Table of Integrals, Series, and Products (eBook)

Daniel Zwillinger (Herausgeber)

eBook Download: PDF | EPUB

2014 | 8. Auflage
1184 Seiten
Elsevier Science (Verlag)
978-0-12-384934-2 (ISBN)

Infos

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

Inhalt

Front Cover 1
Table of Integrals, Series, and Products 4
Copyright 5
Contents 6
Preface to the Eighth Edition 18
Acknowledgments 20
The Order of Presentation of the Formulas 26
Use of the Tables 30
Bernoulli and Euler Polynomials and Numbers 31
Elliptic Functions and Elliptic Integrals 32
The Jacobi Zeta Function and Theta Functions 33
Exponential and Related Integrals 34
Hermite and Chebyshev Orthogonal Polynomials 35
Bessel Functions 36
Parabolic Cylinder Functions and Whittaker Functions 37
Mathieu Functions 37
Index of Special Functions 38
Notation 42
Note on the Bibliographic References 46
0 Introduction 48
0.1 Finite sums 48
0.11 Progressions 48
0.12 Sums of powers of natural numbers 48
0.13 Sums of reciprocals of natural numbers 50
0.14 Sums of products of reciprocals of natural numbers 50
0.15 Sums of the binomial coefficients 51
0.2 Numerical series and infinite products 54
0.21 The convergence of numerical series 54
0.22 Convergence tests 54
0.23–0.24 Examples of numerical series 56
0.25 Infinite products 62
0.26 Examples of infinite products 62
0.3 Functional series 63
0.30 Definitions and theorems 63
0.31 Power series 64
0.32 Fourier series 67
0.33 Asymptotic series 68
0.4 Certain formulas from differential calculus 69
0.41 Differentiation of a definite integral with respect to a parameter 69
0.42 The nth derivative of a product (Leibniz's rule) 69
0.43 The nth derivative of a composite function 70
0.44 Integration by substitution 71
1 Elementary Functions 72
1.1 Power of Binomials 72
1.11 Power series 72
1.12 Series of rational fractions 73
1.2 The Exponential Function 73
1.21 Series representation 73
1.22 Functional relations 74
1.23 Series of exponentials 74
1.3–1.4 Trigonometric and Hyperbolic Functions 75
1.30 Introduction 75
1.31 The basic functional relations 75
1.32 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argu ... 78
1.33 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers... 80
1.34 Certain sums of trigonometric and hyperbolic functions 83
1.35 Sums of powers of trigonometric functions of multiple angles 84
1.36 Sums of products of trigonometric functions of multiple angles 85
1.37 Sums of tangents of multiple angles 86
1.38 Sums leading to hyperbolic tangents and cotangents 86
1.39 The representation of cosines and sines of multiples of the angle as finite products 87
1.41 The expansion of trigonometric and hyperbolic functions in power series 89
1.42 Expansion in series of simple fractions 90
1.43 Representation in the form of an infinite product 91
1.44–1.45 Trigonometric (Fourier) series 92
1.46 Series of products of exponential and trigonometric functions 97
1.47 Series of hyperbolic functions 98
1.48 Lobachevskiy's ``Angle of parallelism'' .(x) 98
1.49 The hyperbolic amplitude (the Gudermannian) gd x 99
1.5 The Logarithm 99
1.51 Series representation 99
1.52 Series of logarithms (cf. 1.431) 102
1.6 The Inverse Trigonometric and Hyperbolic Functions 102
1.61 The domain of definition 102
1.62–1.63 Functional relations 103
1.64 Series representations 107
2 Indefinite Integrals of Elementary Functions 110
2.0 Introduction 110
2.00 General remarks 110
2.01 The basic integrals 111
2.02 General formulas 112
2.1 Rational Functions 113
2.10 General integration rules 113
2.11–2.13 Forms containing the binomial a+bxk 115
2.14 Forms containing the binomial 1 ± xn 121
2.15 Forms containing pairs of binomials: a+bx and a+ßx 125
2.16 Forms containing the trinomial a+bxk+c x2k 125
2.17 Forms containing the quadratic trinomial a+bx+cx2 and powers of x 126
2.18 Forms containing the quadratic trinomial a+bx+cx2 and the binomial a+ßx 128
2.2 Algebraic functions 129
2.20 Introduction 129
2.21 Forms containing the binomial a+bxk and vx 130
2.22–2.23 Forms containing n(a + bx)k 131
The square root 131
Cube root 133
2.24 Forms containing a+bx and the binomial a+ßx 135
2.25 Forms containing a+bx+cx2 139
Integration techniques 139
2.26 Forms containing a+bx+cx2 and integral powers of x 141
2.2712 Forms containing a+c x2 and integral powers of x 146
2.28 Forms containing a+bx+c x2 and first-and second-degree polynomials 150
2.29 Integrals that can be reduced to elliptic or pseudo-elliptic integrals 151
2.3 The Exponential Function 153
2.31 Forms containing eax 153
2.32 The exponential combined with rational functions of x 153
2.4 Hyperbolic Functions 157
2.41–2.43 Powers of sinh x, cosh x, tanh x, and coth x 157
Powers of hyperbolic functions and hyperbolic functions of linear functions of the argument 167
2.44–2.45 Rational functions of hyperbolic functions 172
2.46 Algebraic functions of hyperbolic functions 179
2.47 Combinations of hyperbolic functions and powers 187
2.48 Combinations of hyperbolic functions, exponentials, and powers 196
2.5–2.6 Trigonometric Functions 198
2.50 Introduction 198
2.51–2.52 Powers of trigonometric functions 199
2.53–2.54 Sines and cosines of multiple angles and of linear and more complicated functions of the argument 208
2.55–2.56 Rational functions of the sine and cosine 218
2.57 Integrals containing a ± b sin x or a ± b cos x 226
2.58–2.62 Integrals reducible to elliptic and pseudo-elliptic integrals 231
2.63–2.65 Products of trigonometric functions and powers 261
2.66 Combinations of trigonometric functions and exponentials 274
2.67 Combinations of trigonometric and hyperbolic functions 278
2.7 Logarithms and Inverse-Hyperbolic Functions 284
2.71 The logarithm 284
2.72–2.73 Combinations of logarithms and algebraic functions 285
2.74 Inverse hyperbolic functions 289
2.75 Logarithms and exponential functions 289
2.8 Inverse Trigonometric Functions 289
2.81 Arcsines and arccosines 289
2.82 The arcsecant, the arccosecant, the arctangent and the arccotangent 290
2.83 Combinations of arcsine or arccosine and algebraic functions 291
2.84 Combinations of the arcsecant and arccosecant with powers of x 292
2.85 Combinations of the arctangent and arccotangent with algebraic functions 293
3–4 Definite Integrals of Elementary Functions 296
3.0 Introduction 296
3.01 Theorems of a general nature 296
3.02 Change of variable in a definite integral 297
3.03 General formulas 298
3.04 Improper integrals 300
3.05 The principal values of improper integrals 301
3.1–3.2 Power and Algebraic Functions 302
3.11 Rational functions 302
3.12 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials 303
3.13–3.17 Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with ration 303
3.18 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions 362
3.19–3.23 Combinations of powers of x and powers of binomials of the form (a+ßx) 365
3.24–3.27 Powers of x, of binomials of the form a+ßxp and of polynomials in x 371
3.3–3.4 Exponential Functions 383
3.31 Exponential functions 383
3.32–3.34 Exponentials of more complicated arguments 385
3.35 Combinations of exponentials and rational functions 389
3.36–3.37 Combinations of exponentials and algebraic functions 393
3.38–3.39 Combinations of exponentials and arbitrary powers 395
3.41–3.44 Combinations of rational functions of powers and exponentials 402
3.45 Combinations of powers and algebraic functions of exponentials 412
3.46–3.48 Combinations of exponentials of more complicated arguments and powers 413
3.5 Hyperbolic Functions 421
3.51 Hyperbolic functions 421
3.52–3.53 Combinations of hyperbolic functions and algebraic functions 424
3.54 Combinations of hyperbolic functions and exponentials 431
3.55–3.56 Combinations of hyperbolic functions, exponentials, and powers 435
3.6–4.1 Trigonometric Functions 439
3.61 Rational functions of sines and cosines and trigonometric functions of multiple angles 439
3.62 Powers of trigonometric functions 444
3.63 Powers of trigonometric functions and trigonometric functions of linear functions 446
3.64–3.65 Powers and rational functions of trigonometric functions 450
3.66 Forms containing powers of linear functions of trigonometric functions 454
3.67 Square roots of expressions containing trigonometric functions 457
3.68 Various forms of powers of trigonometric functions 462
3.69–3.71 Trigonometric functions of more complicated arguments 466
3.72–3.74 Combinations of trigonometric and rational functions 474
3.75 Combinations of trigonometric and algebraic functions 485
3.76–3.77 Combinations of trigonometric functions and powers 487
3.78–3.81 Rational functions of x and of trigonometric functions 498
3.82–3.83 Powers of trigonometric functions combined with other powers 510
3.84 Integrals containing 1 k2 sin2 x, 1 k2 cos2 x, and similar expressions 523
3.85–3.88 Trigonometric functions of more complicated arguments combined with powers 526
3.89–3.91 Trigonometric functions and exponentials 536
3.92 Trigonometric functions of more complicated arguments combined with exponentials 544
3.93 Trigonometric and exponential functions of trigonometric functions 546
3.94–3.97 Combinations involving trigonometric functions, exponentials, and powers 548
3.98–3.99 Combinations of trigonometric and hyperbolic functions 560
4.11–4.12 Combinations involving trigonometric and hyperbolic functions and powers 567
4.13 Combinations of trigonometric and hyperbolic functions and exponentials 573
4.14 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 576
4.2–4.4 Logarithmic Functions 577
4.21 Logarithmic functions 577
4.22 Logarithms of more complicated arguments 580
4.23 Combinations of logarithms and rational functions 585
4.24 Combinations of logarithms and algebraic functions 588
4.25 Combinations of logarithms and powers 590
4.26-4.27 Combinations involving powers of the logarithm and other powers 593
4.28 Combinations of rational functions of ln x and powers 603
4.29–4.32 Combinations of logarithmic functions of more complicated arguments and powers 606
4.33–4.34 Combinations of logarithms and exponentials 622
4.35–4.36 Combinations of logarithms, exponentials, and powers 624
4.37 Combinations of logarithms and hyperbolic functions 629
4.38–4.41 Logarithms and trigonometric functions 631
4.42–4.43 Combinations of logarithms, trigonometric functions, and powers 644
4.44 Combinations of logarithms, trigonometric functions, and exponentials 650
4.5 Inverse Trigonometric Functions 650
4.51 Inverse trigonometric functions 650
4.52 Combinations of arcsines, arccosines, and powers 650
4.53–4.54 Combinations of arctangents, arccotangents, and powers 652
4.55 Combinations of inverse trigonometric functions and exponentials 655
4.56 A combination of the arctangent and a hyperbolic function 656
4.57 Combinations of inverse and direct trigonometric functions 656
4.58 A combination involving an inverse and a direct trigonometric function and a power 657
4.59 Combinations of inverse trigonometric functions and logarithms 657
4.6 Multiple Integrals 658
4.60 Change of variables in multiple integrals 658
4.61 Change of the order of integration and change of variables 659
4.62 Double and triple integrals with constant limits 661
4.63–4.64 Multiple integrals 664
5 Indefinite Integrals of Special Functions 670
5.1 Elliptic Integrals and Functions 670
5.11 Complete elliptic integrals 670
5.12 Elliptic integrals 672
5.13 Jacobian elliptic functions 674
5.14 Weierstrass elliptic functions 677
5.2 The Exponential Integral Function 678
5.21 The exponential integral function 678
5.22 Combinations of the exponential integral function and powers 678
5.23 Combinations of the exponential integral and the exponential 679
5.3 The Sine Integral and the Cosine Integral 679
5.4 The Probability Integral and Fresnel Integrals 680
5.5 Bessel Functions 680
5.6 Orthogonal Polynomials 681
5.7 Hypergeometric Functions 681
6–7 Definite Integrals of Special Functions 684
6.1 Elliptic Integrals and Functions 684
6.11 Forms containing F(x, k) 684
6.12 Forms containing E(x, k) 685
6.13 Integration of elliptic integrals with respect to the modulus 686
6.14–6.15 Complete elliptic integrals 686
6.16 The theta function 689
6.17 Generalized elliptic integrals 690
6.2–6.3 The Exponential Integral Function and Functions Generated by It 691
6.21 The logarithm integral 691
6.22–6.23 The exponential integral function 693
6.24–6.26 The sine integral and cosine integral functions 695
6.27 The hyperbolic sine integral and hyperbolic cosine integral functions 700
6.28–6.31 The probability integral 700
6.32 Fresnel integrals 704
6.4 The Gamma Function and Functions Generated by It 706
6.41 The gamma function 706
6.42 Combinations of the gamma function, the exponential, and powers 707
6.43 Combinations of the gamma function and trigonometric functions 710
6.44 The logarithm of the gamma function 711
6.45 The incomplete gamma function 712
6.46–6.47 The function .(x) 713
6.5–6.7 Bessel Functions 714
6.51 Bessel functions 714
6.52 Bessel functions combined with x and x2 719
6.53–6.54 Combinations of Bessel functions and rational functions 725
6.55 Combinations of Bessel functions and algebraic functions 729
6.56–6.58 Combinations of Bessel functions and powers 730
6.59 Combinations of powers and Bessel functions of more complicated arguments 744
6.61 Combinations of Bessel functions and exponentials 749
6.62–6.63 Combinations of Bessel functions, exponentials, and powers 753
6.64 Combinations of Bessel functions of more complicated arguments, exponentials, and powers 763
6.65 Combinations of Bessel and exponential functions of more complicated arguments and powers 765
6.66 Combinations of Bessel, hyperbolic, and exponential functions 767
Bessel and hyperbolic functions 767
Bessel, hyperbolic, and algebraic functions 769
Exponential, hyperbolic, and Bessel functions 770
6.67–6.68 Combinations of Bessel and trigonometric functions 771
6.69–6.74 Combinations of Bessel and trigonometric functions and powers 781
6.75 Combinations of Bessel, trigonometric, and exponential functions and powers 797
6.76 Combinations of Bessel, trigonometric, and hyperbolic functions 800
6.77 Combinations of Bessel functions and the logarithm, or arctangent 801
6.78 Combinations of Bessel and other special functions 802
6.79 Integration of Bessel functions with respect to the order 803
6.8 Functions Generated by Bessel Functions 807
6.81 Struve functions 807
6.82 Combinations of Struve functions, exponentials, and powers 808
6.83 Combinations of Struve and trigonometric functions 809
6.84–6.85 Combinations of Struve and Bessel functions 810
6.86 Lommel functions 814
6.87 Thomson functions 815
6.9 Mathieu Functions 817
6.91 Mathieu functions 817
6.92 Combinations of Mathieu, hyperbolic, and trigonometric functions 817
6.93 Combinations of Mathieu and Bessel functions 821
6.94 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems 821
7.1–7.2 Associated Legendre Functions 823
7.11 Associated Legendre functions 823
7.12–7.13 Combinations of associated Legendre functions and powers 824
7.14 Combinations of associated Legendre functions, exponentials, and powers 830
7.15 Combinations of associated Legendre and hyperbolic functions 832
7.16 Combinations of associated Legendre functions, powers, and trigonometric functions 833
7.17 A combination of an associated Legendre function and the probability integral 835
7.18 Combinations of associated Legendre and Bessel functions 836
7.19 Combinations of associated Legendre functions and functions generated by Bessel functions 841
7.21 Integration of associated Legendre functions with respect to the order 842
7.22 Combinations of Legendre polynomials, rational functions, and algebraic functions 843
7.23 Combinations of Legendre polynomials and powers 845
7.24 Combinations of Legendre polynomials and other elementary functions 846
7.25 Combinations of Legendre polynomials and Bessel functions 848
7.3–7.4 Orthogonal Polynomials 849
7.31 Combinations of Gegenbauer polynomials C.n(x) and powers 849
7.32 Combinations of Gegenbauer polynomials Cvn(x) and elementary functions 852
7.325* Complete System of Orthogonal Step Functions 852
7.33 Combinations of the polynomials Cvn(x) and Bessel functions. Integration of Gegenbauer functions with respect to the ... 853
Integration of Gegenbauer functions with respect to the index 854
7.34 Combinations of Chebyshev polynomials and powers 854
7.35 Combinations of Chebyshev polynomials and elementary functions 856
7.36 Combinations of Chebyshev polynomials and Bessel functions 857
7.37–7.38 Hermite polynomials 857
7.39 Jacobi polynomials 861
7.41–7.42 Laguerre polynomials 863
7.5 Hypergeometric Functions 867
7.51 Combinations of hypergeometric functions and powers 867
7.52 Combinations of hypergeometric functions and exponentials 869
7.53 Hypergeometric and trigonometric functions 872
7.54 Combinations of hypergeometric and Bessel functions 872
7.6 Confluent Hypergeometric Functions 875
7.61 Combinations of confluent hypergeometric functions and powers 875
7.62–7.63 Combinations of confluent hypergeometric functions and exponentials 877
7.64 Combinations of confluent hypergeometric and trigonometric functions 884
7.65 Combinations of confluent hypergeometric functions and Bessel functions 885
7.66 Combinations of confluent hypergeometric functions, Bessel functions, and powers 886
7.67 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers 889
Combinations of Struve functions and confluent hypergeometric functions 893
7.68 Combinations of confluent hypergeometric functions and other special functions 894
Combinations of confluent hypergeometric functions and associated Legendre functions 894
A combination of confluent hypergeometric functions and orthogonal polynomials 895
A combination of hypergeometric and confluent hypergeometric functions 896
7.69 Integration of confluent hypergeometric functions with respect to the index 896
7.7 Parabolic Cylinder Functions 896
7.71 Parabolic cylinder functions 896
7.72 Combinations of parabolic cylinder functions, powers, and exponentials 897
7.73 Combinations of parabolic cylinder and hyperbolic functions 898
7.74 Combinations of parabolic cylinder and trigonometric functions 899
7.75 Combinations of parabolic cylinder and Bessel functions 900
Combinations of parabolic cylinder and Struve functions 903
7.76 Combinations of parabolic cylinder functions and confluent hypergeometric functions 904
7.77 Integration of a parabolic cylinder function with respect to the index 904
7.8 Meijer's and MacRobert's Functions (G and E) 905
7.81 Combinations of the functions G and E and the elementary functions 905
7.82 Combinations of the functions G and E and Bessel functions 909
7.83 Combinations of the functions G and E and other special functions 911
8–9 Special Functions 914
8.1 Elliptic Integrals and Functions 914
8.11 Elliptic integrals 914
Series representations 916
Trigonometric series 918
8.12 Functional relations between elliptic integrals 919
8.13 Elliptic functions 921
8.14 Jacobian elliptic functions 922
8.15 Properties of Jacobian elliptic functions and functional relationships between them 926
Functional relations 928
8.16 The Weierstrass function Ã(u) 929
8.17 The functions .(u) and s(u) 932
Functional relations and properties 932
8.18–8.19 Theta functions 933
Functional relations and properties 934
q-series and products, q = exp (-pK' K) 936
8.2 The Exponential Integral Function and Functions Generated by It 939
8.21 The exponential integral function Ei(x) 939
Series and asymptotic representations 940
The hyperbolic sine integral shi x and the hyperbolic cosine integral chi x 942
8.23 The sine integral and the cosine integral: si x and ci x 942
8.24 The logarithm integral li(x) 943
Integral representations 943
8.25 The probability integral F(x), the Fresnel integrals S(x), C(x), the error function erf(x), and the complementary err ... 943
Integral representations 944
Asymptotic representations 945
8.26 Lobachevskiy's function L(x) 947
8.3 Euler’s Integrals of the First and Second Kinds and Functions Generated by Them 948
8.31 The gamma function (Euler's integral of the second kind): G(z) 948
Integral representations 948
8.32 Representation of the gamma function as series and products 950
Infinite-product representation 950
8.33 Functional relations involving the gamma function 951
Special cases 952
Particular values 953
8.34 The logarithm of the gamma function 954
8.35 The incomplete gamma function 955
8.36 The psi function .(x) 958
Series representation 959
Infinite-product representation 960
8.37 The function ß(x) 962
Series representation 963
Functional relations 963
8.38 The beta function (Euler's integral of the first kind): B(x,y) 964
Integral representation 964
Series representation 965
8.39 The incomplete beta function Bx(p,q) 966
8.4–8.5 Bessel Functions and Functions Associated with Them 966
8.40 Definitions 966
Modified Bessel functions of imaginary argument I .(z) and K.(z) 967
8.41 Integral representations of the functions J.(z) and N.(z) 968
8.42 Integral representations of the functions H(1).(z) and H(2).(z) 970
8.43 Integral representations of the functions I.(z) and K.(z) 972
The function I .(z) 972
The function K.(z) 973
8.44 Series representation 974
The function J.(z) 974
The function Y.(z) 974
The functions I .(z) and Kn(z) 975
8.45 Asymptotic expansions of Bessel functions 976
“Approximation by tangents” 977
8.46 Bessel functions of order equal to an integer plus one-half 980
The function J.(z) 980
The function Yn+12(z) 981
The functions H(1,2)n-1/2(z),In+1/2(z), Kn+1/2(z) 981
8.47–8.48 Functional relations 982
Relations between Bessel functions of the first, second, and third kinds 984
8.49 Differential equations leading to Bessel functions 987
8.51–8.52 Series of Bessel functions 989
The series SJk(z) 990
The series Sak Jk(kx) and Sak Jk'(kx) 991
The series Sak J0(kx) 992
The series Sak Z0(kx) sinkx and Sak Z0(kx) cos kx 993
8.53 Expansion in products of Bessel functions 995
8.54 The zeros of Bessel functions 997
8.55 Struve functions 998
8.56 Thomson functions and their generalizations 1000
Series representation 1000
Asymptotic representation 1000
8.57 Lommel functions 1001
Integral representations 1002
Definition 1003
8.58 Anger and Weber functions J.(z) and E.(z) 1004
8.59 Neumann’s and Schläfli's polynomials: On(z) and Sn(z) 1005
8.6 Mathieu Functions 1006
8.60 Mathieu's equation 1006
8.61 Periodic Mathieu functions 1007
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
8.63 Mathieu functions with a purely imaginary argument 1008
8.64 Non-periodic solutions of Mathieu's equation 1009
8.65 Mathieu functions for negative q 1009
8.66 Representation of Mathieu functions as series of Bessel functions 1010
8.67 The general theory 1013
8.7–8.8 Associated Legendre Functions 1014
8.70 Introduction 1014
8.71 Integral representations 1016
8.72 Asymptotic series for large values of |.| 1018
8.73–8.74 Functional relations 1020
8.75 Special cases and particular values 1024
Special values of the indices 1024
Special values of Legendre functions 1025
8.76 Derivatives with respect to the order 1025
8.77 Series representation 1026
The analytic continuation for |z| > >
8.78 The zeros of associated Legendre functions 1028
8.79 Series of associated Legendre functions 1028
Addition theorems 1029
8.81 Associated Legendre functions with integer indices 1030
Functional relations 1031
8.82–8.83 Legendre functions 1031
Integral representations 1032
Special cases and particular values 1033
Functional relationships 1034
8.84 Conical functions 1036
Functional relations 1036
8.85 Toroidal functions 1037
8.9 Orthogonal Polynomials 1038
8.90 Introduction 1038
8.91 Legendre polynomials 1039
Functional relations 1041
8.91910 Series of products of Legendre and Chebyshev polynomials 1044
8.92 Series of Legendre polynomials 1044
8.93 Gegenbauer polynomials C.n(t) 1046
Functional relati 1047
8.94 The Chebyshev polynomials Tn(x) and Un(x) 1049
Functional relations 1050
8.95 The Hermite polynomials Hn(x) 1052
Functional relations 1052
Series of Hermite polynomials 1053
8.96 Jacobi's polynomials 1054
8.97 The Laguerre polynomials 1056
9.1 Hypergeometric Functions 1061
9.10 Definition 1061
9.11 Integral representations 1061
9.12 Representation of elementary functions in terms of a hypergeometric functions 1062
9.13 Transformation formulas and the analytic continuation of functions defined by hypergeometric series 1064
9.14 A generalized hypergeometric series 1067
9.15 The hypergeometric differential equation 1067
9.16 Riemann's differential equation 1070
9.17 Representing the solutions to certain second-order differential equations using a Riemann scheme 1073
9.18 Hypergeometric functions of two variables 1074
9.19 A hypergeometric function of several variables 1078
9.2 Confluent Hypergeometric Functions 1078
9.20 Introduction 1078
9.21 The functions F(a,. z) and .(a,.
Functional relations 1079
9.22–9.23 The Whittaker functions M.,µ( z ) and W.,µ( z ) 1080
Integral representations 1081
Asymptotic representations 1082
Functional relations 1082
Connections with other functions 1083
9.24–9.25 Parabolic cylinder functions Dp(z) 1084
Integral representations 1084
Functional relations 1086
Connections with other functions 1087
9.26 Confluent hypergeometric series of two variables 1087
9.3 Meijer's G-Function 1088
9.30 Definition 1088
9.31 Functional relations 1090
9.32 A differential equation for the G-function 1091
9.33 Series of G-functions 1091
9.34 Connections with other special functions 1091
9.4 MacRobert's E-Function 1092
9.41 Representation by means of multiple integrals 1092
9.42 Functional relations 1092
9.5 Riemann's Zeta Functions .(z,q), and .(z), and the Functions F(z,s,v) and .(s) 1093
9.51 Definition and integral representations 1093
9.52 Representation as a series or as an infinite product 1094
9.53 Functional relations 1095
9.54 Singular points and zeros 1096
9.55 The Lerch function F(z, s, v) 1096
Functional relations 1096
Series representation 1097
Integral representation 1097
Limit relationships 1097
Relations to other functions 1097
9.56 The function . ( s ) 1098
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
9.61 Bernoulli numbers 1098
Properties and functional relations 1098
9.62 Bernoulli polynomials 1099
9.63 Euler numbers 1101
Properties of the Euler numbers 1101
9.64 The functions .(x), .(x,a), µ(x,ß), µ(x,ß,a), .(x,y) 1101
9.6510 Euler polynomials 1102
9.7 Constants 1103
9.71 Bernoulli numbers 1103
9.72 Euler numbers 1103
9.73 Euler's and Catalan's constants 1104
Euler’s constant 1104
Catalan’s constant 1104
9.7410 Stirling numbers 1104
10 Vector Field Theory 1108
10.1–10.8 Vectors, Vector Operators, and Integral Theorems 1108
10.11 Products of vectors 1108
10.12 Properties of scalar product 1108
10.13 Properties of vector product 1108
10.14 Differentiation of vectors 1109
10.21 Operators grad, div, and curl 1109
10.31 Properties of the operator . 1110
10.41 Solenoidal fields 1111
10.51–10.61 Orthogonal curvilinear coordinates 1111
Special Orthogonal Curvilinear Coordinates and their Metrical Coefficients h1, h2, h3 1113
10.71–10.72 Vector integral theorems 1114
10.81 Integral rate of change theorems 1116
11 Integral Inequalities 1118
11.11 Mean Value Theorems 1118
11.111 First mean value theorem 1118
11.112 Second mean value theorem 1118
11.113 First mean value theorem for infinite integrals 1118
11.114 Second mean value theorem for infinite integrals 1119
11.21 Differentiation of Definite Integral Containing a Parameter 1119
11.211 Differentiation when limits are finite 1119
11.212 Differentiation when a limit is infinite 1119
11.31 Integral Inequalities 1119
11.311 Cauchy–Schwarz–Buniakowsky inequality for integrals 1119
11.312 Hölder's inequality for integrals 1119
11.313 Minkowski's inequality for integrals 1120
11.314 Chebyshev's inequality for integrals 1120
11.315 Young's inequality for integrals 1120
11.316 Steffensen's inequality for integrals 1120
11.317 Gram's inequality for integrals 1120
11.318 Ostrowski's inequality for integrals 1121
11.41 Convexity and Jensen's Inequality 1121
11.411 Jensen's inequality 1121
11.412 Carleman's inequality for integrals 1121
11.51 Fourier Series and Related Inequalities 1121
11.511 Riemann–Lebesgue lemma 1122
11.512 Dirichlet lemma 1122
11.513 Parseval's theorem for trigonometric Fourier series 1122
11.514 Integral representation of the nth partial sum 1122
11.515 Generalized Fourier series 1122
11.516 Bessel's inequality for generalized Fourier series 1123
11.517 Parseval's theorem for generalized Fourier series 1123
12 Fourier, Laplace, and Mellin Transforms 1124
12.1– 12.4 Integral Transforms 1124
12.11 Laplace transform 1124
12.12 Basic properties of the Laplace transform 1124
12.13 Table of Laplace transform pairs 1125
12.21 Fourier transform 1134
12.22 Basic properties of the Fourier transform 1135
12.23 Table of Fourier transform pairs 1135
12.24 Table of Fourier transform pairs for spherically symmetric functions 1137
12.31 Fourier sine and cosine transforms 1138
12.32 Basic properties of the Fourier sine and cosine transforms 1138
12.33 Table of Fourier sine transforms 1139
12.34 Table of Fourier cosine transforms 1143
12.35 Relationships between transforms 1146
12.4110 Mellin transform 1146
12.42 Basic properties of the Mellin transform 1147
12.43 Table of Mellin transforms 1148
Bibliographic References 1152
Supplementary References 1156
General reference books 1156
Asymptotic expansions 1157
Bessel functions 1157
Complex analysis 1157
Error function and Fresnel integrals 1158
Exponential integrals, gamma function and related functions 1158
Hypergeometric and confluent hypergeometric functions 1158
Integral transforms 1158
Jacobian and Weierstrass elliptic functions and related functions 1159
Legendre and related functions 1160
Mathieu functions 1160
Orthogonal polynomials and functions 1160
Parabolic cylinder functions 1161
Probability function 1161
Riemann zeta function 1161
Struve functions 1161
Index of Functions and Constants 1162
Index of Concepts 1172

Textauszug

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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Produkt-Details

• Erscheint lt. Verlag: 18.9.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• ISBN-10: 0-12-384934-9 / 0123849349
• ISBN-13: 978-0-12-384934-2 / 9780123849342