, and exponentials (page 595)
4.5 Inverse Trigonometric Functions (page 596)
4.51 Inverse trigonometric functions (page 596)
4.52 Combinations of arcsines, arccosines, and powers (page 596)
4.53-4.54 Combinations of arctangents, arccotangents, and powers (page 597)
4.55 Combinations of inverse trigonometric functions and exponentials (page 601)
4.56 A combination of the arctangent and a hyperbolic function (page 601)
4.57 Combinations of inverse and direct trigonometric functions (page 601)
4.58 A combination involving an inverse and a direct trigonometric function and a power (page 603)
4.59 Combinations of inverse trigonometric functions and logarithms (page 603)
4.6 Multiple Integrals (page 604)
4.60 Change of variables in multiple integrals (page 604)
4.61 Change of the order of integration and change of variables (page 604)
4.62 Double and triple integrals with constant limits (page 607)
4.63-4.64 Multiple integrals (page 609)

5 Indefinite Integrals of Special Functions (page 615)
5.1 Elliptic Integrals and Functions (page 615)
5.11 Complete elliptic integrals (page 615)
5.12 Elliptic integrals (page 616)
5.13 Jacobian elliptic functions (page 618)
5.14 Weierstrass elliptic functions (page 622)
5.2 The Exponential Integral Function (page 622)
5.21 The exponential integral function (page 622)
5.22 Combinations of the exponential integral function and powers (page 622)
5.23 Combinations of the exponential integral and the exponential (page 622)
5.3 The Sine Integral and the Cosine Integral (page 623)
5.4 The Probability Integral and Fresnel Integrals (page 623)
5.5 Bessel Functions (page 624)

6-7 Definite Integrals of Special Functions (page 625)
6.1 Elliptic Integrals and Functions (page 625)
6.11 Forms containing F(x,k) (page 625)
6.12 Forms containing E(x,k) (page 626)
6.13 Integration of elliptic integrals with respect to the modulus (page 626)
6.14-6.15 Complete elliptic integrals (page 626)
6.16 The theta function (page 627)
6.17 Generalized elliptic integrals (page 628)
6.2-6.3 The Exponential Integral Function and Functions Generated by It (page 630)
6.21 The logarithm integral (page 630)
6.22-6.23 The exponential integral function (page 631)
6.24-6.26 The sine integral and cosine integral functions (page 633)
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions (page 638)
6.28-6.31 The probability integral (page 638)
6.32 Fresnel integrals (page 642)
6.4 The Gamma Function and Functions Generated by It (page 644)
6.41 The gamma function (page 644)
6.42 Combinations of the gamma function, the exponential, and powers (page 645)
6.43 Combinations of the gamma function and trigonometric functions (page 648)
6.44 The logarithm of the gamma function* (page 649)
6.45 The incomplete gamma function (page 650)
6.46-6.47 The function ${ \psi(x)}$ (page 651)
6.5-6.7 Bessel Functions (page 652)
6.51 Bessel functions (page 653)
6.52 Bessel functions combined with x and x2 (page 657)
6.53-6.54 Combinations of Bessel functions and rational functions (page 662)
6.55 Combinations of Bessel functions and algebraic functions (page 666)
6.56-6.58 Combinations of Bessel functions and powers (page 667)
6.59 Combinations of powers and Bessel functions of more complicated arguments (page 681)
6.61 Combinations of Bessel functions and exponentials (page 686)
6.62-6.63 Combinations of Bessel functions, exponentials, and powers (page 691)
6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers (page 701)
6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers (page 703)
6.66 Combinations of Bessel, hyperbolic, and exponential functions (page 705)
6.67-6.68 Combinations of Bessel and trigonometric functions (page 709)
6.69-6.74 Combinations of Bessel and trigonometric functions and powers (page 719)
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers (page 735)
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions (page 739)
6.77 Combinations of Bessel functions and the logarithm, or arctangent (page 739)
6.78 Combinations of Bessel and other special functions (page 740)
6.79 Integration of Bessel functions with respect to the order (page 741)
6.8 Functions Generated by Bessel Functions (page 745)
6.81 Struve functions (page 745)
6.82 Combinations of Struve functions, exponentials, and powers (page 747)
6.83 Combinations of Struve and trigonometric functions (page 748)
6.84-6.85 Combinations of Struve and Bessel functions (page 748)
6.86 Lommel functions (page 752)
6.87 Thomson functions (page 754)
6.9 Mathieu Functions (page 755)
6.91 Mathieu functions (page 755)
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions (page 756)
6.93 Combinations of Mathieu and Bessel functions (page 759)
6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems (page 759)
7.1-7.2 Associated Legendre Functions (page 762)
7.11 Associated Legendre functions (page 762)
7.12-7.13 Combinations of associated Legendre functions and powers (page 763)
7.14 Combinations of associated Legendre functions, exponentials, and powers (page 769)
7.15 Combinations of associated Legendre and hyperbolic functions (page 771)
7.16 Combinations of associated Legendre functions, powers, and trigonometric functions (page 772)
7.17 A combination of an associated Legendre function and the probability integral (page 774)
7.18 Combinations of associated Legendre and Bessel functions (page 774)
7.19 Combinations of associated Legendre functions and functions generated by Bessel functions (page 780)
7.21 Integration of associated Legendre functions with respect to the order (page 781)
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions (page 782)
7.23 Combinations of Legendre polynomials and powers (page 784)
7.24 Combinations of Legendre polynomials and other elementary functions (page 785)
7.25 Combinations of Legendre polynomials and Bessel functions (page 787)
7.3-7.4 Orthogonal Polynomials (page 788)
7.31 Combinations of Gegenbauer polynomials $C_{n}^{\nu }{(x)}$ and powers (page 788)
7.32 Combinations of Gegenbauer polynomials $C_{n}^{\nu }{(x)}$ and some elementary functions (page 790)
7.33 Combinations of the polynomials $C_{n}^{\nu }{(x)}$ and Bessel functions. Integration of Gegenbauer functions with respect to the index. (page 791)
7.34 Combinations of Chebyshev polynomials and powers (page 793)
7.35 Combinations of Chebyshev polynomials and some elementary functions (page 794)
7.36 Combinations of Chebyshev polynomials and Bessel functions (page 795)
7.37-7.38 Hermite polynomials (page 796)
7.39 Jacobi polynomials (page 800)
7.41-7.42 Laguerre polynomials (page 801)
7.5 Hypergeometric Functions (page 806)
7.51 Combinations of hypergeometric functions and powers (page 806)
7.52 Combinations of hypergeometric functions and exponentials (page 807)
7.53 Hypergeometric and trigonometric functions (page 810)
7.54 Combinations of hypergeometric and Bessel functions (page 810)
7.6 Confluent Hypergeometric Functions (page 814)
7.61 Combinations of confluent hypergeometric functions and powers (page 814)
7.62-7.63 Combinations of confluent hypergeometric functions and exponentials (page 815)
7.64 Combinations of confluent hypergeometric and trigonometric functions (page 822)
7.65 Combinations of confluent hypergeometric functions and Bessel functions (page 824)
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers (page 824)
7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers (page 828)
7.68 Combinations of confluent hypergeometric functions and other special functions (page 832)
7.69 Integration of confluent hypergeometric functions with respect to the index (page 834)
7.7 Parabolic Cylinder Functions (page 835)
7.71 Parabolic cylinder functions (page 835)
7.72 Combinations of parabolic cylinder functions, powers, and exponentials (page 835)
7.73 Combinations of parabolic cylinder and hyperbolic functions (page 837)
7.74 Combinations of parabolic cylinder and trigonometric functions (page 837)
7.75 Combinations of parabolic cylinder and Bessel functions (page 838)
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions (page 841)
7.77 Integration of a parabolic cylinder function with respect to the index (page 842)
7.8 Meijer's and MacRobert's Functions (G and E) (page 843)
7.81 Combinations of the functions G and E and the elementary functions (page 843)
7.82 Combinations of the functions G and E and Bessel functions (page 847)
7.83 Combinations of the functions G and E and other special functions (page 849)

8-9 Special Functions (page 851)
8.1 Elliptic integrals and functions (page 851)
8.11 Elliptic integrals (page 851)
8.12 Functional relations between elliptic integrals (page 854)
8.13 Elliptic functions (page 856)
8.14 Jacobian elliptic functions (page 857)
8.15 Properties of Jacobian elliptic functions and functional relationships between them (page 861)
8.16 The Weierstrass function $\wp (u)$ (page 865)
8.17 The functions $\zeta $(u) and $\sigma $(u) (page 868)
8.18-8.19 Theta functions (page 869)
8.2 The Exponential Integral Function and Functions Generated by It (page 875)
8.21 The exponential integral function Ei(x) (page 875)
8.22 The hyperbolic sine integral shi(x) and the hyperbolic cosine integral chi(x) (page 878)
8.23 The sine integral and the cosine integral: si(x) and ci(x) (page 878)
8.24 The logarithm integral li(x) (page 879)
8.25 The probability integral, the Fresnel integrals ${ {\rm\Phi}(x)}$, S(x), C(x), the error function erf(x), and the complementary error function erfc(x), (page 879)
8.26 Lobachevskiy's function L(x), (page 883)
8.3 Euler's Integrals of the First and Second Kinds (page 883)
8.31 The gamma function (Euler's integral of the second kind): $\Gamma (z)$ (page 883)
8.32 Representation of the gamma function as series and products (page 885)
8.33 Functional relations involving the gamma function (page 886)
8.34 The logarithm of the gamma function (page 888)
8.35 The incomplete gamma function (page 890)
8.36 The psi function ${ \psi(x)}$ (page 892)
8.37 The function $ {\beta}(x)$ (page 896)
8.38 The beta function (Euler's integral of the first kind): B(x,y) (page 897)
8.39 The incomplete beta function Bx(p,q) (page 900)
8.4-8.5 Bessel Functions and Functions Associated with Them (page 900)
8.40 Definitions (page 900)
8.41 Integral representations of the functions $J_{\nu }(z)$ and $N_{\nu }(z)$ (page 901)
8.42 Integral representations of the functions $H^{(1)}_{\nu }(z)$ and $H^{(2)}_{\nu }(z)$ (page 904)
8.43 Integral representations of the functions $I_{\nu }(z)$ and $K_{\nu }(z)$ (page 906)
8.44 Series representation (page 908)
8.45 Asymptotic expansions of Bessel functions (page 909)
8.46 Bessel functions of order equal to an integer plus one-half (page 913)
8.47-8.48 Functional relations (page 915)
8.49 Differential equations leading to Bessel functions (page 921)
8.51-8.52 Series of Bessel functions (page 923)
8.53 Expansion in products of Bessel functions (page 930)
8.54 The zeros of Bessel functions (page 931)
8.55 Struve functions (page 932)
8.56 Thomson functions and their generalizations (page 934)
8.57 Lommel functions (page 935)
8.58 Anger and Weber functions J $_{\nu }(z)$ and E $_{\nu }(z)$ (page 938)
8.59 Neumann's and Schläfli's polynomials: On(z) and Sn(z) (page 939)
8.6 Mathieu Functions (page 940)
8.60 Mathieu's equation (page 940)
8.61 Periodic Mathieu functions (page 940)
8.62 Recursion relations for the coefficients A 2r(2n), A 2r+1(2n+1), B 2r+1(2n+1), B 2r+2(2n+2) (page 941)
8.63 Mathieu functions with a purely imaginary argument (page 942)
8.64 Non-periodic solutions of Mathieu's equation (page 943)
8.65 Mathieu functions for negative q (page 943)
8.66 Representation of Mathieu functions as series of Bessel functions (page 944)
8.67 The general theory (page 947)
8.7-8.8 Associated Legendre Functions (page 948)
8.70 Introduction (page 948)
8.71 Integral representations (page 950)
8.72 Asymptotic series for large values of $\vert\nu \vert$ (page 952)
8.73-8.74 Functional relations (page 954)
8.75 Special cases and particular values (page 957)
8.76 Derivatives with respect to the order (page 959)
8.77 Series representation (page 959)
8.78 The zeros of associated Legendre functions (page 961)
8.79 Series of associated Legendre functions (page 962)
8.81 Associated Legendre functions with integral indices (page 964)
8.82-8.83 Legendre functions (page 965)
8.84 Conical functions (page 970)
8.85 Toroidal functions (page 971)
8.9 Orthogonal Polynomials (page 972)
8.90 Introduction (page 972)
8.91 Legendre polynomials (page 973)
8.919 Series of products of Legendre and Chebyshev polynomials (page 977)
8.92 Series of Legendre polynomials (page 978)
8.93 Gegenbauer polynomials $C_{n}^{\lambda }{(t)}$ (page 980)
8.94 The Chebyshev polynomials Tn(x) and Un(x) (page 983)
8.95 The Hermite polynomials Hn(x),
8.96 Jacobi's polynomials (page 988)
8.97 The Laguerre polynomials (page 990)
9.1 Hypergeometric Functions (page 995)
9.10 Definition (page 995)
9.11 Integral representations (page 995)
9.12 Representation of elementary functions in terms of a hypergeometric functions (page 995)
9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series (page 998)
9.14 A generalized hypergeometric series (page 1000)
9.15 The hypergeometric differential equation (page 1000)
9.16 Riemann's differential equation (page 1004)
9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme (page 1007)
9.18 Hypergeometric functions of two variables (page 1008)
9.19 A hypergeometric function of several variables (page 1012)
9.2 Confluent Hypergeometric Functions (page 1012)
9.20 Introduction (page 1012)
9.21 The functions $\Phi (\alpha ,\gamma ;z)$ and $\Psi (\alpha ,\gamma ;z)$ (page 1013)
9.22-9.23 The Whittaker functions $M_{\lambda ,\mu }{ \left ( z \right ) }$ and $W_{\lambda ,\mu }{ \left ( z \right ) }$ (page 1014)
9.24-9.25 Parabolic cylinder functions Dp(z) (page 1018)
9.26 Confluent hypergeometric series of two variables (page 1021)
9.3 Meijer's G-Function (page 1022)
9.30 Definition (page 1022)
9.31 Functional relations (page 1023)
9.32 A differential equation for the G-function (page 1024)
9.33 Series of G-functions (page 1024)
9.34 Connections with other special functions (page 1024)
9.4 MacRobert's E-Function (page 1025)
9.41 Representation by means of multiple integrals (page 1025)
9.42 Functional relations (page 1025)
9.5 Riemann's Zeta Functions $\zeta (z,q)$, and $\zeta (z)$, and the Functions $\Phi (z,s,v)$ and $\xi (s)$ (page 1026)
9.51 Definition and integral representations (page 1026)
9.52 Representation as a series or as an infinite product (page 1026)
9.53 Functional relations (page 1027)
9.54 Singular points and zeros (page 1028)
9.55 The Lerch function $\Phi (z,s,v)$ (page 1028)
9.56 The function ${\xi \left ( s \right ) }$ (page 1029)
9.6 Bernoulli numbers and polynomials, Euler numbers (page 1030)
9.61 Bernoulli numbers (page 1030)
9.62 Bernoulli polynomials (page 1031)
9.63 Euler numbers (page 1032)
9.64 The functions ${\nu (x)}$, ${\nu (x,\alpha )}$, ${\mu (x,\beta )}$, ${\mu (x,\beta ,\alpha )}$, ${\lambda (x,y)}$ (page 1033)
9.65 Euler polynomials (page 1033)
9.7 Constants (page 1035)
9.71 Bernoulli numbers (page 1035)
9.72 Euler numbers (page 1035)
9.73 Euler's and Catalan's constants (page 1036)
9.74 Stirling numbers (page 1036)

10 Vector Field Theory (page 1039)
10.1-10.8 Vectors, Vector Operators, and Integral Theorems (page 1039)
10.11 Products of vectors (page 1039)
10.12 Properties of scalar product (page 1039)
10.13 Properties of vector product (page 1039)
10.14 Differentiation of vectors (page 1039)
10.21 Operators grad, div, and curl (page 1040)
10.31 Properties of the operator ${\rm\nabla}$ (page 1040)
10.41 Solenoidal fields (page 1041)
10.51-10.61 Orthogonal curvilinear coordinates (page 1042)
10.71-10.72 Vector integral theorems (page 1045)
10.81 Integral rate of change theorems (page 1047)

11 Algebraic Inequalities (page 1049)
11.1-11.3 General Algebraic Inequalities (page 1049)
11.11 Algebraic inequalities involving real numbers (page 1049)
11.21 Algebraic inequalities involving complex numbers (page 1050)
11.31 Inequalities for sets of complex numbers (page 1051)

12 Integral Inequalities (page 1053)
12.11 Mean value theorems (page 1053)
12.111 First mean value theorem. (page 1053)
12.112 Second mean value theorem. (page 1053)
12.113 First mean value theorem for infinite integrals. (page 1053)
12.114 Second mean value theorem for infinite integrals. (page 1054)
12.21 Differentiation of definite integral containing a parameter (page 1054)
12.211 Differentiation when limits are finite. (page 1054)
12.212 Differentiation when a limit is infinite. (page 1054)
12.31 Integral inequalities (page 1054)
12.311 Cauchy-Schwarz-Buniakowsky inequality for integrals. (page 1054)
12.312 Hölder's inequality for integrals. (page 1054)
12.313 Minkowski's inequality for integrals. (page 1055)
12.314 Chebyshev's inequality for integrals. (page 1055)
12.315 Young's inequality for integrals. (page 1055)
12.316 Steffensen's inequality for integrals. (page 1055)
12.317 Gram's inequality for integrals. (page 1055)
12.318 Ostrowski's inequality for integrals. (page 1055)
12.41 Convexity and Jensen's inequality (page 1056)
12.411 Jensen's inequality. (page 1056)
12.51 Fourier series and related inequalities (page 1056)
12.511 Riemann-Lebesgue lemma (page 1056)
12.512 Dirichlet lemma (page 1057)
12.513 Parseval's theorem for trigonometric Fourier series (page 1057)
12.514 Integral representation of the nth partial sum. (page 1057)
12.515 Generalized Fourier series (page 1057)
12.516 Bessel's inequality for generalized Fourier series (page 1057)
12.517 Parseval's theorem for generalized Fourier series. (page 1057)

13 Matrices and related results (page 1059)
13.11-13.12 Special matrices (page 1059)
13.111 Diagonal matrix (page 1059)
13.112 Identity matrix and null matrix (page 1059)
13.113 Reducible and irreducible matrices (page 1059)
13.114 Equivalent matrices (page 1059)
13.115 Transpose of a matrix (page 1059)
13.116 Adjoint matrix (page 1059)
13.117 Inverse matrix (page 1060)
13.118 Trace of a matrix (page 1060)
13.119 Symmetric matrix (page 1060)
13.120 Skew-symmetric matrix (page 1060)
13.121 Triangular matrices (page 1060)
13.122 Orthogonal matrices (page 1060)
13.123 Hermitian transpose of a matrix (page 1060)
13.124 Hermitian matrix (page 1060)
13.125 Unitary matrix (page 1060)
13.126 Eigenvalues and eigenvectors (page 1061)
13.127 Nilpotent matrix (page 1061)
13.128 Idempotent matrix (page 1061)
13.129 Positive definite (page 1061)
13.130 Non-negative definite (page 1061)
13.131 Diagonally dominant (page 1061)
13.21 Quadratic forms (page 1061)
13.211 Sylvester's law of inertia (page 1062)
13.212 Rank (page 1062)
13.213 Signature (page 1062)
13.214 Positive definite and semidefinite quadratic form (page 1062)
13.215 Basic theorems on quadratic forms (page 1062)
13.31 Differentiation of matrices (page 1063)
13.41 The matrix exponential (page 1064)
3.411 Basic properties (page 1064)

14 Determinants (page 1065)
14.11 Expansion of second- and third-order determinants (page 1065)
14.12 Basic properties (page 1065)
14.13 Minors and cofactors of a determinant (page 1065)
14.14 Principal minors (page 1066)
14.15 Laplace expansion of a determinant (page 1066)
14.16 Jacobi's theorem (page 1066)
14.17 Hadamard's theorem (page 1066)
14.18 Hadamard's inequality (page 1067)
14.21 Cramer's rule (page 1067)
14.31 Some special determinants (page 1068)
14.311 Vandermonde's determinant (alternant). (page 1068)
14.312 Circulants. (page 1068)
14.313 Jacobian determinant. (page 1068)
14.314 Hessian determinants. (page 1069)
14.315 Wronskian determinants. (page 1069)
14.316 Properties. (page 1069)
14.317 Gram-Kowalewski theorem on linear dependence. (page 1070)

15 Norms (page 1071)
15.1-15.9 Vector Norms (page 1071)
15.11 General properties (page 1071)
15.21 Principal vector norms (page 1071)
15.211 The norm $\vert\vert{\bf x}\vert\vert _{1}$ (page 1071)
15.212 The norm $\vert\vert{\bf x}\vert\vert _{2}$ (Euclidean or L2 norm) (page 1071)
15.213 The norm $\vert\vert{\bf x}\vert\vert _{\infty }$ (page 1071)
15.31 Matrix norms (page 1072)
15.311 General properties (page 1072)
15.312 Induced norms (page 1072)
15.313 Natural norm of unit matrix (page 1072)
15.41 Principal natural norms (page 1072)
15.411 Maximum absolute column sum norm (page 1072)
15.412 Spectral norm (page 1072)
15.413 Maximum absolute row sum norm (page 1072)
15.51 Spectral radius of a square matrix (page 1073)
15.511 Inequalities concerning matrix norms and the spectral radius (page 1073)
15.512 Deductions from Gerschgorin's theorem (see 15.814) (page 1073)
15.61 Inequalities involving eigenvalues of matrices (page 1074)
15.611 Cayley-Hamilton theorem (page 1074)
15.612 Corollaries (page 1074)
15.71 Inequalities for the characteristic polynomial (page 1074)
15.711 Named and unnamed inequalities (page 1075)
15.712 Parodi's theorem (page 1076)
15.713 Corollary of Brauer's theorem (page 1076)
15.714 Ballieu's theorem (page 1076)
15.715 Routh-Hurwitz theorem (page 1076)
15.81-15.82 Named theorems on eigenvalues (page 1076)
15.811 Schur's inequalities (page 1077)
15.812 Sturmian separation theorem (page 1077)
15.813 Poincare's separation theorem (page 1077)
15.814 Gerschgorin's theorem (page 1078)
15.815 Brauer's theorem (page 1078)
15.816 Perron's theorem (page 1078)
15.817 Frobenius theorem (page 1078)
15.818 Perron-Frobenius theorem (page 1078)
15.819 Wielandt's theorem (page 1078)
15.820 Ostrowski's theorem (page 1079)
15.821 First theorem due to Lyapunov (page 1079)
15.822 Second theorem due to Lyapunov (page 1079)
15.823 Hermitian matrices and diophantine relations involving circular functions of rational angles due to Calogero and Perelomov (page 1079)
15.91 Variational principles (page 1081)
15.911 Rayleigh quotient (page 1081)
15.912 Basic theorems (page 1081)

16 Ordinary differential equations (page 1083)
16.1-16.9 Results relating to the solution of ordinary differential equations (page 1083)
16.11 First-order equations (page 1083)
16.111 Solution of a first-order equation (page 1083)
16.112 Cauchy problem (page 1083)
16.113 Approximate solution to an equation (page 1083)
16.114 Lipschitz continuity of a function (page 1084)
16.21 Fundamental inequalities and related results (page 1084)
16.211 Gronwall's lemma (page 1084)
16.212 Comparison of approximate solutions of a differential equation (page 1084)
16.31 First-order systems (page 1085)
16.311 Solution of a system of equations (page 1085)
16.312 Cauchy problem for a system (page 1085)
16.313 Approximate solution to a system (page 1085)
16.314 Lipschitz continuity of a vector (page 1085)
16.315 Comparison of approximate solutions of a system (page 1086)
16.316 First-order linear differential equation (page 1086)
16.317 Linear systems of differential equations (page 1086)
16.41 Some special types of elementary differential equations (page 1087)
16.411 Variables separable (page 1087)
16.412 Exact differential equations (page 1087)
16.413 Conditions for an exact equation (page 1087)
16.414 Homogeneous differential equations (page 1087)
16.51 Second-order equations (page 1088)
16.511 Adjoint and self-adjoint equations (page 1088)
16.512 Abel's identity (page 1088)
16.513 Lagrange identity (page 1089)
16.514 The Riccati equation (page 1089)
16.515 Solutions of the Riccati equation (page 1089)
16.516 Solution of a second-order linear differential equation (page 1090)
16.61-16.62 Oscillation and non-oscillation theorems for second-order equations (page 1090)
16.611 First basic comparison theorem (page 1090)
16.622 Second basic comparison theorem (page 1091)
16.623 Interlacing of zeros (page 1091)
16.624 Sturm separation theorem (page 1091)
16.625 Sturm comparison theorem (page 1091)
16.626 Szegö's comparison theorem (page 1091)
16.627 Picone's identity (page 1092)
16.628 Sturm-Picone theorem (page 1092)
16.629 Oscillation on the half line (page 1092)
16.71 Two related comparison theorems (page 1093)
16.711 Theorem 1 (page 1093)
16.712 Theorem 2 (page 1093)
16.81-16.82 Non-oscillatory solutions (page 1093)
16.811 Kneser's non-oscillation theorem (page 1094)
16.822 Comparison theorem for non-oscillation (page 1094)
16.823 Necessary and sufficient conditions for non-oscillation (page 1094)
16.91 Some growth estimates for solutions of second-order equations (page 1094)
16.911 Strictly increasing and decreasing solutions (page 1094)
16.912 General result on dominant and subdominant solutions (page 1095)
16.913 Estimate of dominant solution (page 1095)
16.914 A theorem due to Lyapunov (page 1096)
16.92 Boundedness theorems (page 1096)
16.921 All solutions of the equation ... (page 1096)
16.922 If all solutions of the equation ... (page 1096)
16.923 If $a(x)\rightarrow \infty $ monotonically as $x\rightarrow \infty $, then all solutions of ... (page 1096)
16.924 Consider the equation ... (page 1096)
16.93 Growth of maxima of |y| (page 1097)

17 Fourier, Laplace, and Mellin Transforms (page 1099)
17.1- 17.4 Integral Transforms (page 1099)
17.11 Laplace transform (page 1099)
17.12 Basic properties of the Laplace transform (page 1099)
17.13 Table of Laplace transform pairs (page 1100)
17.21 Fourier transform (page 1109)
17.22 Basic properties of the Fourier transform (page 1110)
17.23 Table of Fourier transform pairs (page 1110)
17.24 Table of Fourier transform pairs for spherically symmetric functions (page 1112)
17.31 Fourier sine and cosine transforms (page 1113)
17.32 Basic properties of the Fourier sine and cosine transforms (page 1113)
17.33 Table of Fourier sine transforms (page 1114)
17.34 Table of Fourier cosine transforms (page 1118)
17.35 Relationships between transforms (page 1121)
17.41 Mellin transform (page 1121)
17.42 Basic properties of the Mellin transform (page 1122)
17.43 Table of Mellin cosine transforms (page 1122)

18 The z-transform (page 1127)
18.1-18.3 Definition, Bilateral, and Unilateral z-Transforms (page 1127)
18.1 Definitions (page 1127)
18.2 Bilateral z-transform (page 1127)
18.3 Unilateral z-transform (page 1129)

References (page 1133)
Supplemental references (page 1137)
Function and constant index (page 1143)
General index (page 1153)