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Luke Y.L. — Mathematical Functions and Their Approximations

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Название: Mathematical Functions and Their Approximations

Автор: Luke Y.L.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1975

Количество страниц: 585

Добавлена в каталог: 02.07.2008

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Предметный указатель

$\tau$ — Method      491 495 499
Airy functions      312 313 403 406 Approximation Expansion)
Anger — Weber functions      413 426
Approximation of functions based on orthogonality properties of Chebyshev polynomials with respect to summation      469 475
Approximation of functions based on orthogonality properties of rational and polynomial      413 427 490 505
Approximation, $10^{x}$ , BCP      75
Approximation, Bessel functions, $2(\nu+1)I_{\nu+1}(z)/zI_{\nu}(z)$ , RFP      364 365
Approximation, Bessel functions, $I_{n}(z), K_{n}(z)$ , BCRF      411 412
Approximation, Bessel functions, $I_{n}, P$       411
Approximation, Bessel functions, $J_{n}(z)$ , TR      397 399
Approximation, Bessel functions, $J_{\nu}(z)$ , $I_{\nu}(z)$ , BCP      411 412
Approximation, Bessel functions, $J_{\nu}, I_{\nu}$ , BCP      411 412
Approximation, Bessel functions, $K_{\nu}(z)$ , BCP and BCRF      411 412
Approximation, Bessel functions, $K_{\nu}(z)$ , repeated integrals of      411
Approximation, Bessel functions, $K_{\nu}(z), J_{\nu}(z), Y_{\nu}(z), H_{\nu}(z), BCP$       412
Approximation, Bessel functions, R      380 395
Approximation, Bessel functions, RF      361 366 379
Approximation, Bessel functions, RFP      412
Approximation, best in sense of $\tau$ -method      491 493—495
Approximation, best in sense of Chebyshev      490 493
Approximation, best in sense of least square      491
Approximation, best in sense of mean square      493
Approximation, best in sense of Pade      493—495
Approximation, binomial function, $(1+1/z)^{-c}, RFP$       27—31
Approximation, binomial function, $(1+1/z)^{\pm\frac{1}{2}}$ , BCP and BCRF      75
Approximation, binomial function, $(1+x)^{-1}$ , BCP      76
Approximation, binomial function, $x^{x}$ , $x^{1/x}$ , RF      28
Approximation, binomial function, , BCP      76
Approximation, binomial function, Newton-Raphson process      29
Approximation, binomial function, RFP      28 30 75
Approximation, Debye functions, RFP      76
Approximation, economized      496
Approximation, error functions, Erf(z), Erfi(z), BCRF      152
Approximation, error functions, Erfc(z), BCRF      151 152
Approximation, error functions, inverse      152
Approximation, error functions, P      512
Approximation, error functions, repeated integrals, R      101 102
Approximation, error functions, RFP      124 128
Approximation, error functions, TR      134 137
Approximation, exponential function, $e^{-w}=2z+1\pm2(z^{2}+z)^{\frac{1}{2}}$ , RFP      31 33
Approximation, exponential function, $e^{z}$ , BCP and BCRF      75 76
Approximation, exponential function, exp(), as a sum of exponentials      76
Approximation, exponential function, RF      76
Approximation, exponential function, RFP      46 51 386
Approximation, exponential integral, BCRF      150
Approximation, exponential integral, RF      106 110 150
Approximation, exponential integral, RFP      88 93 107 110 114 150
Approximation, Fresnel integrals, BCRF      153
Approximation, Fresnel integrals, P      152 153
Approximation, Fresnel integrals, RF      139
Approximation, functions, defined by a differential equation, P and RF      554
Approximation, gamma function, $arg \Gamma(1+iy)$ , BCRF      23
Approximation, gamma function, $ln \Gamma(x)$ , BCRF      22
Approximation, gamma function, $\Gamma(x+1)$ , BCP      22 33
Approximation, gamma function, $\Gamma(x+1)$ , RF      7 23
Approximation, gamma function, $\Gamma(x+1)$ , RFP      7 100
Approximation, gamma function, logarithmic derivative, BCP, BCRF, RFP      23
Approximation, gamma function, RF      13 16
Associated Bessel function      413 427
Asymptotic expansions, large variable      315
Asymptotic expansions, references to      154 (see also modifiers such as Bessel functions)
Basic series      446
Ber, bei functions      312 404 Approximation
Bernoulli and generalized Bernoulli polynomials      505 507
Bernoulli numbers      506
Bessel functions, Airy functions, connection with      312 313
Bessel functions, approximation, see Approximation associated      413 427
Bessel functions, asymptotic expansions, large variable      315
Bessel functions, ber, bei functions, connection with      312
Bessel functions, bibliographic and numerical data      403 412
Bessel functions, computation of, by use of recurrence formulas      380 395 509
Bessel functions, computation of, by use of trapezoidal type integration formulas      395 399
Bessel functions, confluent hypergeometric function, connection with      294 311
Bessel functions, definitions, connecting relations, power series      311 313
Bessel functions, derivatives with respect to the order      360 405
Bessel functions, difference-differential properties      313 314
Bessel functions, expansion, expansions in series of      (see Expansion)
Bessel functions, G-function, connection with      302 307 309
Bessel functions, incomplete      414 417 426
Bessel functions, inequalities      366 396 399 403
Bessel functions, integral representations      396 398
Bessel functions, integrals involving      187 315 316 403 405 408 411 412 414 426 509 510
Bessel functions, ker, kei functions, connection with      312
Bessel functions, large order      393 398
Bessel functions, multiplications theorems      185 360
Bessel functions, products of      219 304 310 314
Bessel functions, ratio $2(\nu+1)I_{\nu+1}(z)/z I_{\nu}(z)$ , inequalities for      366 400
Bessel functions, repeated integrals of $K_{\nu}(z)$       397 398
Bessel functions, series of      403 405 406
Bessel functions, tables, description of and references to      404 411
Bessel functions, Wronskians      314
Bessel functions, zeros and extrema of      365 403 406 407 412
Bessel polynomials      229 433
Bessel's inequality      430
Best approximation      (see Approximation)
Beta function, complete      298
Beta function, incomplete      297 299
Beta transform      160
Binomial coefficient      11 12
Binomial function      24 35 Expansion Inequalities)
Catalan' constant      511 515
Chebyshev polynomials of the first kind      (see also Jacobi polynomials)
Chebyshev polynomials of the first kind, approximations in series of      490 493
Chebyshev polynomials of the first kind, approximations in series of (based on orthogonality properties with respect to summation)      469 473 491
Chebyshev polynomials of the first kind, coefficients for expansion of integrals of functions in series of      464 469
Chebyshev polynomials of the first kind, connection with Pade approximation for square root      29
Chebyshev polynomials of the first kind, definition and basic properties      434 436 453 463
Chebyshev polynomials of the first kind, evaluation of series of by use of backward recurrence formula      478 481 482
Chebyshev polynomials of the first kind, expansion of $x^{m}d^{r}T_{n}(x)/dx^{r}$ in series of      454 455 459 461
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      448 452
Chebyshev polynomials of the first kind, expansions in series of (based on orthogonality property with respect to integration)evaluation of coefficients      443 448
Chebyshev polynomials of the first kind, expressed in powers of x      453 458 459 462 463
Chebyshev polynomials of the first kind, integrals involving      455 456 459 460 464 469
Chebyshev polynomials of the first kind, Jacobi polynomial, connection with      434 453
Chebyshev polynomials of the first kind, minimax, least and mean square properties      490 493
Chebyshev polynomials of the first kind, orthogonality property with respect to integration      434 436 455 459
Chebyshev polynomials of the first kind, powers of x, expressed in terms of      454 457 459 461
Chebyshev polynomials of the first kind, solution of differential and integral equations by expansion in series of      464 469 500 504
Chebyshev polynomials of the first kind, summation      432 469 473
Chebyshev polynomials of the second kind      (see also Jacobi polynomials)
Chebyshev polynomials of the second kind, approximations in series of (based on orthogonality properties with respect to summation)      473 475
Chebyshev polynomials of the second kind, connection with Pade approximation for square root      29
Chebyshev polynomials of the second kind, definition and basic properties      434 436 453 463
Chebyshev polynomials of the second kind, integrals involving      455 456
Chebyshev polynomials of the second kind, Jacobi polynomial, connection with      434 453
Chebyshev polynomials of the second kind, orthogonality property with respect to integration      434 436 455
Chebyshev polynomials of the second kind, summation      432 473 475
Chebyshev, best approximation in sense of      490 493
Christoffel — Darboux formulas      432 433
Circular functions      (see Sine and cosine and Tangent)
Clebsch — Gordan coefficients      168
Collocation      498
Computation and check of the tables      509 512
Computation of series of functions where functions satisfy a linear finite difference equation      475 482
Computation, by use of recurrence formulas      98 100 102 219 220 320 380 395 464 468 475 482 483 489 509—512
Confluence principle and theorems      179 183
Constants, mathematical table of      514 515
Continued fractions      (see Approximation (RFP))
Converging factors      145 148 404
Cosine      (see Sine and cosine)
Cosine integral      (see Sine and cosine integrals)
Cotangent      (see Tangent)
Coulomb wave functions      297 300
Cylinder function      311
D operator      507 509
Debye functions      76
Delta $(\delta)$ operator      507 509
Difference equations      (see Recurrence formulas)
Differential, integral and functional equations      (see Approximation Expansion)

Dilogarithm      67 511
Dixon's equation      164
Dominance in asymptotic expansions      201
Dominance in solutions of difference equations      483
E-function      176
Economized approximation      496
Elliptic functions and integrals      279 299 511
Error functions      88 94 102 119 138 145 148 509 510 Expansion)
Euler transforms      189 190
Euler — Mascheroni constant, y      1 512 515
Expansion, Airy functions      312 313
Expansion, Bessel functions, $J_{n}(z), I_{n}(z)$ , BF      317
Expansion, Bessel functions, $J_{\nu}(z), I_{\nu}(z), CP$       317 319 500 504
Expansions      (see The discussion below)
Exponential function      31 33 42 52 74 76 509 514
Exponential integral      88 93 96 99 103 114 143 145 509 510
Fourier coefficients      430 442
Fourier series, convergence of      443
Fourier transform, of G-function      189
Fresnel integrals      139 142 148 219 303 Expansion)
G-function      154 256
G-function, analytic continuation      178 179
G-function, approximation of      (see Approximation)
G-function, asymptotic expansion, large parameter      216 218
G-function, definition      170 171
G-function, differential equation and solutions      192 193
G-function, elementary properties      176 178
G-function, expansions      (see Expansion)
G-function, expressed as a named function      306 310
G-function, hypergeometric functions and named functions, connection with      300 310
G-function, inequalities      255 256
G-function, integral representations      162 170 187 190
G-function, integrals involving      186 189
G-function, large variable      199 206
G-function, multiplication theorems      184 185
G-function, named functions expressed in terms of      300 306
G-function, references to      154
Gamma function      1 23
Gamma function, approximation      (see Approximation)
Gamma function, asymptotic expansion      7 10 11 12
Gamma function, definite integrals expressed in terms of      161 163 186
Gamma function, expansion      (see Expansion)
Gamma function, inequalities      17 20
Gamma function, logarithmic derivative of      (see Psi-function)
Gamma function, numerical values for certain fractional arguments      515
Gamma function, power series and other expansions      1 7
Gamma function, references      20
Gamma function, tables, references to      21 22 74 147
Gegenbauer polynomial      434 436
h-transform      187
Hankel functions      305 306 309 312 315 316 319 367
Hankel transforms      187 444
Hermite polynomials      434 436
Hyperbolic functions      (see Sine and cosine and tangent and cotangent)
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, approximation of      (see Approximation)
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, asymptotic expansion, large parameter(s)      182 183 292 293
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, bibliographic and numerical data      296 297
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, confluence      180 284
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, contiguous relations      285 286
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, difference-differential properties      285 286
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, differential equations and solutions      287 290
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, expansions in series of Jacobi polynomials, Chebyshev polynomials, etc.      (see Expansion)
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, G-function, connection with      301 307
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, inequalities      295
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, integral representations      284
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, integrals of      296
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, Kummer relations      288
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, large variable      291 292
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, products of      286 287 303 308 310
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, references      296
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, tables, description of and references to      296 297
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, uniform      85 89
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, Whittaker functions      284 297 Hypergeometric Hypergeometric Incomplete
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, Wronskians      291
Hypergeometric function (confluent), $_{1}F_{1}$ , U-function, zeros      90 296
Hypergeometric function (Gaussian), $_{2}F_{1}$       257 283
Hypergeometric function (Gaussian), $_{2}F_{1}$ , analytic continuation      263 265
Hypergeometric function (Gaussian), $_{2}F_{1}$ , approximation of      (see Approximation)
Hypergeometric function (Gaussian), $_{2}F_{1}$ , bibliographic and numerical data      279 283
Hypergeometric function (Gaussian), $_{2}F_{1}$ , confluence principle      180
Hypergeometric function (Gaussian), $_{2}F_{1}$ , contiguous relations      258 259
Hypergeometric function (Gaussian), $_{2}F_{1}$ , difference-differential properties      257 259
Hypergeometric function (Gaussian), $_{2}F_{1}$ , differential equations and solutions      260 270
Hypergeometric function (Gaussian), $_{2}F_{1}$ , evaluation of, for special values of argument      271 273
Hypergeometric function (Gaussian), $_{2}F_{1}$ , expansions in series of Jacobi polynomials, Chebyshev polynomials, etc.      (see Expansion)
Hypergeometric function (Gaussian), $_{2}F_{1}$ , inequalities      278
Hypergeometric function (Gaussian), $_{2}F_{1}$ , integral representations      259 260
Hypergeometric function (Gaussian), $_{2}F_{1}$ , Kummer relations      262
Hypergeometric function (Gaussian), $_{2}F_{1}$ , named functions, expressed in terms of      298 299 434
Hypergeometric function (Gaussian), $_{2}F_{1}$ , orthogonal polynomials, classical, connection with      434
Hypergeometric function (Gaussian), $_{2}F_{1}$ , quadratic transformations      270 271
Hypergeometric function (Gaussian), $_{2}F_{1}$ , references      279
Hypergeometric function (Gaussian), $_{2}F_{1}$ , tables, description of and references to      279 283
Hypergeometric function (Gaussian), $_{2}F_{1}$ , Wronskians      270
Hypergeometric function (generalized), $_{p}F_{q}$       154 256
Hypergeometric function (generalized), $_{p}F_{q}$ , analytic continuation      179
Hypergeometric function (generalized), $_{p}F_{q}$ , approximation of      (see Approximation)
Hypergeometric function (generalized), $_{p}F_{q}$ , asymptotic expansions, large parameter(s)      182 183 452
Hypergeometric function (generalized), $_{p}F_{q}$ , confluence principle and theorems      179 183
Hypergeometric function (generalized), $_{p}F_{q}$ , contiguous relations      159
Hypergeometric function (generalized), $_{p}F_{q}$ , convergence of series      155
Hypergeometric function (generalized), $_{p}F_{q}$ , differential equations and solutions      190 191
Hypergeometric function (generalized), $_{p}F_{q}$ , differential-difference properties      159
Hypergeometric function (generalized), $_{p}F_{q}$ , evaluation of for special values of argument and parameters      163 170 271 273 437 438 441
Hypergeometric function (generalized), $_{p}F_{q}$ , expansion theorem for large parameter      182 183
Hypergeometric function (generalized), $_{p}F_{q}$ , expansions in series of Jacobi polynomials, Chebyshev polynomials, etc.      (see Expansion)
Hypergeometric function (generalized), $_{p}F_{q}$ , G-function, connection with      174 175 300 301 306
Hypergeometric function (generalized), $_{p}F_{q}$ , inequalities      155 252 256
Hypergeometric function (generalized), $_{p}F_{q}$ , integral representations and integrals involving      160 163 187
Hypergeometric function (generalized), $_{p}F_{q}$ , large variable      206 213
Hypergeometric function (generalized), $_{p}F_{q}$ , polynomial      156
Hypergeometric function (generalized), $_{p}F_{q}$ , products of      157 158
Hypergeometric function (generalized), $_{p}F_{q}$ , special functions, references to      154
Hypergeometric function (generalized), $_{p}F_{q}$ , zeros      257
Incomplete gamma function      77 153
Incomplete gamma function, bibliographic and numerical data      143 149
Incomplete gamma function, computation of      97 103
Incomplete gamma function, inequalities      95 97
Incomplete gamma function, tables, references to      143 149
Inequalities for Bessel functions      366 396 403
Inequalities for binomial functions      34 35
Inequalities for error functions      96 137 138
Inequalities for exponential function      51 52
Inequalities for exponential integral      96 97
Inequalities for G-functions, a certain class of      255 256
Inequalities for gamma functions      17 18
Inequalities for hyperbolic tangent      60
Inequalities for hypergeometric functions, a certain class of      252 255 278 279 295
Inequalities for incomplete gamma functions      96 97
Inequalities for logarithmic derivative of the gamma function      19
Inequalities for logarithmic functions      41 42 73 74
Inequalities for sinh, inverse of      73
Inequalities for tangent and its inverse      60 61 72 73
Inequalities Inverse tangent integral      67 511
Integral equations, solution of by expansion in series of Chebyshev polynomials of the first kind      (see Expansion)
Integrals      (see appropriate modifiors such as Bessel functions integrals)
Integration, numerical      (see Approximation integrals)
Inverse circular and hyperbolic functions      61 (see also Approximation Expansion)
Jacobi and associated Jacobi function      244 433 449
Jacobi polynomial, connection with      434
Jacobi polynomials, application of in approximation of functions by polynomials and rational functions      497 498
Jacobi polynomials, asymptotic expansion for large order      438 439
Jacobi polynomials, definition and basic properties      235 237 242 243 434 452
Jacobi polynomials, evaluation and estimation of coefficients for a given function when expanded in series of      443 452
Jacobi polynomials, expansion of functions in series of      442 443 446 448
Jacobi polynomials, extended and generalized      227 231 235 447
Jacobi polynomials, integrals involving      439 440
Jacobi polynomials, orthogonality property      434 436
K — Transform      187
Ker and kei functions (also known as Kelvin functions)      312 404 Approximation Expansion)

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