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Erdelyi A. Higher Transcendental Functions, Vol. 3
Erdelyi A.  Higher Transcendental Functions, Vol. 3









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: Higher Transcendental Functions, Vol. 3

: Erdelyi A.

:

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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: 13.07.2005

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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, Burnsides      35
Automorphic functions, general theorems for      27 ff.
Automorphic functions, Siegels      36 ff.
Automorphic functions, simple      9
Automorphic functions, Whittakers      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.
Darbouxs 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 Lames 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
Eulers function $\phi(n)$      168 193
Eulers identities      176 ff.
Floquets 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 Lagranges 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, Hilberts      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
Heuns equation      57 ff. 98
Hille Hardy formula      272
Hills equation      133
Hills 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
Jacobis identities      177 182
Jacobsthals sums      187
Jordans function J, (n)      168
Kloostermans sums      188
L-series      192 194
Lagranges polynomials      see Polynomials
Lagranges 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.
Lames equation      47 52 55 59 61 86
Lames equation, algebraic forms      56 ff.
Lames equation, asymptotic behavior of the characteristic values of h in      68
Lames equation, characteristic values of h in      63 ff.
Lames equation, degenerate cases of      74 ff.
Lames equation, imaginary transformation of      69 ff.
Lames equation, Jacobian form of      55
Lames equation, solutions of      62 ff.
Lames equation, trigonometric form of      56
Lames equation, Weierstrassian form of      56
Lames wave equation      97 159
Lames wave equation, power series expansions of solutions of      159 ff.
Lames wave equation, solutions of      159 ff.
Lames wave equation, subnormal solutions of      160
Laplaces equation      45 ff. 49 80 83
Lattice points      196 ff.
Legendre polynomials      see Polynomials
Legendre Jacobi symbol      183 ff. 186 193
Legendres equation      134 234
Liouvilles 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
Mathieus equation      75 92 97 134
Mathieus equation, algebraic      98
Mathieus equation, approximations to solutions of      105 ff.
Mathieus equation, associated      98
Mathieus equation, asymptotic forms of solutions of      106 ff.
Mathieus equation, characteristic exponent of      99 ff. 106
Mathieus equation, expansions of solutions of      100 ff. 103
Mathieus equation, integral equations satisfied by solutions of      109 ff.
Mathieus equation, integral relations for solutions of      107 ff. 110
Mathieus equation, modified      92 120
Mathieus equation, solutions of the first kind of      99 108 129
Mathieus equation, solutions of the third kind of      99 129
Mathieus equation, stability chart for      101
Mathieus equation, stable and unstable regions of      101
Mathieus equation, subnormal solutions of      107
Meissels formula      172
Mittag-Lefflers function $E_{\alpha}(z)$      206 ff. 215
Mittag-Lefflers function $E_{\alpha}(z)$, functions related to      211 ff.
Mittag-Lefflers 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 Laplaces 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, Lagranges      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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