Erdelyi A. — Higher Transcendental Functions, Vol. 1 :: Электронная библиотека попечительского совета мехмата МГУ

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Erdelyi A. — Higher Transcendental Functions, Vol. 1

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Название: Higher Transcendental Functions, Vol. 1

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.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1953

Количество страниц: 324

Добавлена в каталог: 13.07.2005

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Предметный указатель

$\psi$ -function      15
Automorphic functions      98
Basic hypergometric series      195 203
Basic hypergometric series of several variables      245
Bateman — Pasternack polynomials      192
Bateman's k-function      267 274
Bernoulli numbers      30 35
Bernoulli numbers expressed by zeta function      34 38
Bernoulli numbers of higher order      39
Bernoulli polynomials      36 43
Bernoulli polynomials of higher order      39
Bernoulli polynomials relation with generalized zeta function      27
Bessel functions as special case of E-function      204
Bessel functions as special case of G-function      2 16
Bessel functions, connection with confluent hypergeometric functions      252 265
Bessel functions, connection with Meijer's G-function      216
Bessel functions, expansion of confluent hypergeometric function in terms of      276
Bessel's differential equation      249
beta function      9 (see also “Incomplete beta function”)
Beta function, contour integrals for      14
Binomial coefficients relations involving      85
Binomial series, generalization of      196
Binomial series, truncated      87 101
Confluence of singularities      248
Confluent hypergeometric differential equation      248
Confluent hypergeometric differential equation, fundamental systems of solutions      258
Confluent hypergeometric differential equation, general solution of      253
Confluent hypergeometric differential equation, transformation of      251
Confluent hypergeometric functions      248
Confluent hypergeometric functions, associated      254
Confluent hypergeometric functions, asymptotic behavior of      277
Confluent hypergeometric functions, behavior at origin      262
Confluent hypergeometric functions, branch cut for      263
Confluent hypergeometric functions, connection with Bessel functions      265
Confluent hypergeometric functions, contiguous      254
Confluent hypergeometric functions, derivative of      254 258
Confluent hypergeometric functions, elementary relations for      253 257
Confluent hypergeometric functions, expansions of      275
Confluent hypergeometric functions, integral addition theorem for      271 285
Confluent hypergeometric functions, integral representations of      255 271
Confluent hypergeometric functions, integrals involving      283 287
Confluent hypergeometric functions, Laplace transforms containing      269
Confluent hypergeometric functions, logarithmic case      260
Confluent hypergeometric functions, multiplication formula for      282
Confluent hypergeometric functions, products of      286
Confluent hypergeometric functions, reduction of $\mathrm{A}_0y''+\mathrm{A}_1y'+\mathrm{A}_2y=0$ , A=ax+b, and other types of linear differential equation, to      249
Confluent hypergeometric functions, relations between      259
Confluent hypergeometric functions, series involving      284 287
Confluent hypergeometric functions, special values of parameters      265 269
Confluent hypergeometric functions, Wronskian      253 259
Confluent hypergeometric functions, zeros of      288
Conical functions      174
Convergence, associated radii of      227
Dougall — Ramanujan formula      191
E-function      see “MacRobert's E -function”
Elliptic functions, connection with basic hypergeometric series      196
Elliptic modular functions      99
Error functions, connection with confluent hypergeometric function      266
Euler 's constant      1
Euler numbers      40
Euler numbers of higher order      43
Euler polynomials      40 44
Euler polynomials of higher order      43
Euler's dilogarithm      31
Euler's identity      197
Euler's transformation of hypergeometric function      64 253
Exponent differences      90
Exponential integral, connection with confluent hypergeometric function      266
Fresnel integrals, connection with confluent hypergeometric function      267
G-function      see “Meijer's G -function”
Gamma function      1 (see also “Incomplete gamma function”)
Gamma function, asymptotic expansions      47
Gamma function, behavior of, at poles      46
Gamma function, Binet's expression for $\psi(z)$       18 21 22
Gamma function, contour integrals for      13
Gamma function, Dougall's formula      7
Gamma function, duplication formula for      5
Gamma function, expansions for $\Gamma(1+z)$ , $\psi(1+z)$ , $\Gamma(z)$ and $\mathrm{G}(1+z)$       45
Gamma function, expressions for $\log\Gamma(z)$       20 48
Gamma function, functional equations for      3
Gamma function, G(z)      20 44
Gamma function, Gauss' formula for $\psi(p/q)$       18
Gamma function, infinite products expressed by      5
Gamma function, infinite series in terms of $\psi$ -function      19
Gamma function, infinite series involving      7
Gamma function, integrals containing      49
Gamma function, Joncquiere's relation for F(z,s)      31
Gamma function, Kummer's series      23
Gamma function, Legendre's duplication formula      5
Gamma function, Lipschitz's formula      28
Gamma function, logarithmic derivative of      15 23 44
Gamma function, Multiplication formula of Gauss      4
Gamma function, singularities of      2
Gamma function, various notations for      52
Gauss' series      see “Hypergeometric series”
Gegenbauer polynomials      275
Gegenbauer's differential equation      178
Gegenbauer's functions      2 78
Generalized hypergeometric differential equations      184 203 210
Generalized hypergeometric functions      see “Generalized hypergeometric series”
Generalized hypergeometric series      182 202
Generalized hypergeometric series as special case of G-function      215
Generalized hypergeometric series of unit argument      183 188 190
Generalized hypergeometric series, $_1\mathrm{F}_1(a;c;x)$       248
Generalized hypergeometric series, basic      195
Generalized hypergeometric series, contiguous      187
Generalized hypergeometric series, convergence of      182 184
Generalized hypergeometric series, cubic transformations for $_3\mathrm{F}_2$       190 3 2
Generalized hypergeometric series, differential equations for      184
Generalized hypergeometric series, Dixon's theorem      189
Generalized hypergeometric series, Dougall's theorem      189
Generalized hypergeometric series, identities involving      185
Generalized hypergeometric series, integrals involving      192
Generalized hypergeometric series, linear transformations of $_{q+1}\mathrm{F}_{q}$       188 190
Generalized hypergeometric series, nearly-poised      188 190
Generalized hypergeometric series, Pochhammer — Barnes notation      184
Generalized hypergeometric series, quadratic transformations for $_3\mathrm{F}_2$       190
Generalized hypergeometric series, recurrence relations for      185
Generalized hypergeometric series, Saalschuetzian      188
Generalized hypergeometric series, special cases of      190 192
Generalized hypergeometric series, terminating      182
Generalized hypergeometric series, Watson's theorem      189
Generalized hypergeometric series, well-poised      188 190
Generalized hypergeometric series, Whipples' theorem      189
H-function      204
Hankel-transform of $_2\mathrm{F}_2$       192
Heine's hypergeometric series      183
Hermite polynomials connection with confluent hypergeometric function      267
Hypergeometric differential equation      56 (see also “Hypergeometric series and functions” “Generalized “Hypergeometric
Hypergeometric differential equation for Legendre polynomials      122
Hypergeometric differential equation solutions in the degenerate case of      71
Hypergeometric equations as special case of Riemann's equation      91
Hypergeometric equations, associated      95
Hypergeometric equations, degenerate case of      69
Hypergeometric equations, degenerate solution of      69
Hypergeometric equations, full solution      74
Hypergeometric equations, group of      93
Hypergeometric equations, singularities      56
Hypergeometric equations, transformation of      92
Hypergeometric function      57
Hypergeometric function as continued fractions      87
Hypergeometric function, a relation for $_2\matrm{F}_1(1,z,1+z,-1)$       20
Hypergeometric function, analytic continuation of      62 108
Hypergeometric function, associated      58
Hypergeometric function, asymptotic expansions of      75
Hypergeometric function, bilinear relations for      84
Hypergeometric function, connection with Legendre functions      141

Hypergeometric function, contiguous      57
Hypergeometric function, cubic transformation of      67 113
Hypergeometric function, degenerate case of      68
Hypergeometric function, derivatives of      58 102
Hypergeometric function, duplication formula for      83
Hypergeometric function, elementary functions as      101
Hypergeometric function, elementary relations for      102
Hypergeometric function, full solution of      74
Hypergeometric function, Gauss relations for contiguous      103
Hypergeometric function, generating function for F(b,a-b,c,z)      82
Hypergeometric function, Goursats tables of quadratic transformations for      111
Hypergeometric function, integral representations of      59 62 67 78 114
Hypergeometric function, integrals involving      79—81 205
Hypergeometric function, Kummer's relations between      105
Hypergeometric function, Legendre's relation      85
Hypergeometric function, multiplication formulas for      83
Hypergeometric function, products of      82
Hypergeometric function, quadratic transformation of      64 110
Hypergeometric function, special cases of      89 101 122 150
Hypergeometric function, special values of      104
Hypergeometric function, transformations of higher degree of      67
Hypergeometric function, uniformization of      99
Hypergeometric function, zeros of      99
Hypergeometric functions of several variables      see also “Hypergeometric series of several variables”
Hypergeometric functions of several variables as solutions of partial differential equations      231
Hypergeometric functions of several variables, analytic continuation of      241
Hypergeometric functions of several variables, Horn's list      224
Hypergeometric functions of several variables, integral representations of      229
Hypergeometric functions of several variables, reducible      238
Hypergeometric functions of several variables, reduction of, to hypergeometric functions of one variable      237
Hypergeometric functions of several variables, transformations of      239
Hypergeometric series      57 (see also “Hypergeometric functions”)
Hypergeometric series of several variables      222 (see also “Hypergeometric function of several variables”)
Hypergeometric series of several variables, complete      224
Hypergeometric series of several variables, confluent      225
Hypergeometric series of several variables, convergence of      227
Hypergeometric series of several variables, expansion of      243
Hypergeometric series of several variables, Horn's list of      224
Hypergeometric series of several variables, symbolic forms      243
Hypergeometric series, convergence of      57
Hypergeometric series, terminating      57
Hypergeometric series, truncated      191
Hyperspherical harmonics      245
Incomplete beta function      87
Incomplete gamma functions, connection with confluent hypergeometric function      266
Integral cosine, connection with confluent hypergeometric function      266
Integral sine, connection with confluent hypergeometric function      266
Jacobi polynomials as special case of hypergeometric function      81
Jacobi polynomials generating function      81
Jacobi polynomials of two variables      244
Jacobi polynomials, bilinear generating function of      244
Jacobian theta functions      99
k-function      see “Bateman's k-function”
Knar's formula      6
Kummer's formulas for hypergeometric series      64
Kummer's series for $\log \Gamma(z)$       23
Kummer's series for confluent hypergeometric function      248
Kummer's series of hypergeometric function      92 105
Kummer's transformation of confluent hypergeometric function      253
Kummer's transformation of Whittaker's function      264
Laguerre functions, connection with confluent hypergeometric function      268
Laguerre polynomials, connection with confluent hypergeometric function      268
Laguerre polynomials, expansion of confluent hypergeometric function in terms of      275
Laplace integrals for confluent hypergeometric function      255
Laplace transforms      269
Laplace transforms of generalized hypergeometric series      192
Laplace's equation      120 173
Legendre polynomials      120 150
Legendre polynomials, Christoffel's formula      153
Legendre polynomials, generating function of      154
Legendre polynomials, Heine's formula      168
Legendre polynomials, integrals involving      152
Legendre polynomials, Neumann's integral relation for      154
Legendre polynomials, orthogonality of      121 151
Legendre polynomials, relations between      152
Legendre polynomials, Rodrigues' formula for      151
Legendre polynomials, Wronskian      152
Legendre's differential equation      121
Legendre's differential equation as special case of hypergeometric equation      122
Legendre's functions      122
Legendre's functions of integral order      148
Legendre's functions of special order and degree      150
Legendre's functions on the cut (for —1<x<1)      143 161
Legendre's functions, addition theorem for      168
Legendre's functions, associated      148
Legendre's functions, asymptotic expansions for      162
Legendre's functions, behavior of, at singular point      163
Legendre's functions, derivatives of      148 181
Legendre's functions, Dougall's expansion      167
Legendre's functions, expansions in terms of      165
Legendre's functions, Ferrer's associated      179
Legendre's functions, integral representations for      155
Legendre's functions, integrals involving      169
Legendre's functions, Olbricht's solutions of      123
Legendre's functions, recurrence relations for      160
Legendre's functions, relations between      140 149
Legendre's functions, relations of, with hypergeometric functions      123 141 148
Legendre's functions, trigonometric expansions for Legendre functions on the cut      146
Legendre's functions, Whipple's formula      141
Legendre's functions, Wronskian of      123
Lerch's theorem      86
Lerch's transformation formula for $\Phi(z,s,v)$       29
Logarithmic integral, connection with confluent hypergeometric function      267
MacRobert's E-function      203
MacRobert's E-function, integrals involving      206
MacRobert's E-function, recurrence relations for      205
Mascheroni's constant      see “Euler's constant”
Mehler — Dirichlet formula      159
Meijer's G-f unction      206
Meijer's G-f unction, asymptotic expansions for      211
Meijer's G-f unction, differential equation for      210
Meijer's G-f unction, integral transforms of      214
Meijer's G-f unction, multiplication theorems      213
Meijer's G-f unction, series with      213
Meijer's G-f unction, special cases of      215
Mellin — Barnes integrals      49 232 256
Mellin's formula for infinite products      6
P-function      204
Papperitz equation      90
Parabolic cylinder functions, connection with confluent hypergeometric functions      267
Poisson's — Charlier polynomials      268
Polygamma functions      44 0
Q-function      204
Riemann's $\xi$ -function      35
Riemann's P-equation      65 89
Riemann's zeta function      see “Zeta function”
Ring functions      see “Toroidal functions”
Rogers — Ramanujan identities      197
Saalschuetz's formula      66
Schwarz's s-function      96 183
Schwarz's table      98
Schwarzian derivatives      96
Steepest descent, method of      279 281
Stirling series      47
Summation formula of Plana      22
Toroidal coordinates      173
Toroidal functions      173
Toronto functions      268
Whittaker functions      264 270 274 291
Whittaker functions as special case of E-function      205
Whittaker functions as special case of G-function      216
Wronskian of confluent hypergeometric functions      253
Wronskian of Legendre functions      123
Wronskian of Legendre polynomials      152
Zeta function      32
Zeta function, asymptotic expansion of $\zeta(s,v)$       48
Zeta function, generalized      24 27

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