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Luke Y.L. — The special functions and their approximations (volume 1)

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Название: The special functions and their approximations (volume 1)

Автор: Luke Y.L.

Аннотация:

These volumes are designed to provide scientific workers with a self-contained and unified development for many of the mathematical functions which arise in applied problems, as well as the attendant mathematical theory for their approximations. These functions are often called the special functions of mathematical physics or more simply the special functions.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1969

Количество страниц: 349

Добавлена в каталог: 15.07.2005

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Предметный указатель

— Function      148
-Function      see Chapter V (135—208)
-Function, analytic continuation      148 149 194
-Function, asymptotic expansion, exponential, recurrence formula for coefficients in      200 202 205—208
-Function, asymptotic expansion, large variable      178—180 189—194
-Function, definition      139 142—145
-Function, differential equation and solutions      181 182
-Function, elementary properties      149—152
-Function, expansion of in series of -functions      147 148 152—157 183—189
-Function, expansion of in series of generalized hypergeometric functions      139 142 145—147
-Function, hypergeometric functions and named functions, connection with      225—234
-Function, integral representations      59 143—145 159 164 175 177
-Function, integrals involving, Euler and related transforms      170—177
-Function, integrals involving, Founer transform      166 169
-Function, integrals involving, Laplace and inverse Laplace transforms      166—169
-Function, integrals involving, Mellin and inverse Mellin transforms      157—159
-Function, integrals involving, other transform pairs      177
-Function, integrals involving, product of two -functions      159 166
-Function, multiplication theorems      152—157
-Function, named functions expressed in terms of      225—230
-functions; special cases of ${}_{p}F_{q}$ Hypergeometric function (generalized), ${}_{p}F_{q}$ , analytic continuation      149
— Transform      165
— Transform      165
$\psi$ -function      see Hypergeometric function (confluent) Whittaker
Airy functions      205 217
Anger function      218
Approximation, best in Chebyshev sense      303 305
Approximation, best in least square sense      270
Approximation, best in mean square sense      268 305
Associated Bessel function      219
Asymptotic expansion      see Chapter I 1-7; asymptotic
Asymptotic expansion, definition      2
Asymptotic expansion, elementary properties      3 4
Asymptotic expansion, Watson’s lemma      4 7
Basic series      291
Ber and Bei functions      216
Bernoulli and generalized Bernoulli polynomials, definition and elementary properties      18—22
Bernoulli and generalized Bernoulli polynomials, Fourier series for $B_{n}(x)$       23
Bernoulli and generalized Bernoulli polynomials, integrals of      22 23
Bernoulli and generalized Bernoulli polynomials, recurrence formulas      20 22 35
Bernoulli and generalized Bernoulli polynomials, table of $B_{2n}^{(2a)}(x)$       34
Bernoulli and generalized Bernoulli polynomials, table of $B_{n}(x)$       20
Bernoulli and generalized Bernoulli polynomials, table of $B_{n}^{(a)}(x)$       19
Bernoulli numbers, definition      19
Bernoulli numbers, expansion of cot z, tan z, csc z, and ln cos z in series involving      23
Bernoulli numbers, table of      20
Bessel functions, -functions, connection with      226—233
Bessel functions, associated      219
Bessel functions, asymptotic expansions      204 215
Bessel functions, confluent hypergeometric function, connection with      120 135 213
Bessel functions, definitions, connecting relations, power series      39 40 212 213
Bessel functions, derivatives with respect to order at half an odd integer      216
Bessel functions, difference-differential properties      48 214
Bessel functions, differential equation      120 214
Bessel functions, expansion of exponential function in series of      287
Bessel functions, integrals involving      115 165 177 219 220 287
Bessel functions, Jacobi polynomial, connection with      52
Bessel functions, products      216 228—230 232—234
Bessel functions, Wronskians      214 215
Bessel functions, zeros of      205
Bessel polynomials      274
Beta function, complete      15 16 18
Beta function, incomplete      210
Beta transform      58 59 170—175 286
Binomial coefficient      9 35
Binomial function      38 40 49 209 210
Chebyshev polynomials of first kind      see also Jacobi polynomials
Chebyshev polynomials of first kind, approximations in series of (based on orthogonality properties with respect to summation)      308 312
Chebyshev polynomials of first kind, definition and basic properties      273 296 297 300 301
Chebyshev polynomials of first kind, difference-differential properties      297—299 301 302 316 321 324
Chebyshev polynomials of first kind, evaluation of series of, by use of backward recurrence formula      325—329
Chebyshev polynomials of first kind, expansion of $x^{m}T*_{n}(x)$ in series of      301
Chebyshev polynomials of first kind, expansion of $x^{m}T_{n}(x)$ in series of      298
Chebyshev polynomials of first kind, expansion of a ${}_{p}F_}q}$ in series of      296
Chebyshev polynomials of first kind, expansions in series of (based on orthogonality property with respect to integration), asymptotic estimate of coefficients      293—296
Chebyshev polynomials of first kind, expansions in series of (based on orthogonality property with respect to integration),differential and integral properties      314—325
Chebyshev polynomials of first kind, expansions in series of (based on orthogonality property with respect to integration),evaluation of coefficients      286—293
Chebyshev polynomials of first kind, integrals involving      293—295 299 300—302 316-325
Chebyshev polynomials of first kind, Jacobi polynomial, connection with      273 296
Chebyshev polynomials of first kind, minimax and mean square properties      303—307
Chebyshev polynomials of first kind, orthogonality property with respect to integration      273 299 301
Chebyshev polynomials of first kind, orthogonality property with respect to summation      307 310 311
Chebyshev polynomials of second kind      see also Jacobi polynomials
Chebyshev polynomials of second kind, approximations in series of (based on orthogonality properties with respect to summation)      312—314
Chebyshev polynomials of second kind, definition and basic properties      273 296—300
Chebyshev polynomials of second kind, integrals involving      299 300
Chebyshev polynomials of second kind, Jacobi polynomial, connection with      273
Chebyshev polynomials of second kind, orthogonality property with respect to integration      299
Chebyshev polynomials of second kind, orthogonality property with respect to summation      312 313
Chebyshev, best approximation in sense of      303
Christoffel — Darboux formulas      272
Confluence principle and theorems      48—57
Confluent hypergeometric function      see Hypergeometric function (confluent)
Cosecant      23 210
Cosine      23 39 210
Cosine integral      135 221—223 227
Cotangent      23
Coulomb wave functions      135 212
Cylinder function      204 213
D operator      24—26
Darboux, method of generating functions      254—259
Delta ( $\delta$ ) operator      24 26
Difference equations, use of in backward direction      317—319 325—329
Dixon’s theorem      103 104
Elliptic integrals      211
Error functions      135 223 224
Euler — Mascheroni constant, y      9
Expansions      see also Particular functions evaluation of in
Expansions, difference equation      325—329
Exponential function      38 40 48 209 287 288
Exponential integral      3 7 135 221 222
Fresnel integrals      135 224 227
Gamma function      see Chapter II (8—37)
Gamma function, asymptotic expansion, $\Gamma(z)$ and ln $\Gamma(z)$       31—33
Gamma function, asymptotic expansion, ratio of products of gamma functions      33—37
Gamma function, asymptotic expansion, recurrence formula for coefficients in asymptotic expansion for ratio of products of gamma functions      205—208
Gamma function, definite integrals expressed in terms of      15 16 60 61 157 177
Gamma function, definition and elementary properties      8—10
Gamma function, expansion for $\Gamma(z)$ and ln $\Gamma(z)$ in series of Chebyshev polynomials of first kind      28 29
Gamma function, integral representations      8 14 18
Gamma function, logarithmic derivative of      see Psi-function
Gamma function, multiplication formula      11—12
Gamma function, power series and other expansions      26 31
Gegenbauer polynomial      273 279
H — Transform      165
Hankel functions      135 204 213 215 230 234
Hankel transform      165 287
Hermite polynomials      135 273
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions      see Chapter IV (115—134) Hypergeometric
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, -function, connection with      225 226 228 231 234
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, asymptotic expansion, large variable      127 128
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, confluence principle      48
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, contiguous relations      48 118 119 126
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, definition      40 47 49
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, difference-differential properties      48 117—119 125 126
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, differential equation      119 120
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, differential equation, solutions, complete      121—124
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, differential equation, solutions, degenerate      120 121
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, differential equation, solutions, logarithmic      122 123
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, differential equation, solutions, ordinary      119
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, elementary relations      117 118
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, evaluation of certain ${}_{1}F_{1}$ for special value of argument      113
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, expansion in series of Bessel functions      129—133
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, expressed as named function      224

Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, integral representations      115—117
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, Kummer relations      121 124 126
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, named functions expressed in terms of      211—213 220—223
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, other notations and related functions      134 135
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, products      211 228 233 234
Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, Wronskians      124
Hypergeometric function (Gaussian), ${}_{2}F_{1}$       see Chapter III (38—114) Hypergeometric function (generalized)
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , -function, connection with      139
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , analytic continuation      68—71
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , asymptotic expansions for large parameters)      235—242
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , confluence principle      49
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , contiguous relations      47 89
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , convergence of series      65 68
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , definition      39 41
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , difference-differential properties      44—47 88 275 276
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , differential equation      64 93
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , differential equation, solutions, complete      65 72—84
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , differential equation, solutions, degenerate      65 66 69 77 84
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , differential equation, solutions, logarithmic      75—84
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , differential equation, solutions, ordinary      64 65 67—71
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , elementary relations      44—46
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , evaluation of, for special values of argument      99—103 114
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , expansion in series of ${}_{1}F_{1}$ ’s      130
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , expressed as named function      224
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , integral representations      57 58 62 63 89—91
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , integrals involving      170 172—174
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , Kummer relations      67 68 85—92
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , named functions expressed in terms of      209—211
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , polynomial      40
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , quadratic transformations      92—98
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , truncated series      42 109 110
Hypergeometric function (Gaussian), ${}_{2}F_{1}$ , Wronskians      84
Hypergeometric function (generalized), ${}_{p}F_{q}$       see Chapters III (38—114) IV V
Hypergeometric function (generalized), ${}_{p}F_{q}$ , -function, connection with      139 142 145—147 225 230 231
Hypergeometric function (generalized), ${}_{p}F_{q}$ , asymptotic expansion, exponential, recurrence formula for coefficients in      200 202 205 208
Hypergeometric function (generalized), ${}_{p}F_{q}$ , asymptotic expansion, large parameter(s)      51 55 56 133 242—266
Hypergeometric function (generalized), ${}_{p}F_{q}$ , asymptotic expansion, large variable      195—203
Hypergeometric function (generalized), ${}_{p}F_{q}$ , confluence principle and theorems      49 51 53—56
Hypergeometric function (generalized), ${}_{p}F_{q}$ , contiguous relations      48
Hypergeometric function (generalized), ${}_{p}F_{q}$ , convergence of series      43 44
Hypergeometric function (generalized), ${}_{p}F_{q}$ , definition      41 42 136
Hypergeometric function (generalized), ${}_{p}F_{q}$ , difference-differential properties      44 48
Hypergeometric function (generalized), ${}_{p}F_{q}$ , differential equation      136 138 247
Hypergeometric function (generalized), ${}_{p}F_{q}$ , differential equation, solutions, complete      147 148
Hypergeometric function (generalized), ${}_{p}F_{q}$ , differential equation, solutions, logarithmic      140—143
Hypergeometric function (generalized), ${}_{p}F_{q}$ , differential equation, solutions, ordinary      137—139
Hypergeometric function (generalized), ${}_{p}F_{q}$ , elementary relations      44
Hypergeometric function (generalized), ${}_{p}F_{q}$ , evaluation of ${}_{p+1}F_{p}$ for special values of argument,       99 101 102 114
Hypergeometric function (generalized), ${}_{p}F_{q}$ , evaluation of ${}_{p+1}F_{p}$ for special values of argument,       103—111 113 259
Hypergeometric function (generalized), ${}_{p}F_{q}$ , evaluation of ${}_{p+1}F_{p}$ for special values of argument,       112—114
Hypergeometric function (generalized), ${}_{p}F_{q}$ , evaluation of ${}_{p+1}F_{p}$ for special values of argument, general p      26 113 114 257—259
Hypergeometric function (generalized), ${}_{p}F_{q}$ , evaluation of certain ${}_{2}F_{2}$ for special values of argument      113
Hypergeometric function (generalized), ${}_{p}F_{q}$ , expansion in series of Chebyshev polynomials of first kind      296
Hypergeometric function (generalized), ${}_{p}F_{q}$ , expansion theorem for large parameter      51 55 56
Hypergeometric function (generalized), ${}_{p}F_{q}$ , expressed as named function      224 225
Hypergeometric function (generalized), ${}_{p}F_{q}$ , integral representations      58 63
Hypergeometric function (generalized), ${}_{p}F_{q}$ , integrals involving      58—62 164 168 169
Hypergeometric function (generalized), ${}_{p}F_{q}$ , multiplication theorem      155
Hypergeometric function (generalized), ${}_{p}F_{q}$ , named functions expressed in terms of      209—224
Hypergeometric function (generalized), ${}_{p}F_{q}$ , nearly poised      103
Hypergeometric function (generalized), ${}_{p}F_{q}$ , polynomial      42
Hypergeometric function (generalized), ${}_{p}F_{q}$ , truncated series      42 210
Hypergeometric function (generalized), ${}_{p}F_{q}$ , well poised      103 104
Incomplete gamma function and related functions      38 40 135 220—224
Incomplete gamma function and related functions Hypergeometric function (confluent), ${}_{1}F_{1}$ , $\psi$ -function, Whittaker functions, asymptotic expansion, large parameter(s)      129 133 134
Incomplete gamma function Gamma function, analytic continuation      10 11
Integrals      see appropriate modifiers such as Bessel functions integrals
Jacobi function      274
Jacobi polynomials, asymptotic expansion for large order      53 54 237 250—259 278 279
Jacobi polynomials, Bessel function, connection with      52
Jacobi polynomials, Chebyshev polynomials Orthogonal polynomials, classical, orthogonal properties of      273
Jacobi polynomials, definition and basic properties      273— 275 280
Jacobi polynomials, difference-differential properties      275 276
Jacobi polynomials, evaluation and estimation of coefficients of given when expanded in series of      286 296
Jacobi polynomials, evaluation and estimation of coefficients of given when expanded in series of, asymptotic estimates of coefficients      293
Jacobi polynomials, evaluation and estimation of coefficients of given when expanded in series of, coefficients as integral transform      286—290
Jacobi polynomials, evaluation and estimation of coefficients of given when expanded in series of, coefficients, when f(x) is defined by Taylor series      290—293
Jacobi polynomials, expansion of $x^{p}$ in series of      277 284 285 291
Jacobi polynomials, expansion of $x^{\mu}e^{zx}$ in series of      285 287
Jacobi polynomials, expansion of functions in series of      283—286 290—293
Jacobi polynomials, extended and generalized, asymptotic expansion for large order      53 54 247—263 281—283
Jacobi polynomials, extended and generalized, definition      247
Jacobi polynomials, extended and generalized, expansion of functions in series of      291
Jacobi polynomials, extended and generalized, generating function      254
Jacobi polynomials, generating functions      254 278
Jacobi polynomials, inequalities      280
Jacobi polynomials, integrals involving      276—277 281 282 286
Jacobi polynomials, orthogonal polynomials, connection with other      273
Jacobi polynomials, orthogonality property      276
Ker and Kei functions      216
Kontorovich — Lebedev transforms      177
Laguerre polynomial, asymptotic expansion for large order      264—265
Laguerre polynomial, confluent hypergeometric function, connection with      135
Laguerre polynomial, definition and orthogonality property      273
Laguerre polynomial, expansion of functions in series of      291 292
Laguerre polynomial, generalized and extended, asymptotic expansion for large order      263—266
Laguerre polynomial, generalized and extended, definition      263
Laguerre polynomial, generalized and extended, expansion of functions in series of      291 292
Laguerre polynomial, Jacobi polynomial, connection with      273
Legendre functions      178 211
Legendre polynomials      273 279
Logarithm      38 40 210
Lommel functions      217 227 234
Mehler transforms      178
Named functions expressed as -functions      225—230
Named functions expressed as -functions, as ${}_{p}F_{q}$ 's      209—224
Order symbols      1
Orthogonal functions      267—270
Orthogonal polynomials      see Chapter VIII (267—329)
Orthogonal polynomials, definition and basic properties      267—272
Parabolic cylinder functions      135 212
Pi ( $\pi$ ), asymptotic expansion for      35 36
Polynomials      see modifiers such as Bessel orthogonal
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, asymptotic expansion      33
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, definition and elementary properties      12 13
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, expansion Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, in series of Chebyshev polynomials of the first kind      28 29
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, integral representations      13—15 28
Psi ( $\psi$ )-function or logarithmic derivative of the gamma function, power series expansion      26
Recurrence formulas, use of in backward direction      317—319 325—329
Rummer’s formula      57
Rummer’s solutions      67
Saalsch $\ddot{u}$ tz’s formula      103
Sine      39 210
Sine integral      135 221—223 227
Sine, inverse of      210
Stokes phenomenon      128 199
Struve functions, asymptotic expansion      219
Struve functions, definition      217
Struve functions, difference-differential properties      218
Struve functions, expressed as -function      227 233
Struve functions, integrals involving      165 220
Tangent      23
Tangent, inverse of or arc tan      38 40 210
Ultraspherical polynomial      273 279
Vandermonde’s theorem      99
Watson’s formula      104
Watson’s lemma      4—7
Weber function      218
Whipple’s formula      104
Whittaker functions      see also Hypergeometric function confluent
Whittaker functions, definition      134
Whittaker functions, expressed as -function      225 226 228 231 233 234
Wronskians      84 124
Zeta-function (Riemann)      27

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