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| Zhang S., Jin J. — Computation of Special Functions |
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| Предметный указатель |
Airy functions 325—340
Airy functions, curves 326 327
Airy functions, differential equation 325
Airy functions, integral of 329 333
Airy functions, numerical computation 329—336
Airy functions, relation with Bessel functions 327 328
Airy functions, tables 337 338
Airy functions, zeros 339
Airy's equation 325
Arithmetic-geometric mean method for complementary elliptic integrals 664
Arithmetic-geometric mean method for elliptic integrals 663
Arithmetic-geometric mean method for Jacobian elliptic functions 671
Associated Legendre functions, arbitrary degree 104—112
Associated Legendre functions, arbitrary degree, curves 105—107
Associated Legendre functions, arbitrary degree, definition 104 105
Associated Legendre functions, arbitrary degree, numerical computation 108
Associated Legendre functions, arbitrary degree, recurrence relations 108
Associated Legendre functions, arbitrary degree, special values 107
Associated Legendre functions, arbitrary degree, tables 122 123
Associated Legendre functions, arbitrary degree, zeros 124
Associated Legendre functions, differential equation 77
Associated Legendre functions, first kind 89—96
Associated Legendre functions, first kind, curves 90—92
Associated Legendre functions, first kind, definition 89
Associated Legendre functions, first kind, numerical computation 94
Associated Legendre functions, first kind, orthogonal relations 93
Associated Legendre functions, first kind, recurrence relations 93 94
Associated Legendre functions, first kind, special values 92
Associated Legendre functions, first kind, symmetry relations 93
Associated Legendre functions, first kind, tables 118 119
Associated Legendre functions, second kind 96—104
Associated Legendre functions, second kind, curves 98—100
Associated Legendre functions, second kind, definition 96
Associated Legendre functions, second kind, numerical computation 101
Associated Legendre functions, second kind, recurrence relations 100
Associated Legendre functions, second kind, special values 98
Associated Legendre functions, second kind, symmetry relations 100
Associated Legendre functions, second kind, tables 120 121
Associated Legendre functions, Wronskian relation 101
Associated Legendre polynomials 78 89
Associated Legendre polynomials, tables 118 119
Bernoulli numbers 1—6
Bernoulli numbers, numerical calculation 5
Bernoulli numbers, recurrence relation 1
Bernoulli numbers, table 10
Bernoulli polynomials 1 2
Bernoulli polynomials, generating function 1
Bessel functions 126—201
Bessel functions, asymptotic expansions 131
Bessel functions, curves 128 129
Bessel functions, differential equation 126
Bessel functions, integral of 252—260
Bessel functions, integral of, asymptotic expressions 253 257
Bessel functions, integral of, curves 269
Bessel functions, integral of, numerical computation 254—256 258—260
Bessel functions, integral of, polynomial approximation 255 259
Bessel functions, integral of, series expansions 253 257
Bessel functions, integral of, table 270
Bessel functions, integral representation 41
Bessel functions, large-argument approximation 128
Bessel functions, large-order asymptotic expressions 172
Bessel functions, negative arguments 149
Bessel functions, negative orders 162
Bessel functions, Neumann series 177
Bessel functions, normalization relations 131
Bessel functions, numerical computation, arbitrary orders 161—175
Bessel functions, numerical computation, complex arguments 149—161
Bessel functions, numerical computation, real arguments 140—148
Bessel functions, numerical computation, zeroth and first orders 131—140
Bessel functions, polynomial approximation 137
Bessel functions, recurrence relations 130
Bessel functions, series expansions 131
Bessel functions, small-argument approximation 128
Bessel functions, tables 185—196
Bessel functions, Wronskian relations 130 157 172 178
Bessel functions, zeros 180—182
Bessel functions, zeros, tables 196—200
Bessel's equation 126
beta function 53—55
Beta function, integral representation 40
Beta function, recurrence relations 54
Chebyshev polynomials 13—18
Chebyshev polynomials, curves 15 16
Chebyshev polynomials, first kind 14
Chebyshev polynomials, Fourier series 17
Chebyshev polynomials, generation function 14
Chebyshev polynomials, numerical computation 23
Chebyshev polynomials, orthogonal relations 17
Chebyshev polynomials, recurrence relations 17
Chebyshev polynomials, reflection formulas 17
Chebyshev polynomials, Rodrigues's formulas 14
Chebyshev polynomials, second kind 14
Chebyshev polynomials, special values 16
Chebyshev polynomials, table 25
Chebyshev's equation, first kind 13
Chebyshev's equation, second kind 14
Confluent hypergeometric equation 385
Confluent hypergeometric functions 385—411
Confluent hypergeometric functions, asymptotic formulas 388
Confluent hypergeometric functions, curves 389—393
Confluent hypergeometric functions, differentiation formulas 387
Confluent hypergeometric functions, integral representation 42
Confluent hypergeometric functions, numerical computation 398
Confluent hypergeometric functions, recurrence relations 389
Confluent hypergeometric functions, series expansion 386 388
Confluent hypergeometric functions, special values 387
Confluent hypergeometric functions, tables 412—423
Confluent hypergeometric functions, transformation formulas 387
Confluent hypergeometric functions, Wronskian relations 387
Cosine integral 644—653
Cosine integral, asymptotic expansion 647
Cosine integral, curve 645
Cosine integral, numerical computation 647
Cosine integral, polynomial approximation 649
Cosine integral, rational approximation 649
Cosine integral, series expansion 646
Cosine integral, table 652
Elliptic cylinder functions 475
Elliptic functions see "Jacobian elliptic functions"
Elliptic integrals 654—666
Elliptic integrals, arithmetic-geometric mean method 663
Elliptic integrals, complementary 656
Elliptic integrals, complete 655
Elliptic integrals, curves 656—659
Elliptic integrals, numerical computation 661 665
Elliptic integrals, polynomial approximation 661
Elliptic integrals, series expansions 659
Elliptic integrals, tables 673—677
Error function 620—624
Error function, asymptotic expansion 621
Error function, curve 622
Error function, numerical computation 621
Error function, polynomial approximation 622
Error function, series expansion 621
Error function, tables 637 638
Error function, zeros 635 642
Euler integral, first kind 53
Euler integral, second kind 44
Euler numbers 6—9
Euler numbers, numerical calculation 8
Euler numbers, recurrence relation 6
Euler numbers, table 10
Euler polynomials 6 7
Euler polynomials, generating function 6
Euler's constant 45
Exponential integrals 680—693
Exponential integrals, asymptotic expansions 682
Exponential integrals, continued fraction 683
Exponential integrals, curves 681 682
Exponential integrals, numerical computation 684
| Exponential integrals, rational approximations 683
Exponential integrals, recurrence relation 681
Exponential integrals, series expansions 682
Exponential integrals, tables 689—692
Fourier — Chebyshev series 17
Fourier — Hermite series 22
Fourier — Laguerre series 20
Fourier — Legendre series 81—94
Fresnel integrals 625—636
Fresnel integrals, asymptotic expansions 630
Fresnel integrals, curves 627 628
Fresnel integrals, modified 626
Fresnel integrals, numerical computation 630
Fresnel integrals, series expansions 629
Fresnel integrals, symmetry relations 629
Fresnel integrals, tables 639—641
Fresnel integrals, zeros 635 642
Gamma function 44—53
Gamma function, asymptotic expansion 48
Gamma function, curves 45
Gamma function, integral representation 40
Gamma function, multiplication formulas 47
Gamma function, numerical computation 49
Gamma function, recurrence relations 46
Gamma function, reflection formulas 47
Gamma function, series expansion 47
Gamma function, Stirling's formulas 49
Gamma function, tables 67—71
Gauss's equation 366
Gaussian probability integral 624
Gaussian probability integral, polynomial approximation 625
Gaussian quadrature 28
Gaussian quadrature, Gauss — Chebyshev quadrature 32
Gaussian quadrature, Gauss — Hermite quadrature 37
Gaussian quadrature, Gauss — Laguerre quadrature 32
Gaussian quadrature, Gauss — Legendre quadrature 28
Generating function for Bernoulli polynomial 1
Generating function for Chebyshev polynomial 14
Generating function for Euler polynomial 6
Generating function for Hermite polynomial 21
Generating function for Laguerre polynomial 18
Hankel functions 127
Hankel functions, large-argument approximation 129
Hankel functions, numerical computation 235—239
Hankel functions, spherical 274
Helmholtz equation 697
Hermite polynomials 20—23
Hermite polynomials, curves 22
Hermite polynomials, Fourier series 22
Hermite polynomials, generating function 21
Hermite polynomials, numerical computation 23
Hermite polynomials, orthogonal relation 22
Hermite polynomials, recurrence relation 22
Hermite polynomials, Rodrigues's formula 21
Hermite polynomials, special values 21
Hermite polynomials, symmetry relation 22
Hermite polynomials, table 27
Hermite's equation 21
Hypergeometric equation 366
Hypergeometric functions 366—385
Hypergeometric functions, asymptotic formula 369
Hypergeometric functions, differentiation formulas 368
Hypergeometric functions, integral representation 41
Hypergeometric functions, numerical computation 374
Hypergeometric functions, recurrence relations 372
Hypergeometric functions, series expansion 367
Hypergeometric functions, special values 368
Hypergeometric functions, transformation formulas 369
Incomplete beta function 64—66
Incomplete beta function, continued fraction 65
Incomplete beta function, curves 64
Incomplete beta function, integral representation 40
Incomplete beta function, numerical computation 66
Incomplete beta function, recurrence relations 65
Incomplete beta function, series expansion 65
Incomplete beta function, tables 75
Incomplete gamma function 61—64
Incomplete gamma function, asymptotic expression 62
Incomplete gamma function, continued fraction 63
Incomplete gamma function, curves 61 62
Incomplete gamma function, numerical computation 63
Incomplete gamma function, recurrence relations 62
Incomplete gamma function, series expansion 62
Incomplete gamma function, tables 74
Jacobian elliptic functions 666—672
Jacobian elliptic functions, arithmetic-geometric mean method 671
Jacobian elliptic functions, basic formulas 667 669
Jacobian elliptic functions, curves 668
Jacobian elliptic functions, numerical computation 670
Jacobian elliptic functions, series expansion 670
Jacobian elliptic functions, tables 678
Kelvin functions 307—324
Kelvin functions, asymptotic expansion 312
Kelvin functions, curves 311
Kelvin functions, differential equation 307
Kelvin functions, negative arguments 311
Kelvin functions, negative orders 311
Kelvin functions, numerical computation 315—320
Kelvin functions, polynomial approximation 318
Kelvin functions, recurrence relations 311
Kelvin functions, series expression 308
Kelvin functions, special values 311
Kelvin functions, tables 322 323
Kelvin functions, zeros 321
Kelvin's equation 307
Kummer's equation 385
Kummer's transformation formulas 387
Laguerre polynomials 18—20
Laguerre polynomials, curves 19
Laguerre polynomials, Fourier series 20
Laguerre polynomials, generating function 18
Laguerre polynomials, numerical computation 23
Laguerre polynomials, orthogonal relation 20
Laguerre polynomials, recurrence relations 20
Laguerre polynomials, Rodrigues's formula 18
Laguerre polynomials, special values 20
Laguerre polynomials, table 26
Laguerre's equation 18
Lambda functions 182—184
Lambda functions, curves 183
Lambda functions, differential equation 182
Lambda functions, recurrence relation 183
Lambda functions, series expansion 182
Legendre functions 77—125
Legendre functions, arbitrary degree 104
Legendre functions, differential equation 78
Legendre functions, first kind 78—83
Legendre functions, first kind, curves 79 80
Legendre functions, first kind, definition 78
Legendre functions, first kind, integral of 82
Legendre functions, first kind, numerical computation 81
Legendre functions, first kind, orthogonal relations 80
Legendre functions, first kind, recurrence relations 80
Legendre functions, first kind, Rodrigues's formula 78
Legendre functions, first kind, special values 79
Legendre functions, first kind, symmetry relations 80
Legendre functions, second kind 83—89
Legendre functions, second kind, curves 84 85
Legendre functions, second kind, definition 83
Legendre functions, second kind, numerical computation 86
Legendre functions, second kind, recurrence relations 85
Legendre functions, second kind, special values 84
Legendre functions, second kind, symmetry relations 85
Legendre functions, tables 116 117
Legendre functions, Wronskian relation 86
Legendre polynomials 78
Legendre polynomials, tables 114 115
Legendre's equation 78
Mathieu functions 475—535
Mathieu functions, characteristic values 479
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