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Erdelyi A. — Higher Transcendental Functions, Vol. 3

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Название: Higher Transcendental Functions, Vol. 3

Автор: Erdelyi A.

Аннотация:

The Bulletin of the London Mathematical Society hailed this three-volume series as "The most widely cited mathematical works of all time and a basic reference source for generations of applied mathematicians and physicists throughout the world." Working from extensive notes on familiar special functions by the renowned mathematician Harry Bateman, a team of editors not only finished Bateman's original project but also made significant advances in mathematical analysis. The books, which can be used independently of each other, consist of Volume 1, which focuses on hypergeometric series; Volume 2, an exploration of Bessel functions, orthogonal polynomials, and elliptic functions and integrals; and Volume 3, an examination of automorphic functions, spheroidal and ellipsoidal wave functions, and other functions.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1955

Количество страниц: 301

Добавлена в каталог: 13.07.2005

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Предметный указатель

Absolute invariant      17 ff.
Accessory parameter      57
Algebraic numbers      167
Almost all, almost no      175
Appell polynomials      see “Polynomials”
Arithmetical differentiation      172
Arithmetical functions      169
Arithmetical functions, asymptotic behavior of      174 ff.
Arithmetical functions, explicit expressions for      169 ff.
Arithmetical functions, general theorems on      175
Arithmetical functions, generating functions of      169 ff.
Arithmetical functions, properties of      171 ff.
Arithmetical functions, relations for      171 ff.
Arithmetical integration      172
Automorphic forms      30 ff.
Automorphic forms, metrization of      31
Automorphic functions      1 ff. 7
Automorphic functions of groups of parabolic substitutions      12 ff.
Automorphic functions of infinite cyclic groups      14 ff.
Automorphic functions of several variables      12 35
Automorphic functions of subgroups of the modular group      21 ff.
Automorphic functions of the icosahedral group      11
Automorphic functions of the lambda-group      22 ff.
Automorphic functions of the modular group      17
Automorphic functions, Burnside’s      35
Automorphic functions, general theorems for      27 ff.
Automorphic functions, Siegel’s      36 ff.
Automorphic functions, simple      9
Automorphic functions, Whittaker’s      34
Bernoulli numbers      189 241 252 257 260
Bernoulli polynomials      see “Polynomials”
Bessel polynomials      see “Polynomials”
Binomial polynomials      see “Polynomials”
Character (mod m), imprimitive      194
Character (mod m), primitive      194
Character (mod n)      193 ff.
Character (mod n), principal      193 ff.
Charlier polynomials      see “Polynomials”
Circulant      214
Confluent hypergeometric functions      251 261 275
Congruence      175
Coordinates of confocal cones      48 ff.
Coordinates of confocal cyclides of revolution      50 ff.
Coordinates of confocal elliptic and hyperbolic cylinders      91 ff.
Coordinates of confocal quadrics      44 ff.
Coordinates, ellipsoidal      46 ff. 96
Coordinates, oblate spheroidal      95 ff.
Coordinates, prolate spheroidal      93 ff.
Coordinates, sphero-conal      49 ff. 73
Coprime      168
Cyclides of revolution, coordinates of      50 ff.
Cyclides of revolution, harmonics associated with      84 ff.
Darboux’s method      244
Decomposition      175
Dedelcind — Liouville formula      see “Moebius’ inversion formula”
Discontinuous groups      5
Discontinuous groups, classification of      26 ff.
Discontinuous groups, fundamental regions of      5 ff.
Discontinuous groups, generators of      5
Discontinuous groups, limit points of      6
Divisors, number of      168
Eisenstein series      17 ff.
Ellipsoidal harmonics      48 69 73 8
Ellipsoidal harmonics, integral representations of      83
Ellipsoidal wave functions      91 ff. 97 159
Ellipsoidal wave functions, differential equation of      see “Lame’s wave equation”
Ellipsoidal wave functions, integral equations for      162 ff.
Elliptic functions, Jacobian      45 ff. 51
Elliptic modular functions      16 ff.
Euler numbers      252
Euler polynomials      see “Polynomials”
Euler product of $\Sigma f(n)$       169
Euler’s function $\phi(n)$       168 193
Euler’s identities      176 ff.
Floquet’s theorem      99 119
Fuchsian equations      57 160
Gaussian sums      187 ff.
Gegenbauer function      275 ff.
Gegenbauer polynomials      see “Polynomials”
Generalized Bessel polynomial      see “Polynomials”
Generating function of an Appell set of polynomials      236 255 262
Generating function of Appell polynomials      256
Generating function of Bernoulli numbers      241 252
Generating function of Bernoulli polynomials      234 252
Generating function of Bessel coefficients      250 260
Generating function of Bessel functions      250 260
Generating function of Bessel polynomials      251 266
Generating function of Charlier polynomials      255
Generating function of Euler polynomials      252 ff.
Generating function of Fuler numbers      252
Generating function of Gegenbauer polynomials      246 ff. 262 265 271
Generating function of Hermite polynomials      242 249 263 266 269 272
Generating function of hypergeometric polynomials      247 ff. 251 255 264 266
Generating function of Jacobi polynomials      247 262 267 272
Generating function of Lagrange’s polynomials      267
Generating function of Laguerre polynomials      249 ff. 251 262 272
Generating function of Legendre functions      264 266
Generating function of Legendre polynomials      234 245 261 264 271
Generating function of parabolic cylinder functions      263
Generating function of Poisson — Charlier polynomials      255
Generating function of Stirling numbers      257
Generating function of Stirling polynomials      257 259
Generating function of Tchebycheff polynomials      231 245
Generating function, bilinear, of Gegenbauer polynomials      271
Generating function, bilinear, of Hermite polynomials      272
Generating function, bilinear, of Jacobi polynomials      272
Generating function, bilinear, of Laguerre polynomials      272
Generating function, bilinear, of Legendre polynomials      27
Generating functions      228 ff.
Generating functions and asymptotic representations      243 ff.
Generating functions and orthogonal polynomials      270 ff. (see also “Gegenbauer” “Hermite” “Jacobi” “Laguerre” “Legendre” “Polynomials”)
Generating functions and symbolic relations      240 ff.
Generating functions in number theory      169 ff. 245
Generating functions, bilinear      271 ff.
Generating functions, continuous      274 ff.
Generating functions, general theorems for      235 ff.
Group      167
Group of homographic substitutions      5
Group, alternating      9
Group, discontinuous      see “Discontinuous groups”
Group, dodecahedral      9
Group, finite      9
Group, Fuchsian      26 32
Group, icosahedral      9
Group, Kleinian      27
Group, lambda-      22 34
Group, modular      16 ff. 34
Group, modular, Hilbert’s      35 ff.
Group, modular, of degree n      37 ff.
Group, modular, subgroups of the      21 ff.
Harmonic polynomials      83
Hermite polynomials      see “Polynomials”
Heun functions      60 ff.
Heun functions, integral equations for      72
Heun polynomials      62
Heun’s equation      57 ff. 98
Hille — Hardy formula      272
Hill’s equation      133
Hill’s problem      133
Homographic substitution      2
Homographic transformation      2
Homographic transformation, elliptic      4
Homographic transformation, fixed points of      3 ff.
Homographic transformation, hyperbolic      4
Homographic transformation, loxodromic      4
Homographic transformation, parabolic      4
Hyper geometric polynomials      see “Polynomials”
Hyperbolic functions of order n      206 212 216
Hypergeometric series      20 ff. 23 236 238 259 263 276
Jacobi polynomials      see “Polynomials”

Jacobi’s identities      177 182
Jacobsthal’s sums      187
Jordan’s function J, (n)      168
Kloosterman’s sums      188
L-series      192 194
Lagrange’s polynomials      see “Polynomials”
Lagrange’s theorem on four squares      182
Laguerre polynomials      see “Polynomials”
Lame functions      44 ff. 61 63 97
Lame functions of imaginary periods      69 ff.
Lame functions of periods 2K and 4K      64 ff.
Lame functions of real periods      63 ff. 68
Lame functions of the second kind      67 ff.
Lame functions, algebraic      62 68 71 88
Lame functions, coexistence of      67 71
Lame functions, degenerate      74 ff.
Lame functions, doubly-periodic      71 ff. 81
Lame functions, finite      see “Lame — Wangerin functions”
Lame functions, integral equations for      72 ff.
Lame functions, Legendre function expansions of      67 69
Lame functions, periodic      63 ff. 88
Lame functions, transformation formulas for      70
Lame functions, trigonometric expansions of      65 ff. 68
Lame polynomials      62 67 69 73 81
Lame polynomials, transformation formulas for      70
Lame wave functions      97
Lame wave functions of the first kind      161
Lame wave functions of the second and third kinds      163
Lame wave functions, characteristic curves for      161
Lame wave functions, orthogonal properties of      162
Lame — Wangerin functions      75 ff. 84 86 88
Lame — Wangerin functions, integral equations for      79 ff.
Lame — Wangerin functions, power series representing      77 ff.
Lame — Wangerin functions, series of exponentials representing      78 ff.
Lame’s equation      47 52 55 59 61 86
Lame’s equation, algebraic forms      56 ff.
Lame’s equation, asymptotic behavior of the characteristic values of h in      68
Lame’s equation, characteristic values of h in      63 ff.
Lame’s equation, degenerate cases of      74 ff.
Lame’s equation, imaginary transformation of      69 ff.
Lame’s equation, Jacobian form of      55
Lame’s equation, solutions of      62 ff.
Lame’s equation, trigonometric form of      56
Lame’s equation, Weierstrassian form of      56
Lame’s wave equation      97 159
Lame’s wave equation, power series expansions of solutions of      159 ff.
Lame’s wave equation, solutions of      159 ff.
Lame’s wave equation, subnormal solutions of      160
Laplace’s equation      45 ff. 49 80 83
Lattice points      196 ff.
Legendre polynomials      see “Polynomials”
Legendre — Jacobi symbol      183 ff. 186 193
Legendre’s equation      134 234
Liouville’s function, $\lambda(n)$       169
Mathieu functions      91 ff. 93 97 99 108 111
Mathieu functions of fractional order      114
Mathieu functions of the first kind      111 ff.
Mathieu functions of the second kind      119 ff.
Mathieu functions, addition theorem of      132
Mathieu functions, approximations to      125
Mathieu functions, asymptotic forms of      125 ff.
Mathieu functions, Bessel function expansions of      117 ff.
Mathieu functions, characteristic curves for      111
Mathieu functions, expansions in series of      132
Mathieu functions, expansions of, in series of parabolic cylinder functions      127 ff.
Mathieu functions, Fourier expansions of      115 ff.
Mathieu functions, infinite series involving      128 ff.
Mathieu functions, integral equations for      114 ff. 117
Mathieu functions, integrals involving      130 132
Mathieu functions, normalization of      111 ff.
Mathieu functions, orthogonal properties of      114 132
Mathieu functions, products of      129 ff. 133
Mathieu functions, symmetry properties of      113
Mathieu’s equation      75 92 97 134
Mathieu’s equation, algebraic      98
Mathieu’s equation, approximations to solutions of      105 ff.
Mathieu’s equation, associated      98
Mathieu’s equation, asymptotic forms of solutions of      106 ff.
Mathieu’s equation, characteristic exponent of      99 ff. 106
Mathieu’s equation, expansions of solutions of      100 ff. 103
Mathieu’s equation, integral equations satisfied by solutions of      109 ff.
Mathieu’s equation, integral relations for solutions of      107 ff. 110
Mathieu’s equation, modified      92 120
Mathieu’s equation, solutions of the first kind of      99 108 129
Mathieu’s equation, solutions of the third kind of      99 129
Mathieu’s equation, stability chart for      101
Mathieu’s equation, stable and unstable regions of      101
Mathieu’s equation, subnormal solutions of      107
Meissel’s formula      172
Mittag-Leffler’s function $E_{\alpha}(z)$       206 ff. 215
Mittag-Leffler’s function $E_{\alpha}(z)$ , functions related to      211 ff.
Mittag-Leffler’s function $E_{\alpha}(z)$ , generalizations of      210 ff. 215
Modified Mathieu functions      120 ff.
Modified Mathieu functions of the first kind      93 120
Modified Mathieu functions of the second kind      120 ff.
Modified Mathieu functions of the third kind      93 122
Modified Mathieu functions, asymptotic forms of      122 125
Modified Mathieu functions, Bessel function expansions of      120 ff.
Modified Mathieu functions, integral equations for      124
Modified Mathieu functions, integral relations for      122 ff. 130
Modular equations      24 ff.
Modular forms      18 30 36 39 178
Modular functions      see “Elliptic modular functions”
Modular functions of the nth degree      36 ff.
Modular group      see “Group” “Modular”
Moebius’ function $\mu(n)$       169
Moebius’ inversion formula      171 ff.
Moebius’ inversion formula, generalizations of      172
Multiplicative functions      169 171 185
Multiplicative functions, completely      169
Multiply-periodic functions      12 ff.
Non-associative algebra      230
Normal solutions      52
Normal solutions of Laplace’s equation      47 49 86
Normal solutions of the wave equation      92 94 97 150
Number theory, functions of      167 ff.
Orthogonal polynomials      see “Polynomials”
P-symbol      57 ff. 77
P-symbol, transformations of      58 ff.
Parabolic cylinder functions      263 275
Partitions      175 ff.
Partitions, asymptotic formulas for      178 ff.
Partitions, asymptotic theory of      211
Partitions, congruence properties of      178 ff.
Partitions, enumerating function of      176
Partitions, enumeration of      176
Partitions, generating functions for      176
Partitions, restricted      175
Partitions, theorems on      178
Partuto numerorum      189
Poisson — Charlier polynomials      see “Polynomials”
Polynomials, Appell      256
Polynomials, Appell set of      235 ff. 255 261
Polynomials, Bernoulli      234 ff. 252 259
Polynomials, Bessel      251
Polynomials, binomial      248 ff. 258
Polynomials, Charlier      255 273
Polynomials, Euler      252 ff. 260
Polynomials, Gegenbauer      246 ff. 262 265 271 274
Polynomials, generalized Bessel      251 266
Polynomials, Hermite      240 242 249 256 263 266 267 269 271 274
Polynomials, hypergometric      247 ff. 251 255 264 266 273 “Jacobi” “Legendre
Polynomials, Jacobi      247 262 267 271 274
Polynomials, Lagrange’s      267
Polynomials, Laguerre      241 249 252 255 262 271 274
Polynomials, Legendre      234 240 245 262 264 271 274
Polynomials, orthogonal      238 (see also “Gegenbauer” “Hermite” “Jacobi” “Laguerre” “Legendre
Polynomials, Poisson — Charlier      255 273
Polynomials, Stirling      257 259
Polynomials, Tchebycheff      231 ff. 245

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