Zwillinger Table of Integrals, Series, and Products

8. Auflage 2014
ISBN: 978-0-12-384934-2
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 1184 Seiten

The eighth edition of the classic Gradshteyn and Ryzhik is an updated completely revised edition of what is acknowledged universally by mathematical and applied science users as the key reference work concerning the integrals and special functions. The book is valued by users of previous editions of the work both for its comprehensive coverage of integrals and special functions, and also for its accuracy and valuable updates. Since the first edition, published in 1965, the mathematical content of this book has significantly increased due to the addition of new material, though the size of the book has remained almost unchanged. The new 8th edition contains entirely new results and amendments to the auxiliary conditions that accompany integrals and wherever possible most entries contain valuable references to their source. - Over 10, 000 mathematical entries - Most up to date listing of integrals, series and products (special functions) - Provides accuracy and efficiency in industry work - 25% of new material not including changes to the restrictions on results that revise the range of validity of results, which lend to approximately 35% of new updates

Autoren/Hrsg.

Zwillinger, Daniel

Inhaltsverzeichnis

1;Front Cover;1
2;Table of Integrals, Series, and Products;4
3;Copyright;5
4;Contents;6
5;Preface to the Eighth Edition;18
6;Acknowledgments;20
7;The Order of Presentation of the Formulas;26
8;Use of the Tables;30
8.1;Bernoulli and Euler Polynomials and Numbers;31
8.2;Elliptic Functions and Elliptic Integrals;32
8.3;The Jacobi Zeta Function and Theta Functions;33
8.4;Exponential and Related Integrals;34
8.5;Hermite and Chebyshev Orthogonal Polynomials;35
8.6;Bessel Functions;36
8.7;Parabolic Cylinder Functions and Whittaker Functions;37
8.8;Mathieu Functions;37
9;Index of Special Functions;38
10;Notation;42
11;Note on the Bibliographic References;46
12;0 Introduction;48
12.1;0.1 Finite sums;48
12.1.1;0.11 Progressions;48
12.1.2;0.12 Sums of powers of natural numbers;48
12.1.3;0.13 Sums of reciprocals of natural numbers;50
12.1.4;0.14 Sums of products of reciprocals of natural numbers;50
12.1.5;0.15 Sums of the binomial coefficients;51
12.2;0.2 Numerical series and infinite products;54
12.2.1;0.21 The convergence of numerical series;54
12.2.2;0.22 Convergence tests;54
12.2.3;0.23–0.24 Examples of numerical series;56
12.2.4;0.25 Infinite products;62
12.2.5;0.26 Examples of infinite products;62
12.3;0.3 Functional series;63
12.3.1;0.30 Definitions and theorems;63
12.3.2;0.31 Power series;64
12.3.3;0.32 Fourier series;67
12.3.4;0.33 Asymptotic series;68
12.4;0.4 Certain formulas from differential calculus;69
12.4.1;0.41 Differentiation of a definite integral with respect to a parameter;69
12.4.2;0.42 The nth derivative of a product (Leibniz's rule);69
12.4.3;0.43 The nth derivative of a composite function;70
12.4.4;0.44 Integration by substitution;71
13;1 Elementary Functions;72
13.1;1.1 Power of Binomials;72
13.1.1;1.11 Power series;72
13.1.2;1.12 Series of rational fractions;73
13.2;1.2 The Exponential Function;73
13.2.1;1.21 Series representation;73
13.2.2;1.22 Functional relations;74
13.2.3;1.23 Series of exponentials;74
13.3;1.3–1.4 Trigonometric and Hyperbolic Functions;75
13.3.1;1.30 Introduction;75
13.3.2;1.31 The basic functional relations;75
13.3.3;1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argu ...;78
13.3.4;1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers...;80
13.3.5;1.34 Certain sums of trigonometric and hyperbolic functions;83
13.3.6;1.35 Sums of powers of trigonometric functions of multiple angles;84
13.3.7;1.36 Sums of products of trigonometric functions of multiple angles;85
13.3.8;1.37 Sums of tangents of multiple angles;86
13.3.9;1.38 Sums leading to hyperbolic tangents and cotangents;86
13.3.10;1.39 The representation of cosines and sines of multiples of the angle as finite products;87
13.3.11;1.41 The expansion of trigonometric and hyperbolic functions in power series;89
13.3.12;1.42 Expansion in series of simple fractions;90
13.3.13;1.43 Representation in the form of an infinite product;91
13.3.14;1.44–1.45 Trigonometric (Fourier) series;92
13.3.15;1.46 Series of products of exponential and trigonometric functions;97
13.3.16;1.47 Series of hyperbolic functions;98
13.3.17;1.48 Lobachevskiy's ``Angle of parallelism'' .(x);98
13.3.18;1.49 The hyperbolic amplitude (the Gudermannian) gd x;99
13.4;1.5 The Logarithm;99
13.4.1;1.51 Series representation;99
13.4.2;1.52 Series of logarithms (cf. 1.431);102
13.5;1.6 The Inverse Trigonometric and Hyperbolic Functions;102
13.5.1;1.61 The domain of definition;102
13.5.2;1.62–1.63 Functional relations;103
13.5.3;1.64 Series representations;107
14;2 Indefinite Integrals of Elementary Functions;110
14.1;2.0 Introduction;110
14.1.1;2.00 General remarks;110
14.1.2;2.01 The basic integrals;111
14.1.3;2.02 General formulas;112
14.2;2.1 Rational Functions;113
14.2.1;2.10 General integration rules;113
14.2.2;2.11–2.13 Forms containing the binomial a+bxk;115
14.2.3;2.14 Forms containing the binomial 1 ± xn;121
14.2.4;2.15 Forms containing pairs of binomials: a+bx and a+ßx;125
14.2.5;2.16 Forms containing the trinomial a+bxk+c x2k;125
14.2.6;2.17 Forms containing the quadratic trinomial a+bx+cx2 and powers of x;126
14.2.7;2.18 Forms containing the quadratic trinomial a+bx+cx2 and the binomial a+ßx;128
14.3;2.2 Algebraic functions;129
14.3.1;2.20 Introduction;129
14.3.2;2.21 Forms containing the binomial a+bxk and vx;130
14.3.3;2.22–2.23 Forms containing n(a + bx)k;131
14.3.3.1;The square root;131
14.3.3.2;Cube root;133
14.3.4;2.24 Forms containing a+bx and the binomial a+ßx;135
14.3.5;2.25 Forms containing a+bx+cx2;139
14.3.5.1;Integration techniques;139
14.3.6;2.26 Forms containing a+bx+cx2 and integral powers of x;141
14.3.7;2.2712 Forms containing a+c x2 and integral powers of x;146
14.3.8;2.28 Forms containing a+bx+c x2 and first-and second-degree polynomials;150
14.3.9;2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals;151
14.4;2.3 The Exponential Function;153
14.4.1;2.31 Forms containing eax;153
14.4.2;2.32 The exponential combined with rational functions of x;153
14.5;2.4 Hyperbolic Functions;157
14.5.1;2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x;157
14.5.1.1;Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument;167
14.5.2;2.44–2.45 Rational functions of hyperbolic functions;172
14.5.3;2.46 Algebraic functions of hyperbolic functions;179
14.5.4;2.47 Combinations of hyperbolic functions and powers;187
14.5.5;2.48 Combinations of hyperbolic functions, exponentials, and powers;196
14.6;2.5–2.6 Trigonometric Functions;198
14.6.1;2.50 Introduction;198
14.6.2;2.51–2.52 Powers of trigonometric functions;199
14.6.3;2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated functions of the argument;208
14.6.4;2.55–2.56 Rational functions of the sine and cosine;218
14.6.5;2.57 Integrals containing a ± b sin x or a ± b cos x;226
14.6.6;2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals;231
14.6.7;2.63–2.65 Products of trigonometric functions and powers;261
14.6.8;2.66 Combinations of trigonometric functions and exponentials;274
14.6.9;2.67 Combinations of trigonometric and hyperbolic functions;278
14.7;2.7 Logarithms and Inverse-Hyperbolic Functions;284
14.7.1;2.71 The logarithm;284
14.7.2;2.72–2.73 Combinations of logarithms and algebraic functions;285
14.7.3;2.74 Inverse hyperbolic functions;289
14.7.4;2.75 Logarithms and exponential functions;289
14.8;2.8 Inverse Trigonometric Functions;289
14.8.1;2.81 Arcsines and arccosines;289
14.8.2;2.82 The arcsecant, the arccosecant, the arctangent and the arccotangent;290
14.8.3;2.83 Combinations of arcsine or arccosine and algebraic functions;291
14.8.4;2.84 Combinations of the arcsecant and arccosecant with powers of x;292
14.8.5;2.85 Combinations of the arctangent and arccotangent with algebraic functions;293
15;3–4 Definite Integrals of Elementary Functions;296
15.1;3.0 Introduction;296
15.1.1;3.01 Theorems of a general nature;296
15.1.2;3.02 Change of variable in a definite integral;297
15.1.3;3.03 General formulas;298
15.1.4;3.04 Improper integrals;300
15.1.5;3.05 The principal values of improper integrals;301
15.2;3.1–3.2 Power and Algebraic Functions;302
15.2.1;3.11 Rational functions;302
15.2.2;3.12 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials;303
15.2.3;3.13–3.17 Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with ration;303
15.2.4;3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions;362
15.2.5;3.19–3.23 Combinations of powers of x and powers of binomials of the form (a+ßx);365
15.2.6;3.24–3.27 Powers of x, of binomials of the form a+ßxp and of polynomials in x;371
15.3;3.3–3.4 Exponential Functions;383
15.3.1;3.31 Exponential functions;383
15.3.2;3.32–3.34 Exponentials of more complicated arguments;385
15.3.3;3.35 Combinations of exponentials and rational functions;389
15.3.4;3.36–3.37 Combinations of exponentials and algebraic functions;393
15.3.5;3.38–3.39 Combinations of exponentials and arbitrary powers;395
15.3.6;3.41–3.44 Combinations of rational functions of powers and exponentials;402
15.3.7;3.45 Combinations of powers and algebraic functions of exponentials;412
15.3.8;3.46–3.48 Combinations of exponentials of more complicated arguments and powers;413
15.4;3.5 Hyperbolic Functions;421
15.4.1;3.51 Hyperbolic functions;421
15.4.2;3.52–3.53 Combinations of hyperbolic functions and algebraic functions;424
15.4.3;3.54 Combinations of hyperbolic functions and exponentials;431
15.4.4;3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers;435
15.5;3.6–4.1 Trigonometric Functions;439
15.5.1;3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles;439
15.5.2;3.62 Powers of trigonometric functions;444
15.5.3;3.63 Powers of trigonometric functions and trigonometric functions of linear functions;446
15.5.4;3.64–3.65 Powers and rational functions of trigonometric functions;450
15.5.5;3.66 Forms containing powers of linear functions of trigonometric functions;454
15.5.6;3.67 Square roots of expressions containing trigonometric functions;457
15.5.7;3.68 Various forms of powers of trigonometric functions;462
15.5.8;3.69–3.71 Trigonometric functions of more complicated arguments;466
15.5.9;3.72–3.74 Combinations of trigonometric and rational functions;474
15.5.10;3.75 Combinations of trigonometric and algebraic functions;485
15.5.11;3.76–3.77 Combinations of trigonometric functions and powers;487
15.5.12;3.78–3.81 Rational functions of x and of trigonometric functions;498
15.5.13;3.82–3.83 Powers of trigonometric functions combined with other powers;510
15.5.14;3.84 Integrals containing 1 k2 sin2 x, 1 k2 cos2 x, and similar expressions;523
15.5.15;3.85–3.88 Trigonometric functions of more complicated arguments combined with powers;526
15.5.16;3.89–3.91 Trigonometric functions and exponentials;536
15.5.17;3.92 Trigonometric functions of more complicated arguments combined with exponentials;544
15.5.18;3.93 Trigonometric and exponential functions of trigonometric functions;546
15.5.19;3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers;548
15.5.20;3.98–3.99 Combinations of trigonometric and hyperbolic functions;560
15.5.21;4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers;567
15.5.22;4.13 Combinations of trigonometric and hyperbolic functions and exponentials;573
15.5.23;4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers;576
15.6;4.2–4.4 Logarithmic Functions;577
15.6.1;4.21 Logarithmic functions;577
15.6.2;4.22 Logarithms of more complicated arguments;580
15.6.3;4.23 Combinations of logarithms and rational functions;585
15.6.4;4.24 Combinations of logarithms and algebraic functions;588
15.6.5;4.25 Combinations of logarithms and powers;590
15.6.6;4.26-4.27 Combinations involving powers of the logarithm and other powers;593
15.6.7;4.28 Combinations of rational functions of ln x and powers;603
15.6.8;4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers;606
15.6.9;4.33–4.34 Combinations of logarithms and exponentials;622
15.6.10;4.35–4.36 Combinations of logarithms, exponentials, and powers;624
15.6.11;4.37 Combinations of logarithms and hyperbolic functions;629
15.6.12;4.38–4.41 Logarithms and trigonometric functions;631
15.6.13;4.42–4.43 Combinations of logarithms, trigonometric functions, and powers;644
15.6.14;4.44 Combinations of logarithms, trigonometric functions, and exponentials;650
15.7;4.5 Inverse Trigonometric Functions;650
15.7.1;4.51 Inverse trigonometric functions;650
15.7.2;4.52 Combinations of arcsines, arccosines, and powers;650
15.7.3;4.53–4.54 Combinations of arctangents, arccotangents, and powers;652
15.7.4;4.55 Combinations of inverse trigonometric functions and exponentials;655
15.7.5;4.56 A combination of the arctangent and a hyperbolic function;656
15.7.6;4.57 Combinations of inverse and direct trigonometric functions;656
15.7.7;4.58 A combination involving an inverse and a direct trigonometric function and a power;657
15.7.8;4.59 Combinations of inverse trigonometric functions and logarithms;657
15.8;4.6 Multiple Integrals;658
15.8.1;4.60 Change of variables in multiple integrals;658
15.8.2;4.61 Change of the order of integration and change of variables;659
15.8.3;4.62 Double and triple integrals with constant limits;661
15.8.4;4.63–4.64 Multiple integrals;664
16;5 Indefinite Integrals of Special Functions;670
16.1;5.1 Elliptic Integrals and Functions;670
16.1.1;5.11 Complete elliptic integrals;670
16.1.2;5.12 Elliptic integrals;672
16.1.3;5.13 Jacobian elliptic functions;674
16.1.4;5.14 Weierstrass elliptic functions;677
16.2;5.2 The Exponential Integral Function;678
16.2.1;5.21 The exponential integral function;678
16.2.2;5.22 Combinations of the exponential integral function and powers;678
16.2.3;5.23 Combinations of the exponential integral and the exponential;679
16.3;5.3 The Sine Integral and the Cosine Integral;679
16.4;5.4 The Probability Integral and Fresnel Integrals;680
16.5;5.5 Bessel Functions;680
16.6;5.6 Orthogonal Polynomials;681
16.7;5.7 Hypergeometric Functions;681
17;6–7 Definite Integrals of Special Functions;684
17.1;6.1 Elliptic Integrals and Functions;684
17.1.1;6.11 Forms containing F(x, k);684
17.1.2;6.12 Forms containing E(x, k);685
17.1.3;6.13 Integration of elliptic integrals with respect to the modulus;686
17.1.4;6.14–6.15 Complete elliptic integrals;686
17.1.5;6.16 The theta function;689
17.1.6;6.17 Generalized elliptic integrals;690
17.2;6.2–6.3 The Exponential Integral Function and Functions Generated by It;691
17.2.1;6.21 The logarithm integral;691
17.2.2;6.22–6.23 The exponential integral function;693
17.2.3;6.24–6.26 The sine integral and cosine integral functions;695
17.2.4;6.27 The hyperbolic sine integral and hyperbolic cosine integral functions;700
17.2.5;6.28–6.31 The probability integral;700
17.2.6;6.32 Fresnel integrals;704
17.3;6.4 The Gamma Function and Functions Generated by It;706
17.3.1;6.41 The gamma function;706
17.3.2;6.42 Combinations of the gamma function, the exponential, and powers;707
17.3.3;6.43 Combinations of the gamma function and trigonometric functions;710
17.3.4;6.44 The logarithm of the gamma function;711
17.3.5;6.45 The incomplete gamma function;712
17.3.6;6.46–6.47 The function .(x);713
17.4;6.5–6.7 Bessel Functions;714
17.4.1;6.51 Bessel functions;714
17.4.2;6.52 Bessel functions combined with x and x2;719
17.4.3;6.53–6.54 Combinations of Bessel functions and rational functions;725
17.4.4;6.55 Combinations of Bessel functions and algebraic functions;729
17.4.5;6.56–6.58 Combinations of Bessel functions and powers;730
17.4.6;6.59 Combinations of powers and Bessel functions of more complicated arguments;744
17.4.7;6.61 Combinations of Bessel functions and exponentials;749
17.4.8;6.62–6.63 Combinations of Bessel functions, exponentials, and powers;753
17.4.9;6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers;763
17.4.10;6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers;765
17.4.11;6.66 Combinations of Bessel, hyperbolic, and exponential functions;767
17.4.11.1;Bessel and hyperbolic functions;767
17.4.11.2;Bessel, hyperbolic, and algebraic functions;769
17.4.11.3;Exponential, hyperbolic, and Bessel functions;770
17.4.12;6.67–6.68 Combinations of Bessel and trigonometric functions;771
17.4.13;6.69–6.74 Combinations of Bessel and trigonometric functions and powers;781
17.4.14;6.75 Combinations of Bessel, trigonometric, and exponential functions and powers;797
17.4.15;6.76 Combinations of Bessel, trigonometric, and hyperbolic functions;800
17.4.16;6.77 Combinations of Bessel functions and the logarithm, or arctangent;801
17.4.17;6.78 Combinations of Bessel and other special functions;802
17.4.18;6.79 Integration of Bessel functions with respect to the order;803
17.5;6.8 Functions Generated by Bessel Functions;807
17.5.1;6.81 Struve functions;807
17.5.2;6.82 Combinations of Struve functions, exponentials, and powers;808
17.5.3;6.83 Combinations of Struve and trigonometric functions;809
17.5.4;6.84–6.85 Combinations of Struve and Bessel functions;810
17.5.5;6.86 Lommel functions;814
17.5.6;6.87 Thomson functions;815
17.6;6.9 Mathieu Functions;817
17.6.1;6.91 Mathieu functions;817
17.6.2;6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions;817
17.6.3;6.93 Combinations of Mathieu and Bessel functions;821
17.6.4;6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems;821
17.7;7.1–7.2 Associated Legendre Functions;823
17.7.1;7.11 Associated Legendre functions;823
17.7.2;7.12–7.13 Combinations of associated Legendre functions and powers;824
17.7.3;7.14 Combinations of associated Legendre functions, exponentials, and powers;830
17.7.4;7.15 Combinations of associated Legendre and hyperbolic functions;832
17.7.5;7.16 Combinations of associated Legendre functions, powers, and trigonometric functions;833
17.7.6;7.17 A combination of an associated Legendre function and the probability integral;835
17.7.7;7.18 Combinations of associated Legendre and Bessel functions;836
17.7.8;7.19 Combinations of associated Legendre functions and functions generated by Bessel functions;841
17.7.9;7.21 Integration of associated Legendre functions with respect to the order;842
17.7.10;7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions;843
17.7.11;7.23 Combinations of Legendre polynomials and powers;845
17.7.12;7.24 Combinations of Legendre polynomials and other elementary functions;846
17.7.13;7.25 Combinations of Legendre polynomials and Bessel functions;848
17.8;7.3–7.4 Orthogonal Polynomials;849
17.8.1;7.31 Combinations of Gegenbauer polynomials C.n(x) and powers;849
17.8.2;7.32 Combinations of Gegenbauer polynomials Cvn(x) and elementary functions;852
17.9;7.325* Complete System of Orthogonal Step Functions;852
17.9.1;7.33 Combinations of the polynomials Cvn(x) and Bessel functions. Integration of Gegenbauer functions with respect to the ... ;853
17.9.1.1;Integration of Gegenbauer functions with respect to the index;854
17.9.2;7.34 Combinations of Chebyshev polynomials and powers;854
17.9.3;7.35 Combinations of Chebyshev polynomials and elementary functions;856
17.9.4;7.36 Combinations of Chebyshev polynomials and Bessel functions;857
17.9.5;7.37–7.38 Hermite polynomials;857
17.9.6;7.39 Jacobi polynomials;861
17.9.7;7.41–7.42 Laguerre polynomials;863
17.10;7.5 Hypergeometric Functions;867
17.10.1;7.51 Combinations of hypergeometric functions and powers;867
17.10.2;7.52 Combinations of hypergeometric functions and exponentials;869
17.10.3;7.53 Hypergeometric and trigonometric functions;872
17.10.4;7.54 Combinations of hypergeometric and Bessel functions;872
17.11;7.6 Confluent Hypergeometric Functions;875
17.11.1;7.61 Combinations of confluent hypergeometric functions and powers;875
17.11.2;7.62–7.63 Combinations of confluent hypergeometric functions and exponentials;877
17.11.3;7.64 Combinations of confluent hypergeometric and trigonometric functions;884
17.11.4;7.65 Combinations of confluent hypergeometric functions and Bessel functions;885
17.11.5;7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers;886
17.11.6;7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers;889
17.11.6.1;Combinations of Struve functions and confluent hypergeometric functions;893
17.11.7;7.68 Combinations of confluent hypergeometric functions and other special functions;894
17.11.7.1;Combinations of confluent hypergeometric functions and associated Legendre functions;894
17.11.7.2;A combination of confluent hypergeometric functions and orthogonal polynomials;895
17.11.7.3;A combination of hypergeometric and confluent hypergeometric functions;896
17.11.8;7.69 Integration of confluent hypergeometric functions with respect to the index;896
17.12;7.7 Parabolic Cylinder Functions;896
17.12.1;7.71 Parabolic cylinder functions;896
17.12.2;7.72 Combinations of parabolic cylinder functions, powers, and exponentials;897
17.12.3;7.73 Combinations of parabolic cylinder and hyperbolic functions;898
17.12.4;7.74 Combinations of parabolic cylinder and trigonometric functions;899
17.12.5;7.75 Combinations of parabolic cylinder and Bessel functions;900
17.12.5.1;Combinations of parabolic cylinder and Struve functions;903
17.12.6;7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions;904
17.12.7;7.77 Integration of a parabolic cylinder function with respect to the index;904
17.13;7.8 Meijer's and MacRobert's Functions (G and E);905
17.13.1;7.81 Combinations of the functions G and E and the elementary functions;905
17.13.2;7.82 Combinations of the functions G and E and Bessel functions;909
17.13.3;7.83 Combinations of the functions G and E and other special functions;911
18;8–9 Special Functions;914
18.1;8.1 Elliptic Integrals and Functions;914
18.1.1;8.11 Elliptic integrals;914
18.1.1.1;Series representations;916
18.1.1.2;Trigonometric series;918
18.1.2;8.12 Functional relations between elliptic integrals;919
18.1.3;8.13 Elliptic functions;921
18.1.4;8.14 Jacobian elliptic functions;922
18.1.5;8.15 Properties of Jacobian elliptic functions and functional relationships between them;926
18.1.5.1;Functional relations;928
18.1.6;8.16 The Weierstrass function Ã(u);929
18.1.7;8.17 The functions .(u) and s(u);932
18.1.7.1;Functional relations and properties;932
18.1.8;8.18–8.19 Theta functions;933
18.1.8.1;Functional relations and properties;934
18.1.8.2;q-series and products, q = exp (-pK' K) ;936
18.2;8.2 The Exponential Integral Function and Functions Generated by It;939
18.2.1;8.21 The exponential integral function Ei(x);939
18.2.1.1;Series and asymptotic representations;940
18.2.2;The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x;942
18.2.3;8.23 The sine integral and the cosine integral: si x and ci x;942
18.2.4;8.24 The logarithm integral li(x);943
18.2.4.1;Integral representations;943
18.2.5;8.25 The probability integral F(x), the Fresnel integrals S(x), C(x), the error function erf(x), and the complementary err ...;943
18.2.5.1;Integral representations;944
18.2.5.2;Asymptotic representations;945
18.2.6;8.26 Lobachevskiy's function L(x);947
18.3;8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them;948
18.3.1;8.31 The gamma function (Euler's integral of the second kind): G(z);948
18.3.1.1;Integral representations;948
18.3.2;8.32 Representation of the gamma function as series and products;950
18.3.2.1;Infinite-product representation;950
18.3.3;8.33 Functional relations involving the gamma function;951
18.3.3.1;Special cases;952
18.3.3.2;Particular values;953
18.3.4;8.34 The logarithm of the gamma function;954
18.3.5;8.35 The incomplete gamma function;955
18.3.6;8.36 The psi function .(x);958
18.3.6.1;Series representation;959
18.3.6.2;Infinite-product representation;960
18.3.7;8.37 The function ß(x);962
18.3.7.1;Series representation;963
18.3.7.2;Functional relations;963
18.3.8;8.38 The beta function (Euler's integral of the first kind): B(x,y);964
18.3.8.1;Integral representation;964
18.3.8.2;Series representation;965
18.3.9;8.39 The incomplete beta function Bx(p,q);966
18.4;8.4–8.5 Bessel Functions and Functions Associated with Them;966
18.4.1;8.40 Definitions;966
18.4.1.1;Modified Bessel functions of imaginary argument I .(z) and K.(z);967
18.4.2;8.41 Integral representations of the functions J.(z) and N.(z);968
18.4.3;8.42 Integral representations of the functions H(1).(z) and H(2).(z);970
18.4.4;8.43 Integral representations of the functions I.(z) and K.(z);972
18.4.4.1;The function I .(z);972
18.4.4.2;The function K.(z);973
18.4.5;8.44 Series representation;974
18.4.5.1;The function J.(z);974
18.4.5.2;The function Y.(z);974
18.4.5.3;The functions I .(z) and Kn(z);975
18.4.6;8.45 Asymptotic expansions of Bessel functions;976
18.4.6.1;“Approximation by tangents”;977
18.4.7;8.46 Bessel functions of order equal to an integer plus one-half;980
18.4.7.1;The function J.(z);980
18.4.7.2;The function Yn+12(z);981
18.4.7.3;The functions H(1,2)n-1/2(z),In+1/2(z), Kn+1/2(z);981
18.4.8;8.47–8.48 Functional relations;982
18.4.8.1;Relations between Bessel functions of the first, second, and third kinds;984
18.4.9;8.49 Differential equations leading to Bessel functions;987
18.4.10;8.51–8.52 Series of Bessel functions;989
18.4.10.1;The series SJk(z);990
18.4.10.2;The series Sak Jk(kx) and Sak Jk'(kx);991
18.4.10.3;The series Sak J0(kx);992
18.4.10.4;The series Sak Z0(kx) sinkx and Sak Z0(kx) cos kx;993
18.4.11;8.53 Expansion in products of Bessel functions;995
18.4.12;8.54 The zeros of Bessel functions;997
18.4.13;8.55 Struve functions;998
18.4.14;8.56 Thomson functions and their generalizations;1000
18.4.14.1;Series representation;1000
18.4.14.2;Asymptotic representation;1000
18.4.15;8.57 Lommel functions;1001
18.4.15.1;Integral representations;1002
18.4.15.2;Definition;1003
18.4.16;8.58 Anger and Weber functions J.(z) and E.(z);1004
18.4.17;8.59 Neumann’s and Schläfli's polynomials: On(z) and Sn(z);1005
18.5;8.6 Mathieu Functions;1006
18.5.1;8.60 Mathieu's equation;1006
18.5.2;8.61 Periodic Mathieu functions;1007
18.5.3;8.62 Recursion relations for the coefficients A (2n)2r, A (2n+1)2r+1, B (2n+1)2r+1, B (2n+2)2r+2;1007
18.5.4;8.63 Mathieu functions with a purely imaginary argument;1008
18.5.5;8.64 Non-periodic solutions of Mathieu's equation;1009
18.5.6;8.65 Mathieu functions for negative q;1009
18.5.7;8.66 Representation of Mathieu functions as series of Bessel functions;1010
18.5.8;8.67 The general theory;1013
18.6;8.7–8.8 Associated Legendre Functions;1014
18.6.1;8.70 Introduction;1014
18.6.2;8.71 Integral representations;1016
18.6.3;8.72 Asymptotic series for large values of |.|;1018
18.6.4;8.73–8.74 Functional relations;1020
18.6.5;8.75 Special cases and particular values;1024
18.6.5.1;Special values of the indices;1024
18.6.5.2;Special values of Legendre functions;1025
18.6.6;8.76 Derivatives with respect to the order;1025
18.6.7;8.77 Series representation;1026
18.6.7.1;The analytic continuation for |z| >>1;1026
18.6.8;8.78 The zeros of associated Legendre functions;1028
18.6.9;8.79 Series of associated Legendre functions;1028
18.6.9.1;Addition theorems;1029
18.6.10;8.81 Associated Legendre functions with integer indices;1030
18.6.10.1;Functional relations;1031
18.6.11;8.82–8.83 Legendre functions;1031
18.6.11.1;Integral representations;1032
18.6.11.1.1;Special cases and particular values;1033
18.6.11.1.2;Functional relationships;1034
18.6.12;8.84 Conical functions;1036
18.6.12.1;Functional relations;1036
18.6.13;8.85 Toroidal functions;1037
18.7;8.9 Orthogonal Polynomials;1038
18.7.1;8.90 Introduction;1038
18.7.2;8.91 Legendre polynomials;1039
18.7.2.1;Functional relations;1041
18.7.3;8.91910 Series of products of Legendre and Chebyshev polynomials;1044
18.7.4;8.92 Series of Legendre polynomials;1044
18.7.5;8.93 Gegenbauer polynomials C.n(t);1046
18.7.5.1;Functional relati;1047
18.7.6;8.94 The Chebyshev polynomials Tn(x) and Un(x);1049
18.7.6.1;Functional relations;1050
18.7.7;8.95 The Hermite polynomials Hn(x);1052
18.7.7.1;Functional relations;1052
18.7.7.2;Series of Hermite polynomials;1053
18.7.8;8.96 Jacobi's polynomials;1054
18.7.9;8.97 The Laguerre polynomials;1056
18.8;9.1 Hypergeometric Functions;1061
18.8.1;9.10 Definition;1061
18.8.2;9.11 Integral representations;1061
18.8.3;9.12 Representation of elementary functions in terms of a hypergeometric functions;1062
18.8.4;9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series;1064
18.8.5;9.14 A generalized hypergeometric series;1067
18.8.6;9.15 The hypergeometric differential equation;1067
18.8.7;9.16 Riemann's differential equation;1070
18.8.8;9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme;1073
18.8.9;9.18 Hypergeometric functions of two variables;1074
18.8.10;9.19 A hypergeometric function of several variables;1078
18.9;9.2 Confluent Hypergeometric Functions;1078
18.9.1;9.20 Introduction;1078
18.9.2;9.21 The functions F(a,.;z) and .(a,.;z);1079
18.9.2.1;Functional relations;1079
18.9.3;9.22–9.23 The Whittaker functions M.,µ( z ) and W.,µ( z ) ;1080
18.9.3.1;Integral representations;1081
18.9.3.2;Asymptotic representations;1082
18.9.3.3;Functional relations;1082
18.9.3.4;Connections with other functions;1083
18.9.4;9.24–9.25 Parabolic cylinder functions Dp(z);1084
18.9.4.1;Integral representations;1084
18.9.4.2;Functional relations;1086
18.9.4.3;Connections with other functions;1087
18.9.5;9.26 Confluent hypergeometric series of two variables;1087
18.10;9.3 Meijer's G-Function;1088
18.10.1;9.30 Definition;1088
18.10.2;9.31 Functional relations;1090
18.10.3;9.32 A differential equation for the G-function;1091
18.10.4;9.33 Series of G-functions;1091
18.10.5;9.34 Connections with other special functions;1091
18.11;9.4 MacRobert's E-Function;1092
18.11.1;9.41 Representation by means of multiple integrals;1092
18.11.2;9.42 Functional relations;1092
18.12;9.5 Riemann's Zeta Functions .(z,q), and .(z), and the Functions F(z,s,v) and .(s);1093
18.12.1;9.51 Definition and integral representations;1093
18.12.2;9.52 Representation as a series or as an infinite product;1094
18.12.3;9.53 Functional relations;1095
18.12.4;9.54 Singular points and zeros;1096
18.12.5;9.55 The Lerch function F(z, s, v);1096
18.12.5.1;Functional relations;1096
18.12.5.2;Series representation;1097
18.12.5.3;Integral representation;1097
18.12.5.4;Limit relationships;1097
18.12.5.5;Relations to other functions;1097
18.12.6;9.56 The function . ( s ) ;1098
18.13;9.6 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions .(x), .(x,a), µ(x,ß), µ(x,ß,a), .(x,y) and Euler ...;1098
18.13.1;9.61 Bernoulli numbers;1098
18.13.1.1;Properties and functional relations;1098
18.13.2;9.62 Bernoulli polynomials;1099
18.13.3;9.63 Euler numbers;1101
18.13.3.1;Properties of the Euler numbers;1101
18.13.4;9.64 The functions .(x), .(x,a), µ(x,ß), µ(x,ß,a), .(x,y);1101
18.13.5;9.6510 Euler polynomials;1102
18.14;9.7 Constants;1103
18.14.1;9.71 Bernoulli numbers;1103
18.14.2;9.72 Euler numbers;1103
18.14.3;9.73 Euler's and Catalan's constants;1104
18.14.3.1;Euler’s constant;1104
18.14.3.2;Catalan’s constant;1104
18.14.4;9.7410 Stirling numbers;1104
19;10 Vector Field Theory;1108
19.1;10.1–10.8 Vectors, Vector Operators, and Integral Theorems;1108
19.1.1;10.11 Products of vectors;1108
19.1.2;10.12 Properties of scalar product;1108
19.1.3;10.13 Properties of vector product;1108
19.1.4;10.14 Differentiation of vectors;1109
19.1.5;10.21 Operators grad, div, and curl;1109
19.1.6;10.31 Properties of the operator .;1110
19.1.7;10.41 Solenoidal fields;1111
19.1.8;10.51–10.61 Orthogonal curvilinear coordinates;1111
19.1.8.1;Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1, h2, h3;1113
19.1.9;10.71–10.72 Vector integral theorems;1114
19.1.10;10.81 Integral rate of change theorems;1116
20;11 Integral Inequalities;1118
20.1;11.11 Mean Value Theorems;1118
20.1.1;11.111 First mean value theorem;1118
20.1.2;11.112 Second mean value theorem;1118
20.1.3;11.113 First mean value theorem for infinite integrals;1118
20.1.4;11.114 Second mean value theorem for infinite integrals;1119
20.2;11.21 Differentiation of Definite Integral Containing a Parameter;1119
20.2.1;11.211 Differentiation when limits are finite;1119
20.2.2;11.212 Differentiation when a limit is infinite;1119
20.3;11.31 Integral Inequalities;1119
20.3.1;11.311 Cauchy–Schwarz–Buniakowsky inequality for integrals;1119
20.3.2;11.312 Hölder's inequality for integrals;1119
20.3.3;11.313 Minkowski's inequality for integrals;1120
20.3.4;11.314 Chebyshev's inequality for integrals;1120
20.3.5;11.315 Young's inequality for integrals;1120
20.3.6;11.316 Steffensen's inequality for integrals;1120
20.3.7;11.317 Gram's inequality for integrals;1120
20.3.8;11.318 Ostrowski's inequality for integrals;1121
20.4;11.41 Convexity and Jensen's Inequality;1121
20.4.1;11.411 Jensen's inequality;1121
20.4.2;11.412 Carleman's inequality for integrals;1121
20.5;11.51 Fourier Series and Related Inequalities;1121
20.5.1;11.511 Riemann–Lebesgue lemma;1122
20.5.2;11.512 Dirichlet lemma;1122
20.5.3;11.513 Parseval's theorem for trigonometric Fourier series;1122
20.5.4;11.514 Integral representation of the nth partial sum;1122
20.5.5;11.515 Generalized Fourier series;1122
20.5.6;11.516 Bessel's inequality for generalized Fourier series;1123
20.5.7;11.517 Parseval's theorem for generalized Fourier series;1123
21;12 Fourier, Laplace, and Mellin Transforms;1124
21.1;12.1– 12.4 Integral Transforms;1124
21.1.1;12.11 Laplace transform;1124
21.1.2;12.12 Basic properties of the Laplace transform;1124
21.1.3;12.13 Table of Laplace transform pairs;1125
21.1.4;12.21 Fourier transform;1134
21.1.5;12.22 Basic properties of the Fourier transform;1135
21.1.6;12.23 Table of Fourier transform pairs;1135
21.1.7;12.24 Table of Fourier transform pairs for spherically symmetric functions;1137
21.1.8;12.31 Fourier sine and cosine transforms;1138
21.1.9;12.32 Basic properties of the Fourier sine and cosine transforms;1138
21.1.10;12.33 Table of Fourier sine transforms;1139
21.1.11;12.34 Table of Fourier cosine transforms;1143
21.1.12;12.35 Relationships between transforms;1146
21.1.13;12.4110 Mellin transform;1146
21.1.14;12.42 Basic properties of the Mellin transform;1147
21.1.15;12.43 Table of Mellin transforms;1148
22;Bibliographic References;1152
23;Supplementary References;1156
23.1;General reference books;1156
23.2;Asymptotic expansions;1157
23.3;Bessel functions;1157
23.4;Complex analysis;1157
23.5;Error function and Fresnel integrals;1158
23.6;Exponential integrals, gamma function and related functions;1158
23.7;Hypergeometric and confluent hypergeometric functions;1158
23.8;Integral transforms;1158
23.9;Jacobian and Weierstrass elliptic functions and related functions;1159
23.10;Legendre and related functions;1160
23.11;Mathieu functions;1160
23.12;Orthogonal polynomials and functions;1160
23.13;Parabolic cylinder functions;1161
23.14;Probability function;1161
23.15;Riemann zeta function;1161
23.16;Struve functions;1161
24;Index of Functions and Constants;1162
25;Index of Concepts;1172

Leseproben

Acknowledgments

The publisher and editors would like to take this opportunity to express their gratitude to the following users of the Table of Integrals, Series, and Products who either directly or through errata published in Mathematics of Computation have generously contributed corrections and addenda to the original printing.

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