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Luke Y.L. — Special Functions and Their Approximations. Volume II

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Íàçâàíèå: Special Functions and Their Approximations. Volume II

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: 506

Äîáàâëåíà â êàòàëîã: 17.11.2013

Îïåðàöèè: Ïîëîæèòü íà ïîëêó | Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
 -function      see "Hypergeometric function (confluent)" -Method      II 66—91 Airy functions      I 205 217 see "Approximation" "Expansion" Anger — Weber functions      I 218 II 39 54 55 Approximation      II 287—291 Approximation, Airy functions      I 217 see Approximation, ber, bei      I 216 see Approximation, Bessel functions, , RFP      II 82 83 Approximation, Bessel functions, , RF      II 259—265 Approximation, Bessel functions, $H_{\nu}^{(1)}(x), BCRF II 290 291 Approximation, Bessel functions, II 229—232 436—448 Approximation, Bessel functions, , by approximate inversion of its Laplace transform II 265 Approximation, Bessel functions, , , TR II 220 Approximation, Bessel functions, , TR II 218 219 265 Approximation, Bessel functions, , , , , BCP II 290 291 Approximation, Bessel functions, , , , , R II 162 165 Approximation, Bessel functions, , , zeros of, RF II 228—233 Approximation, Bessel functions, , zeros of, RF II 228—233 259 264 Approximation, Bessel functions, , RFP II 83 84 Approximation, Bessel functions, , P II 111—116 Approximation, Bessel functions, , RF II 111—116 229 230 436—448 Approximation, Bessel functions, , TR II 221—225 Approximation, Bessel functions, Airy functions I 217 Approximation, Bessel functions, ber, bei, ker, kei, and derivatives, BCP II 291 I 216 Approximation, best in sense of, Chebyshev I 303—305 II 68 Approximation, best in sense of, least square I 270 Approximation, best in sense of, mean square I 268 305 Approximation, best in sense of, other II 72 73 Approximation, beta function (incomplete), P, RF, RFP II 178 Approximation, binomial function, }$, RFP      II 179 Approximation, binomial function, }$, Newton — Raphson process II 183 184 Approximation, binomial function, }$, Newton — Raphson process, P      II 387 388 Approximation, binomial function, }$, Newton — Raphson process, RFP II 182—185 269 395 Approximation, binomial function, }$, Newton — Raphson process, sum of partial fractions      II 183 269 Approximation, binomial function, , RFP      II 397 Approximation, binomial function, , BCP      II 288 Approximation, binomial function, , , RF      II 180 181 Approximation, binomial function, , RFP      II 185 396 Approximation, binomial function, |x|, BCP      II 288 Approximation, binomial function, |x|, RFP      II 185 396 Approximation, cosine      see "Approximation sine Approximation, cosine integral      see "Approximation sine Approximation, Debye functions, RFP      II 273—281 Approximation, derivatives, repeated, R      II 165 Approximation, differential equation, solution of, first order Riccati, RFP      II 77—86 Approximation, differential equation, solution of, generalized second order Riccati, RFP      II 86—91 Approximation, differential equations, integral equations, and other functional equations, solution of, collocation, or selected points      II 69 Approximation, differential equations, integral equations, and other functional equations, solution of, P      II 66—69 Approximation, differential equations, integral equations, and other functional equations, solution of, RF      II 66—69 256—269 Approximation, Duffing's equation, RFP      II 89 90 Approximation, economized      II 66 105 106 111 Approximation, elliptic functions (Jacobian), sn(x,k), cn(x,k), RFP      II 90 91 Approximation, elliptic integrals, first two kinds, complete, BCP      II 291 Approximation, elliptic integrals, three kinds, complete and incomplete, RFP      II 85 86 269—273 Approximation, elliptic integrals, three kinds, complete, TR      II 220 221 Approximation, error functions, , zeros of, RFP      II 227 228 Approximation, error functions, , RFP      II 422—429 Approximation, error functions, , P and RF      II 106 107 Approximation, error functions, , RFP      II 206 207 213 422 423 430—435 Approximation, error functions, , TR      II 225 226 Approximation, error functions, repeated integrals, R      II 165 Approximation, exponential function, , BCP      II 288 Approximation, exponential function, , BCP      II 288 Approximation, exponential function, , P and RF      II 69—73 Approximation, exponential function, , RFP      II 73—75 83 191—193 256 399 Approximation, exponential function, , RFP      II 273 274 Approximation, exponential integral, , P and RF      II 196—198 403 411—414 Approximation, exponential integral, , RFP      II 197 402 405—408 Approximation, exponential integral, , P and RF      II 106 107 Approximation, exponential integral, , RFP      II 206 207 212 213 403 404 415—421 Approximation, exponential integral, ratio involving, RFP      II 265—267 Approximation, exponential integral, repeated integrals, R      II 165 Approximation, Fermi — Dirac integrals, BCP and BCRF      II 291 Approximation, Fresnel integrals, P      II 290 Approximation, Fresnel integrals, RFP      II 207 422—435 Approximation, Fresnel integrals, TR      II 226 Approximation, G-function, P and RF      II 118—132 Approximation, gamma function      see also "Approximation incomplete Approximation, gamma function, , BCRF      II 288 Approximation, gamma function, , BCP      II 287 Approximation, gamma function, logarithmic derivative, , BCP      II 287 Approximation, hypergeometric function, confluent, the -function,       see "Approximation incomplete z)$"/>" Approximation, hypergeometric function, confluent, the -function, , P and RF II 111—116 Approximation, hypergeometric function, confluent, the -function, , RFP II 202 Approximation, hypergeometric function, confluent, the , see "Approximation incomplete z)$"/>" Approximation, hypergeometric function, confluent, the , u'(z)/u(z), , RFP      II 80—83 196 Approximation, hypergeometric function, Gaussian, the ,       II 167—185 Approximation, hypergeometric function, Gaussian, the , , P and RF      II 167—169 175—176 Approximation, hypergeometric function, Gaussian, the , , RFP      II 84 169—173 176—178 Approximation, hypergeometric function, Gaussian, the , u'(z)/u(z), , RFP      II 173—175 Approximation, hypergeometric function, generalized, the , P and RF      II 92—132 150 Approximation, incomplete gamma function      II 186—213 see sine "Approximation error "Approximation exponential "Approximation Fresnel "Approximation hypergeometric "Approximation confluent" Approximation, incomplete gamma function, , P and RF      II 186—188 Approximation, incomplete gamma function, , RFP      II 188—194 Approximation, integrals by use of orthogonal polynomials      II 243—245 Approximation, integrals, Simpson's rule and other Newton — Cotes formulas      II 218 Approximation, integrals, TR      II 214—226 see Laplace "Approximation sine Approximation, ker, kei      I 216 see Approximation, Laplace integral or transform, numerical integration      II 245 Approximation, Laplace integral or transform, P and RF      II 106—118 Approximation, Laplace transform, inversion of, numerical integration      II 251—255 Approximation, Laplace transform, inversion of, orthogonal polynomials      II 249—251 Approximation, Laplace transform, inversion of, RF      II 255—269 Approximation, Laplace transform, inversion of, TR      II 225 Approximation, Legendre functions, R      II 165 Approximation, logarithm, , RFP      II 393 394 Approximation, logarithm, ln(l+z), BCP      II 288 Approximation, logarithm, ln(l+z), P      II 385 386 Approximation, logarithm, ln(l+z), RFP      II 389 390 Approximation, logarithm, ln[(1+z)/(1-z)], RFP      II 391 392 Approximation, Lommel functions, P and RF      II 116—118 Approximation, Mellin — Barnes integral, TR      II 225 Approximation, Painleve's first transcendent, RFP      II 89 Approximation, Power series or ratio of two power series, RFP      II 76 Approximation, psi()-function or logarithmic derivative of the gamma function      see "Approximation gamma Approximation, Riccati differential equations, solutions of, RFP      II 77—91 Approximation, sine and cosine, integrals, , zeros of, RFP      II 228 Approximation, sine and cosine, integrals, , , RF      II 196—198 402—408 415—421 Approximation, sine and cosine, integrals, , , RFP      II 197 403 404 409 410 Approximation, sine and cosine, integrals, , , RF      II 207 403 404 415—421 Approximation, sine and cosine, inverse, arc sin z, RFP      II 393 394 Approximation, sine and cosine, sin z, cos z, BCP      II 288 Approximation, sine and cosine, sin z, cos z, RF      II 74 193 399—401 Approximation, Struve functions, P and RF      II 116—118 Approximation, Struve functions, RF      II 449—452 Approximation, tangent, inverse, arc tan z, BCP      II 288 Approximation, tangent, inverse, arc tan z, P      II 385 386 Approximation, tangent, inverse, arc tan z, RFP      II 391 392 Approximation, tangent, tan z, BCP      II 288 Approximation, tangent, tan z, RF      II 399—401 Approximation, tangent, tan z, RFP      II 83 Approximation, Whittaker function      see "Hypergeometric function confluent" Associated Bessel function      I 219 Asymptotic expansion      I 1—7 see asymptotic" Asymptotic expansion, elementary properties      I 3 4 Asymptotic expansion, expansion definition      I 2 Asymptotic expansion, Watson's lemma      I 4—7 Basic series      I 291 II 283 bcp      see "Best polynomial in the Chebyshev sense" BCRF      see "Best rational function in the Chebyshev sense" Ber, bei functions      I 216 II 291 347—351 see "Approximation" "Expansion" Bernoulli and generalized Bernoulli polynomials, definition and elementary properties      I 18—22 Bernoulli and generalized Bernoulli polynomials, Fourier series for       I 23 Bernoulli and generalized Bernoulli polynomials, integrals of      I 22 23 Bernoulli and generalized Bernoulli polynomials, recurrence formulas      I 20 22 35 Bernoulli and generalized Bernoulli polynomials, table of       I 34 Bernoulli and generalized Bernoulli polynomials, table of       I 19 Bernoulli and generalized Bernoulli polynomials, table of       I 20 Bernoulli numbers, definition      I 19 20 Bernoulli numbers, expansion of cot z, tan z, csc z, ln cos z in series involving      I 23 Bernoulli numbers, table of      I 20 Bessel functions, Airy functions, connection with      I 217 Bessel functions, approximation      see "Approximation" Bessel functions, associated      I 219 Bessel functions, asymptotic expansions      I 203—205 215 Bessel functions, ber, bei functions, connection with      I 216 Bessel functions, computation by use of recurrence formulas      II 162 Bessel functions, confluent hypergeometric function, connection with      I 120 135 213 Bessel functions, definitions, connecting relations, power series      I 39 40 212 213 Bessel functions, derivatives with respect to order      II 52 53 Bessel functions, derivatives with respect to order at half an odd integer      I 216 Bessel functions, difference-differential properties      I 48 214 II 232 Bessel functions, expansions, expansions in series of      see "Expansion" Bessel functions, G-functions, connection with      I 226—234 Bessel functions, integrals involving      I 115 165 177 219 220 287 289 II 33 34 39 53 54 see Bessel functions, Jacobi polynomial, connection with      I 52 Bessel functions, ker, kei functions, connection with      I 216 Bessel functions, products      I 216 228—230 232—234 II 50—53 Bessel functions, Wronskians      I 214 215 Bessel functions, zeros of      I 205 II 228—234 Bessel polynomials      I 274 II 194 253 Best approximation      see "Approximation" Beta function, complete      I 15 16 18 Beta function, incomplete      I 210 II 178 Beta transform      I 58 59 170—176 286 BF      see "Bessel function or Binomial coefficient      I 9 35 Binomial function      I 38 40 49 209 210 II 179—185 see binomial "Expansion powers Catalan's constant      II 293 Chebyshev polynomials of the first kind      I 308—312 see "Approximation" Chebyshev polynomials of the first kind, connection with Pade approximation for square root      II 183 Chebyshev polynomials of the first kind, definition and basic properties      I 273 296 297 300 301 Chebyshev polynomials of the first kind, difference-differential properties      I 297—299 301 302 316 321 324 Chebyshev polynomials of the first kind, evaluation of series of, by use of backward recurrence formula      I 325—329 Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration)      see also "Expansion" Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration), asymptotic estimate of coefficients      I 293—296 Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration), evaluation of coefficients      I 286—293 Chebyshev polynomials of the first kind, expansions in series of, differential and integral properties      I 314—325 Chebyshev polynomials of the first kind, expressed in powers of x      II 294 296 297 Chebyshev polynomials of the first kind, integrals involving      I 293—295 299 300—302 316—325 Chebyshev polynomials of the first kind, Jacobi polynomial, connection with      I 273 296 Chebyshev polynomials of the first kind, minimax and mean square properties      I 303—307 Chebyshev polynomials of the first kind, orthogonality property with respect to integration      I 273 299 301 Chebyshev polynomials of the first kind, orthogonality property with respect to summation      I 307 310 311 Chebyshev polynomials of the first kind, powers of x expressed in terms of      II 295 298 Chebyshev polynomials of the first kind, solution of differential and integral equations by expansions in series of      II 66—69 234—241 Chebyshev polynomials of the second kind      I 312—314 see Chebyshev polynomials of the second kind, connection with Pade approximation for square root      II 183 Chebyshev polynomials of the second kind, definition and basic properties      I 273 296—300 Chebyshev polynomials of the second kind, integrals involving      I 299 300 Chebyshev polynomials of the second kind, Jacobi polynomial, connection with      I 273 296 Chebyshev polynomials of the second kind, orthogonality property with respect to integration      I 273 299 Chebyshev polynomials of the second kind, orthogonality property with respect to summation      I 312 313 Chebyshev, best approximation in sense of      I 303 Christoffel — Darboux formulas      I 272 Collocation      II 69 Computation, by use of recurrence formulas      I 317—319 325—329 II 26—28 159—166 282—285 Confluence principle and theorems      I 48—57 Constants, mathematical, table of      II 292 293 Continued fractions      see "Approximation RFP" Continued fractions, convergence      II 80 Cosecant      I 23 210 Cosine      see "Sine and cosine" Cosine integral      see "Sine and cosine integrals" Cotangent      see "Tangent" Coulomb wave functions      I 135 212 cp      see "Chebyshev polynomials of the first kind" Cylinder function      I 204 213 D operator      I 24—26 Darboux, method of generating functions      I 254—259 Debye functions      II 273—281 Delta () operator      I 24—26 Difference equations (or recurrence formulas), use of in computation      I 317—319 325—329 II 159—166 282—285 Differential, integral and functional equations      see "Approximation" "Expansion" Dilogarithm      II 310 Dixon's theorem      I 104 Duffing's equation      II 89 90 E-function      I 148 Economized approximation      II 66 105 106 111 Elliptic functions      II 90 91 Elliptic integrals      I 211 II 91 see "Expansion" Error functions      I 135 223 224 II 195 201 see "Expansion" Euler — Mascheroni constant,       I 9 II 293 Evaluation of expansions in series of functions where functions satisfy a linear finite difference equation      I 325—329 expansion      II 287—291 Expansion, -function, CP      II 25 28 238—240 Expansion, -function, G-functions      II 25 Expansion, Airy functions      I 217 see Expansion, Anger — Weber functions, BF      II 39 54 55 Expansion, Anger — Weber functions, CP      II 39 Expansion, ber, bei      I 216 see Expansion, Bessel functions, , CP      II 27 Expansion, Bessel functions, , , , , BF      II 37 49 50 52 53 Expansion, Bessel functions, , , , , CP      II 37 38 290 331—333 338—341 347—351 Expansion, Bessel functions, , , , , GP      II 38 Expansion, Bessel functions, , , BF      II 50—53 Expansion, Bessel functions, , , BF      II 49 50 Expansion, Bessel functions, , , CP      II 35 37 234—238 290 352—356 359—367 Expansion, Bessel functions, , , GP      II 37 Expansion, Bessel functions, , , zeros of, CP      II 233 234 Expansion, Bessel functions, , , BF      II 49 Expansion, Bessel functions, , , CP      II 240 290 352—356 359—367 Expansion, Bessel functions, , , BF      II 52 53 Expansion, Bessel functions, Airy functions      I 217 see Expansion, Bessel functions, ber, bei, ker, kei, and derivatives, CP      II 347—351 I 216 Expansion, Bessel functions, integrals involving, , $\int^{x}_{0} t^{-1}[1-J_{0}(t)] dt, BF II 39 54 Expansion, Bessel functions, integrals involving, ,$\int^{x}_{0} t^{-1}[1-J_{0}(t)] dt, CP      II 39 334 Expansion, Bessel functions, integrals involving, , , BF      II 33 34 53 54 Expansion, Bessel functions, integrals involving, , CP      II 240—242 Expansion, Bessel functions, integrals involving, , , , , m=0,1, CP      II 242 342—345 Expansion, Bessel functions, integrals involving, , , , CP      II 368 369 Expansion, Bessel functions, integrals involving, , , , , m=0,1, CP      II 334—336 Expansion, Bessel functions, integrals involving, , , , CP      II 357 358 Expansion, Bessel functions, integrals involving, , , CP      II 337 Expansion, Bessel functions, integrals involving, , , CP, the same integrals with J replaced by I, CP      II 346 Expansion, binomial function      see "Expansion powers Expansion, Chebyshev polynomial of first kind, powers of x      II 294 296 297 Expansion, Chebyshev polynomial of first kind, powers of x, derivatives and integrals of, CP      I 297—299 Expansion, cosine      see "Expansion sine Expansion, cosine integral      see "Expansion sine Expansion, cotangent      see "Expansion tangent" Expansion, differential equations, integral equations, other functional equations, solutions of, CP      II 234—242 Expansion, dilogarithm, , CP      II 310 Expansion, elliptic integrals, 's      II 63 64 Expansion, elliptic integrals, first two kinds (complete and incomplete), sin       II 35 376—382 Expansion, elliptic integrals, first two kinds (complete), CP      II 291 383 384 Expansion, elliptic integrals, Legendre polynomials      II 36 Expansion, error functions, , BF      II 42 43 57 58 Expansion, error functions, , CP      II 42 43 57 289 290 323 324 Expansion, error functions, , CP      II 289 290 324 Expansion, error functions, repeated integrals, CP      II 290 Expansion, exponential function, , , , BF      II 32 Expansion, exponential function, , , , CP      II 32 315 Expansion, exponential function, , BF      II 45 Expansion, exponential function, , JP      I 285 Expansion, exponential function, , BF      II 32 45 46 Expansion, exponential function, , CP      II 32 288 313 314 363 Expansion, exponential function, , GP      I 287 II 32 Expansion, exponential function, , JP      I 285 Expansion, exponential function, , BF      II 33 34 Expansion, exponential function, , CP      II 34 Expansion, exponential integral, ,       II 289 Expansion, exponential integral, , BF      II 41 56 Expansion, exponential integral, , CP      II 41 289 321 Expansion, exponential integral, , P.V. , CP      II 289 322 Expansion, Fresnel integrals, $\int_{0}^{x} t^{-1/2} e^{it} dt, BF II 43 58 Expansion, Fresnel integrals,$\int_{0}^{x} t^{-1/2} e^{it} dt, CP      II 43 290 328 Expansion, Fresnel integrals, , CP      II 27 290 329 330 Expansion, G-function, 's      I 139 142 145—147 Expansion, G-function, CP      II 27 28
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