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

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

Àâòîð: Luke Y.L.

Àííîòàöèÿ:

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.

ßçûê:

Ðóáðèêà: Ìàòåìàòèêà/

Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö

ed2k: ed2k stats

Ãîä èçäàíèÿ: 1969

Êîëè÷åñòâî ñòðàíèö: 349

Äîáàâëåíà â êàòàëîã: 15.07.2005

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
 — 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 Hypergeometric function (generalized), , analytic continuation      149 — Transform      165 — Transform      165 -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       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       34 Bernoulli and generalized Bernoulli polynomials, table of       20 Bernoulli and generalized Bernoulli polynomials, table of       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 in series of      301 Chebyshev polynomials of first kind, expansion of in series of      298 Chebyshev polynomials of first kind, expansion of a 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 () 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, and ln      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 and ln 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), , -function, Whittaker functions      see Chapter IV (115—134) Hypergeometric Hypergeometric function (confluent), , -function, Whittaker functions, -function, connection with      225 226 228 231 234 Hypergeometric function (confluent), , -function, Whittaker functions, asymptotic expansion, large variable      127 128 Hypergeometric function (confluent), , -function, Whittaker functions, confluence principle      48 Hypergeometric function (confluent), , -function, Whittaker functions, contiguous relations      48 118 119 126 Hypergeometric function (confluent), , -function, Whittaker functions, definition      40 47 49 Hypergeometric function (confluent), , -function, Whittaker functions, difference-differential properties      48 117—119 125 126 Hypergeometric function (confluent), , -function, Whittaker functions, differential equation      119 120 Hypergeometric function (confluent), , -function, Whittaker functions, differential equation, solutions, complete      121—124 Hypergeometric function (confluent), , -function, Whittaker functions, differential equation, solutions, degenerate      120 121 Hypergeometric function (confluent), , -function, Whittaker functions, differential equation, solutions, logarithmic      122 123 Hypergeometric function (confluent), , -function, Whittaker functions, differential equation, solutions, ordinary      119 Hypergeometric function (confluent), , -function, Whittaker functions, elementary relations      117 118 Hypergeometric function (confluent), , -function, Whittaker functions, evaluation of certain for special value of argument      113 Hypergeometric function (confluent), , -function, Whittaker functions, expansion in series of Bessel functions      129—133 Hypergeometric function (confluent), , -function, Whittaker functions, expressed as named function      224 Hypergeometric function (confluent), , -function, Whittaker functions, integral representations      115—117 Hypergeometric function (confluent), , -function, Whittaker functions, Kummer relations      121 124 126 Hypergeometric function (confluent), , -function, Whittaker functions, named functions expressed in terms of      211—213 220—223 Hypergeometric function (confluent), , -function, Whittaker functions, other notations and related functions      134 135 Hypergeometric function (confluent), , -function, Whittaker functions, products      211 228 233 234 Hypergeometric function (confluent), , -function, Whittaker functions, Wronskians      124 Hypergeometric function (Gaussian),       see Chapter III (38—114) Hypergeometric function (generalized) Hypergeometric function (Gaussian), , -function, connection with      139 Hypergeometric function (Gaussian), , analytic continuation      68—71 Hypergeometric function (Gaussian), , asymptotic expansions for large parameters)      235—242 Hypergeometric function (Gaussian), , confluence principle      49 Hypergeometric function (Gaussian), , contiguous relations      47 89 Hypergeometric function (Gaussian), , convergence of series      65 68 Hypergeometric function (Gaussian), , definition      39 41 Hypergeometric function (Gaussian), , difference-differential properties      44—47 88 275 276 Hypergeometric function (Gaussian), , differential equation      64 93 Hypergeometric function (Gaussian), , differential equation, solutions, complete      65 72—84 Hypergeometric function (Gaussian), , differential equation, solutions, degenerate      65 66 69 77 84 Hypergeometric function (Gaussian), , differential equation, solutions, logarithmic      75—84 Hypergeometric function (Gaussian), , differential equation, solutions, ordinary      64 65 67—71 Hypergeometric function (Gaussian), , elementary relations      44—46 Hypergeometric function (Gaussian), , evaluation of, for special values of argument      99—103 114 Hypergeometric function (Gaussian), , expansion in series of ’s      130 Hypergeometric function (Gaussian), , expressed as named function      224 Hypergeometric function (Gaussian), , integral representations      57 58 62 63 89—91 Hypergeometric function (Gaussian), , integrals involving      170 172—174 Hypergeometric function (Gaussian), , Kummer relations      67 68 85—92 Hypergeometric function (Gaussian), , named functions expressed in terms of      209—211 Hypergeometric function (Gaussian), , polynomial      40 Hypergeometric function (Gaussian), , quadratic transformations      92—98 Hypergeometric function (Gaussian), , truncated series      42 109 110 Hypergeometric function (Gaussian), , Wronskians      84 Hypergeometric function (generalized),       see Chapters III (38—114) IV V Hypergeometric function (generalized), , -function, connection with      139 142 145—147 225 230 231 Hypergeometric function (generalized), , asymptotic expansion, exponential, recurrence formula for coefficients in      200 202 205 208 Hypergeometric function (generalized), , asymptotic expansion, large parameter(s)      51 55 56 133 242—266 Hypergeometric function (generalized), , asymptotic expansion, large variable      195—203 Hypergeometric function (generalized), , confluence principle and theorems      49 51 53—56 Hypergeometric function (generalized), , contiguous relations      48 Hypergeometric function (generalized), , convergence of series      43 44 Hypergeometric function (generalized), , definition      41 42 136 Hypergeometric function (generalized), , difference-differential properties      44 48 Hypergeometric function (generalized), , differential equation      136 138 247 Hypergeometric function (generalized), , differential equation, solutions, complete      147 148 Hypergeometric function (generalized), , differential equation, solutions, logarithmic      140—143 Hypergeometric function (generalized), , differential equation, solutions, ordinary      137—139 Hypergeometric function (generalized), , elementary relations      44 Hypergeometric function (generalized), , evaluation of for special values of argument,       99 101 102 114 Hypergeometric function (generalized), , evaluation of for special values of argument,       103—111 113 259 Hypergeometric function (generalized), , evaluation of for special values of argument,       112—114 Hypergeometric function (generalized), , evaluation of for special values of argument, general p      26 113 114 257—259 Hypergeometric function (generalized), , evaluation of certain for special values of argument      113 Hypergeometric function (generalized), , expansion in series of Chebyshev polynomials of first kind      296 Hypergeometric function (generalized), , expansion theorem for large parameter      51 55 56 Hypergeometric function (generalized), , expressed as named function      224 225 Hypergeometric function (generalized), , integral representations      58 63 Hypergeometric function (generalized), , integrals involving      58—62 164 168 169 Hypergeometric function (generalized), , multiplication theorem      155 Hypergeometric function (generalized), , named functions expressed in terms of      209—224 Hypergeometric function (generalized), , nearly poised      103 Hypergeometric function (generalized), , polynomial      42 Hypergeometric function (generalized), , truncated series      42 210 Hypergeometric function (generalized), , well poised      103 104 Incomplete gamma function and related functions      38 40 135 220—224 Incomplete gamma function and related functions Hypergeometric function (confluent), , -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 in series of      277 284 285 291 Jacobi polynomials, expansion of 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 '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 (), asymptotic expansion for      35 36 Polynomials      see modifiers such as Bessel orthogonal Psi ()-function or logarithmic derivative of the gamma function, asymptotic expansion      33 Psi ()-function or logarithmic derivative of the gamma function, definition and elementary properties      12 13 Psi ()-function or logarithmic derivative of the gamma function, expansion Psi ()-function or logarithmic derivative of the gamma function, in series of Chebyshev polynomials of the first kind      28 29 Psi ()-function or logarithmic derivative of the gamma function, integral representations      13—15 28 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 Saalschtz’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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