|
|
 |
| Àâòîðèçàöèÿ |
|
|
|
| Ïîèñê ïî óêàçàòåëÿì |
|
|
|
|
|
|
|
 |
|
|
|
| Olver F.W.J. — Asymptotics and Special Functions |
|
|
|
| Ïðåäìåòíûé óêàçàòåëü |
 ,  ,  arcs 147
Abel — Plana formula 290
Abel, N. H. 4
Abel’s identity 142
Abel’s theorem on continuity of power series 26
Airey, J. R. 72 536 543
Airy functions 53 393
Airy functions, asymptotic expansions of 103 105 116—118 392—394 413—414
Airy functions, auxiliary functions for complex arguments 415—416
Airy functions, auxiliary functions for real arguments 394—397 404 433
Airy functions, bounds for 116—117 394
Airy functions, differential equations for 55
Airy functions, graphs 393
Airy functions, history 433
Airy functions, integrals of 342—344 394 430—432
Airy functions, relation to Bessel functions 61 392 393
Airy functions, Wronskians 393 416
Airy functions, zeros 403—405 414—416
Airy integral see “Airy functions”
Airy, G. B. 433
Airy’s equation 55
Airy’s equation, connection formula for solutions 55 507
Airy’s equation, inhomogeneous form 430—432
Airy’s equation, numerically satisfactory solutions 278 393 414
Aitken’s  -transformation 543
Alternating function 68 70
Anger function 84 103
Anger function of large order 84 103 336 352—360
Annular sector 7
Anti-Stokes lines 504 518
Apostol, T. M. 144 152 219
Askey, R. A. 446
Associated Legendre equation 169 185 463 Legendre
Associated Legendre equation, connection formulas for solutions 171 179 490—491
Associated Legendre equation, numerically satisfactory solutions 172—173 178
Associated Legendre functions see “Legendre functions”
Asymptotic approximations see “Asymptotic expansions Asymptotic
Asymptotic behavior of power series on circle of convergence 26 304—306
Asymptotic expansions 4 16 Asymptotics Differential
Asymptotic expansions by integration by parts 66—70 75—79 106—112 342—343
Asymptotic expansions in complete sense of Watson 543
Asymptotic expansions of integrals see “Bleistein’s method Haar’s Laplace’s Method Friedman and Method Method Watson’s
Asymptotic expansions of sequences 309—310 (see also “Asymptotic expansions of
Asymptotic expansions of sums 284—292 295—309
Asymptotic expansions to N terms 17
Asymptotic expansions with logarithmic singularities 322—325 338—339
Asymptotic expansions with nearly coincident saddle points and branch points 344—346 361
Asymptotic expansions with nearly coincident saddle points and poles 346 361
Asymptotic expansions with two nearly coincident saddle points 351 352 357
Asymptotic expansions, compound 118
Asymptotic expansions, convergence of 18—19
Asymptotic expansions, differentiation with respect to parameters 324
Asymptotic expansions, failure of 76—79
Asymptotic expansions, functions with prescribed 22—24 29—30
Asymptotic expansions, fundamental properties 16—19
Asymptotic expansions, generalized 24—27 30 117—118 313
Asymptotic expansions, history 3—4 29—30 543—544
Asymptotic expansions, implied constants 17
Asymptotic expansions, induction of expansions 71
Asymptotic expansions, neglect of exponentially small terms 78 95 105
Asymptotic expansions, nonuniqueness of sum 17—18
Asymptotic expansions, numerical use of 105 519—521 543—544
Asymptotic expansions, operations with 19—22
Asymptotic expansions, Poincare’s definition 4 16—17 24—25 105 543
Asymptotic expansions, transformation of 540—544
Asymptotic expansions, uniqueness property 17
asymptotic relations 4—11 29
Asymptotic scale 25
Asymptotic sequence 25
Asymptotic solutions of differential equations see “Differential equations”
Asymptotic sum 23
Asymptotic variable 17
Asymptotics 27
Asymptotics, paradox in 313
Bachmann, P. 4
Bakhoom, N. G. 84
Balogh, C. B. 382 425
Barnes, E. W. 104 301 321
Bateman Manuscript Project (B.M.P.) 64 188 189 277
Berg, L. 29 329 361
Bernoulli numbers 281—284 291
Bernoulli polynomials 281—284 321
Bernoulli polynomials, relation to Zeta function 283 292
Bertocchi, L. 479
Bessel functions 55 57 242 Hankel Modified
Bessel functions of first, second, and third kinds 242
Bessel functions of half-odd-integer order 59 241 244
Bessel functions of imaginary argument 60
Bessel functions of large argument 130—133 242 269 317—318
Bessel functions of large complex order 434
Bessel functions of large positive order 133—134 334—336 360 382 419—426
Bessel functions, analytic continuation 57 244
Bessel functions, ascending series 57 243
Bessel functions, auxiliary functions for 436—438 453—457
Bessel functions, Bessel’s integral 55
Bessel functions, bounds for 59
Bessel functions, collected properties 435—436
Bessel functions, connection formulas 238 239 242—244
Bessel functions, derivatives with respect to order 243 244
Bessel functions, differential equations for 57 341
Bessel functions, generating function 56
Bessel functions, graphs 242 243
Bessel functions, history 277 278
Bessel functions, integral representations 55—56 58 59 244 340
Bessel functions, integrals of 59 60 244 277 278
Bessel functions, Kapteyn’s inequality 426
Bessel functions, Mehler — Sonine integrals 244
Bessel functions, Neumann’s addition theorem 59
Bessel functions, Nicholson’s integral 340
Bessel functions, Poisson’s integral 59 241
Bessel functions, recurrence relations 58—59 436
Bessel functions, relation to confluent hypergeometric functions 255
Bessel functions, Schlafli’s integrals 58
Bessel functions, sums of squares of 340—342
Bessel functions, Wronskian 244
Bessel functions, zeros 214 244—250 278 408 426 434
Bessel’s equation 57 (see also “Modified Bessel equation”)
Bessel’s equation, inhomogeneous form 274 432
Bessel’s equation, numerically satisfactory solutions 242—243 278
beta function 37 38
Birkhoff, G. D. 391 517
Bleistein, N. 104 105 332 345 361
Bleistein’s method 344—351 361
Boin, P. W. M. 105
Born approximation 451
Borsch — Supan, W. 320
Bound states 497—501
Boundary value problem 197
Bounded variation 27
Boyer, T. R 434
Braaksma, B. L. J. 321
Brillouin, L. 228
British Association for the Advancement of Science (B.A.) 134 376 405 433 526
Bromwich, T. J. I’a. 2 21 29 54
Buchholz, H. 278
Burkhardt, H. 105
Calogero, F. 210 479
Caratheodory, C 189
Carleman, T. 29
Carlini, F. 228
Cauchy principal value 41 72 346
Cauchy, A. L. 4
Chako, N. 105
Chakravarti, P. C 321
Chebyshev polynomials 52 185
Cherry, T. M. 223 433 544
Chester, C 351 361
Chester, Friedman, and Ursell’s method see “Method of Chester Friedman and
Chowla, S. 336
Chuhrukidze, N. K. 479
| Cirulis, T. 105 361
Clark, R. A. 434
Clausen, Th. 169
Clemmow, P. C 361 .
Cochran, J. A. 251 381 434
Cohn, J. H. E. 228
col see “Saddle point”
Compact set 24
Completely monotonic function 68
Conditionally convergent integrals 74
Confluent hypergeometric equation 254 (see also “Whittaker’s equation”)
Confluent hypergeometric equation with exponents differing by integer 259
Confluent hypergeometric equation, asymptotic solutions 256
Confluent hypergeometric equation, connection formulas for solutions 257—259
Confluent hypergeometric equation, numerically satisfactory solutions 259
Confluent hypergeometric functions 255 256 Whittaker
Confluent hypergeometric functions as functions of parameters 255
Confluent hypergeometric functions of large argument 256—258
Confluent hypergeometric functions, Barnes’ integral 302
Confluent hypergeometric functions, converging factor for 531—536
Confluent hypergeometric functions, history 277 278
Confluent hypergeometric functions, integral representations 255—257 302
Confluent hypergeometric functions, recurrence relations 256 259
Confluent hypergeometric functions, Rummer’s transformations 256 257
Confluent hypergeometric functions, Wronskians 255 259
Conical functions 473 474
Connection formulas 166 228 480
Connection formulas at singularity 480—490
Connection formulas for simple turning points 483 491—494 503—506 510—513
Connection formulas for three turning points 516
Connection formulas for two turning points 494—503 513—516
Connection formulas, central 517
Connection formulas, history 516—517
Connection formulas, lateral 517
Connection formulas, stretching-matching method 517
Connection formulas, vanishing of error terms 508—510
contour 29
Convergently beginning series 3
Converging factor 522 543
Coppel, W. A. 202
Copson, E. T. 22 31 64 116 123 137 138 156 334
Cosine integrals 42—43
Cosine integrals, asymptotic expansion of 67 320
Cosine integrals, Laplace transform of 43
Critical points, method of see “Method of stationary phase”
Curtis, A. R. 232
Cylinder functions 248 (see also “Bessel functions”)
Darboux, M. G. 310
Darboux’s method 309—310 321
Darboux’s method, examples 311—315
Davis, P. J. 26 30 64
Dawson’s integral 44
Dawson’s integral, converging factors for 531 544
De Bruijn, N. G. 9 11 29 138 321 329
de Kok, F. 105
Debye, P. 134 137 382
Deleted neighborhood 18
Dieudonne, J. 29 105
Difference operator 299 537
Differential equations see “Connection formulas Irregular LG Singularities Turning-point
Differential equations for products of solutions 169
Differential equations with parameter see “Connection formulas Differential with LG Singularities Turning-point
Differential equations with parameter, admissible cases 474—479
Differential equations with parameter, asymptotic solutions 203—206 208—211 224 371—373 382—386
Differential equations with parameter, atypical cases 477 478
Differential equations with parameter, basic equation 475
Differential equations with parameter, basic functions 475 479
Differential equations with parameter, behavior of coefficients in asymptotic solutions 365 368—371 386
Differential equations with parameter, classification of cases 362 474
Differential equations with parameter, condition solutions are functions of single variable 475 479
Differential equations with parameter, continuity of solutions 143 145
Differential equations with parameter, eigensolutions 214 216
Differential equations with parameter, eigenvalues 214—217 228 494—501
Differential equations with parameter, error bounds for asymptotic solutions 203—204 209 366 385
Differential equations with parameter, formal series solutions 364—366 383 477
Differential equations with parameter, history 189 228 391
Differential equations with parameter, holomorphicity of solutions 144 146 236
Differential equations with parameter, inhomogeneous 386—388
Differential equations with parameter, zeros of solutions 211—214
Differential equations with simple pole see “Singularities of differential equations”
Differential equations with simple pole, asymptotic solutions for complex variables 460—462 478—479
Differential equations with simple pole, asymptotic solutions for real variables 208—211 447—451 478—479
Differential equations with simple pole, coefficients in asymptotic solutions 445 448 462—463
Differential equations with simple pole, connection formulas 483—490
Differential equations with simple pole, error bounds for asymptotic solutions 440—446 457—460
Differential equations with simple pole, formal series solutions 438—440
Differential equations with simple pole, history 478 479
Differential equations with simple pole, inhomogeneous 479
Differential equations with simple pole, zeros of solutions 451—453
Differential equations with three regular singularities 156
Differential equations, Cauchy’s method for existence theorems 148 149
Differential equations, Cayley’s identity 193
Differential equations, dominant solutions 155 199
Differential equations, existence theorems for complex variables 145—148 189
Differential equations, existence theorems for real variables 139—143 189
Differential equations, exponential-type solutions 190
Differential equations, fundamental solutions 141—142 146
Differential equations, identification of solutions 155
Differential equations, inhomogeneous 143 270—274 386—388 429—430
Differential equations, linear independence of solutions 142 146 154
Differential equations, local behavior of solutions 190
Differential equations, method of successive approximations 141 233
Differential equations, normal series 230
Differential equations, normal solution 230
Differential equations, numerically satisfactory solutions 154—155 223 431—432
Differential equations, ordinary point 148 153
Differential equations, oscillatory solutions 190
Differential equations, Picard’s method for existence theorems 141
Differential equations, recessive solution 155 199
Differential equations, subdominant solution 155
Differential equations, subnormal solution 231
Differential equations, trigonometric-type solutions 190
Differential equations, zeros of solutions 211—214 405—408 451—453
Digamma function 39
Dingle, R. B. 517 543
Discontinuities 73
Distinguished point 7 25
Doetsch, G. 104 112 317 323
Domain 29
Dominated Convergence Theorem 54
Doming, J. J. 317
Duncan, C. E. 321
eigenvalue problems 214—217 228 494—501
Elliptic integrals 161
Energy-conservation equation 503
Entire functions, asymptotic expansions for large arguments 307—309 321
Erdelyi, A. 25 30 44 81 86 104 105 115 120 228 276 278 325 361 391 412 433 446
Error function 43—44 64
Error function, asymptotic expansion of 67 112
Error function, complementary 43
Error function, converging factors for 526—527 544
Error function, relation to incomplete Gamma functions 45
Error term 2
Error test 68—69
Error-control function 194 222 399
Error-control function, variation at singularity 200—202 401
Euler — Maclaurin formula 281 284—285
Euler — Maclaurin formula, applications 285—289 292—295
Euler — Maclaurin formula, contour integral for remainder term 289—292
Euler — Maclaurin formula, error bounds for 285
Euler — Maclaurin formula, evaluation of constant term 287 292
Euler — Maclaurin formula, generalized 284
Euler — Maclaurin formula, history 321
Euler, L. 3 63 161
Euler’s constant 34 40 64 292
Euler’s transformation 537
Euler’s transformation, applied to asymptotic expansions 540—543
Evgrafov, M. A. 83 306 315 321 517
Exponential integral 40—41 64
Exponential integral with Euler’s transformation 540—543
|
|
|
| Ðåêëàìà |
|
|
|
|
|
|
|
|
Î ïðîåêòå