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Erdelyi A. — Higher Transcendental Functions, Vol. 2
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Íàçâàíèå: Higher Transcendental Functions, Vol. 2
Àâòîð: Erdelyi A.
Àííîòàöèÿ: 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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Ãîä èçäàíèÿ: 1955
Êîëè÷åñòâî ñòðàíèö: 408
Äîáàâëåíà â êàòàëîã: 13.07.2005
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Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
scalar product of vectors 232 273
Hankel’s symbol 10
scalar product of functions 153 264
length of vector 232
(t), (t) Airy integrals 29
$B_{n;\mu,\nu}(z) Gegenbauer’s polynomial 34
Gegenbauer polynomials 174
Parabolic cylinder function 117
modified exponential integral 143
330
, exponential integral 143
Appell’s polynomials 271
function used in astrophysics and nuclear physics 134
Truncated exponential series 136
Weber’s function 35
Appell’s polynomial 271
, invariants of Weierstrass’ elliptic functions 299 305
Hermite polynomials of several variables 285
, Kelvin’s functions, 6 H (x) error function 147
, Bessel functions of the third kind 4
Hermite polynomial of several variables 285
8
Hermite polynomials 193
Struve’s function 37
erfc x repeated integral of error function 149
$I_{\nu}(z) modified Bessel function of the first kind 5
absolute invariant 375
cut Bessel function of the first kind 21
Anger’s function 35
Bessel function of the first first kind 4
modulus of Jacobi’s elliptic functions and integrals 300 306
, modified Kelvin functions 6
function used in astrophysics and nuclear physics 134
modified Bessel function of the third kind 5
Laguerre polynomial 188
polynomial 140
modified Struve function 38
Gauss transform 195
Neumann’s polynomial 32
Jacobi polynomial 168
345
polynomials associated with Jacobi polynomials 171
Jacobi function of the second kind 170
Legendre function of the second kind 180
Legendre function of the second kind on the cut 181
Lommel’s polynomials 34
Schl fli’s polynomial 34
Lommel’s function 40
Lommel’s function 40
Tchebichef polynomial 183
polynomials of Hermite and Didon 277
Tchebichef polynomial 183
Lommel’s function of two variables 42
polynomials of Hermite and Didon 274 ff.
Lommel’s function of two variables 42
Wronskian 12
spherical surface harmonic 250
Bessel function of the second kind 4
Bessel function 2 48
(z) Gegenbauer’s polynomial 34
error function 147
discriminant 332
Laplace’s operator 2 115 234
317
, 329
330
Euler — Mascheroni constant vol. I p.
Weierstrass’ elliptic function 323 328
complementary incomplete gamma function 133
incomplete gamma function 133
modified incomplete gamma function 140
modified incomplete gamma function 133
loop integral 15
\tau)$ modular function 374
complete elliptic integral 321
complete elliptic integral 321
complete elliptic integral 321
complete elliptic integrals of the second kind 314 317
Incomplete elliptic integral of the second kind 300 313
complete elliptic integrals of the first kind 314 317
multi-dimensional Gauss transform 289 290
Appell’s polynomial 270
Bessel’s differential operator 4
periods of Weierstrass’ elliptic functions 328
Neumann’s polynomial 34
periods of Weierstrass’ elliptic functions 330
complete elliptic integral of the third kind 317
incomplete elliptic integral of the third kind 301 313
Weierstrass’ sigma function 329
sigma functions 330
328
Theta functions 355
Hermite’s theta function 360
Weierstrass’ zeta function 329
A(t) Airy function 200
Abelian integrals 295 ff.
Absolute invariant 375
Airy’s integrals 22
am u Jacobi’s function 322
Anger’s function 35 ff. 84 99 103
Appell series 280 ff.
Approximation of quadratically integrable functions 156
Associated polynomials 162 ff.
Automorphic functions 296 374
Barnes’ integral representations of Bessel functions 21 ff.
Basset’s function see “Modified Bessel function of the third kind”
bei , Kelvin’s functions 6
ber x, Kelvin’s functions 6
Bessel coefficients 6
Bessel coefficients, generating function for 7
Bessel coefficients, integral representations for 13 ff. 81
Bessel function of the first kind 4
Bessel function of the first kind, derivative with respect to the order 7
Bessel function of the first kind, duplication formula for 45
Bessel function of the first kind, inequalities for 14 66
Bessel function of the first kind, series involving 63 ff.
Bessel function of the first kind, series involving, Bessel function of the first kind, zeros of 59 ff.
Bessel function of the second kind 4
Bessel function of the second kind, of integer order 7
Bessel function of the second kind, of order zero 8
Bessel function of the second kind, zeros of 61 ff.
Bessel function of the third kind 4
Bessel function of the third kind, zeros of 62
Bessel functions, addition theorems for 43 ff. 101
Bessel functions, addition theorems for, Bessel functions, analytic continuation of 12 80
Bessel functions, and wave motion 2 ff.
Bessel functions, as limits of Jacobi polynomials 173
Bessel functions, as limits of Laguerre polynomials 191
Bessel functions, asymptotic expansions for 22 ff. 85
Bessel functions, asymptotic expansions for, differential equations for 13
Bessel functions, differentiation formulas for 11
Bessel functions, integral representations for 14 ff. 57 81
Bessel functions, integrals involving 45 ff. 57 90
Bessel functions, notations for 3
Bessel functions, of imaginary order 87 ff.
Bessel functions, of order see “Spherical Bessel functions”
Bessel functions, of order 10 79
Bessel functions, power series for products of 10 ff.
Bessel functions, recurrence relations for 12
Bessel functions, relations with Legendre functions 55 ff.
Bessel functions, series involving 58 63 98
Bessel functions, series involving, Bessel functions, Wronskians of 12 79
Bessel functions, series involving, Bessel functions, zeros of 58 ff.
Bessel polynomials 10
Bessel’s differential equation 3 ff.
Bessel’s differential equation, Bessel’s inequality 157
Bilinear forms 284
Biorthogonal system 265
Birational invariant 295
Birational transformation 295
C (x) Fresnel integral 149
C (x, ) generalized Fresnel integral 149
cd u Glaisher’s function 322
cell 325
Chi x modified cosine integral 146
Christoffel numbers 161
Christoffel — Darboux formula 159 269
Ci x cosine integral 145
Classical orthogonal polynomials 163 ff. (see also “Legendre Gegenbauer Hermite Laguerre
Classical orthogonal polynomials, characterization of 164
Classical orthogonal polynomials, differential equation for 166 ff.
Classical orthogonal polynomials, differential equation for, properties of 164 166
Classical orthogonal polynomials, differentiation formula for 167
cn u Jacobi’s elliptic function 322
Complete elliptic integrals see “Elliptic integrals“
Confluent hypergeometric functions, expansion in terms of parabolic cylinder functions 124
Conformal mappings, involving elliptic functions and integrals 376 ff.
Convergence in mean of generalized Fourier expansions 157
Convolution 45
Cornu’s spiral 151
Cosine integral 145 ff.
cs u Glaisher’s function 322
Cut Bessel functions 22
dc u Glaisher’s function 322
Didon series 280 ff.
Didon series, Dini series of Bessel functions 70 ff.
dn u Jacobi’s elliptic function 322
Doubly-periodic functions 323 ff. (see also “Elliptic functions“)
ds u Giaisher’s function 322
E(u) Jacobi’s function 343
Ei (x) exponential integral 143
Elliptic functions 294 ff. 322 325
Elliptic functions, addition theorems satisfied by 328
Elliptic functions, differential equations satisfied by 327
Elliptic functions, expression of, in terms of , 335
Elliptic functions, expression of, in terms of sigma functions 337
Elliptic functions, expression of, in terms of zeta functions 336 ff.
Elliptic functions, general properties of 325 ff.
Elliptic functions, integrals of, expressed in terms of Weierstrass’ functions 337
Elliptic functions, Jacobian 322 ff. 340
Elliptic functions, Jacobian, addition theorems for 344
Elliptic functions, Jacobian, degenerate cases of 354
Elliptic functions, Jacobian, expressed in terms of theta functions 362
Elliptic functions, Jacobian, expressed in terms of Weierstrass’ functions 340 ff.
Elliptic functions, Jacobian, Landen’s transformation of 372
Elliptic functions, Jacobian, linear transformation of 367 ff.
Elliptic functions, Jacobian, periods, zeros, poles and residues of 341
Elliptic functions, Jacobian, quadratic transformations of 372 ff.
Elliptic functions, Jacobian, special values of 346 ff.
Elliptic functions, Jacobian, with 349 ff.
Elliptic functions, Neville’s notation for 294 342
Elliptic functions, order of 326
Elliptic functions, residues of 326
Elliptic functions, transformations of 365 ff.
Elliptic functions, Weierstrass 323 328
Elliptic functions, Weierstrass’ duplication formula for 333
Elliptic functions, Weierstrass’ expressed in terms of theta functions 360 ff.
Elliptic functions, Weierstrass’, addition theorem for 332 ff.
Elliptic functions, Weierstrass’, degenerate cases of 339 ff.
Elliptic functions, Weierstrass’, differential equation for 331 ff.
Elliptic functions, Weierstrass’, Landen’s transformation of 371 ff.
Elliptic functions, Weierstrass’, linear transformations of 367 ff.
Elliptic functions, Weierstrass’, quadratic transformations of 371 ff.
Elliptic functions, Weierstrass’, with real invariants 338 ff.
Elliptic integrals 294 ff.
Elliptic integrals, addition theorems for 315
Elliptic integrals, complete, expressed in terms of hypergeometric series 318
Elliptic integrals, complete, integration formulas for 322
Elliptic integrals, complete, Legendre’s relation for 320
Elliptic integrals, complete, of the first, second, and third kind 314 317
Elliptic integrals, complete, particular cases of 320
Elliptic integrals, complete, transformations of 318 ff.
Elliptic integrals, differentiation formulas for 317 321
Elliptic integrals, expressed in terms of theta functions 363ff.
Elliptic integrals, expressed in terms of Weierstrass’ functions 337 ff.
Elliptic integrals, interchange theorem for 303 ff. 315
Elliptic integrals, inversion of 322 ff.
Elliptic integrals, Landen’s transformation of 317
Elliptic integrals, Legendre’s form of 300 ff. 314
Elliptic integrals, Legendre’s, evaluation of 308 ff.
Elliptic integrals, linear transformations of 315 ff.
Elliptic integrals, Low’s form of 301
Elliptic integrals, moduli of periodicity of 303
Elliptic integrals, of the first, second, and third Elliptic integrals, kinds 299 ff. 313
Elliptic integrals, periods of 303 314
Elliptic integrals, reduction of 296 ff. 304
Elliptic integrals, reduction to Legendre’s normal form 305 ff.
Elliptic integrals, reduction to Weierstrass’ normal form 304 ff.
Elliptic integrals, singularities of 303 314
Elliptic integrals, Weierstrass’ form of 299 ff.
Elliptic modular functions 374 ff.
Equianharmonic elliptic functions and integrals 306 320
Erf x error function 147
Erfc x complementary error function 147
Erfi x modified error function 147
Error functions 147 ff.
Error functions, connection with parabolic Error functions, cylinder functions 119
Error functions, expansions in terms of Bessel Error functions, functions 148
Error functions, power series expansions of 147
Error functions, repeated integrals of 149
Exponential integrals 143 ff.
Exponential integrals, expressed in terms of confluent Exponential integrals, hypergeometric functions 143
Exponential integrals, expressed in terms of incomplete Exponential integrals, gamma functions 143
Exponential integrals, generalizations of 145
F Incomplete elliptic integral of the first kind 300 313
Field of elliptic functions 327
Field, differential 327
Fourier coefficients (generalized) 156
Fourier series (generalized) 156
Fourier — Bessel series 70 ff. 104
Fresnel integrals 149 ff.
Fresnel integrals, connection with error functions 149
Fresnel integrals, connection with incomplete gamma functions 149
Functions of the parabolic cylinder see “Parabolic cylinder functions“
Functions of the paraboloid of revolution 126 ff.
Fundamental period-parallelogram, 325
Fundamental region of the -group 375
Fundamental region of the modular group 375
Funk — Hecke theorem 247
Gauss transforms 194
Gauss transforms, multi-dimensional 289 ff.
Gegenbauer polynomials 164 174 235
Gegenbauer polynomials, asymptotic behavior as n 198
Gegenbauer polynomials, connection with Legendre functions 177
Gegenbauer polynomials, expressed as hypergeometric functions 175 ff.
Gegenbauer polynomials, generating functions of 177
Gegenbauer polynomials, inequalities for 206 ff.
Gegenbauer polynomials, integral representations for 177
Gegenbauer polynomials, monotonic properties of 208
Gegenbauer polynomials, recurrence formula for 175
Gegenbauer polynomials, Rodrigues’ formula for 175
Gegenbauer polynomials, series of 177 ff. 213
Gegenbauer polynomials, zeros of 203 ff. 387
Gegenbauer’s addition theorem for Bessel functions 43 ff.
Gegenbauer’s polynomials 34
Generalized Dirichlet series 72 ff.
Genus of algebraic curves 295
Glaisher’s notation,of Jacobian elliptic functions 322
Graf’s addition theorem for Bessel functions 44 ff.
Gram’s determinant 155
Gubler’s integral representations, of Bessel functions 17 ff.
Hankel’s function see “Bessel function of the third kind“
Hankel’s infinite integral, involving Bessel functions 49
Hankel’s integral representations of Bessel functions 15 ff.
Hankel’s inversion theorem 73
Hardy’s generalization of 73 ff.
Hardy’s generalization of, Hankel’s symbol ( , m) 10
Harmonic polynomials 237 ff.
Harmonic polynomials, complete set of 239 ff.
Hermite polynomials 164 192
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