Abel v x 26 53
Absolutely convergent series 14 75—80
Accelerating convergent series 403
Accuracy obtainable from terminant expansions 300 459—461 466—467 492—493
Additional stationary point 455 457 459 481
Additivity of contributions from expansion points ix 131—133 141—142 458
Aerodynamics 277
Airey vi viii x 54 217 254 403—404 408 429
Airy functions 22—23 51 240—241 279 291—292
Alternating asymptotic series v vi 401—402
Anger function 80 108 202—203 230 250 376—378 380 386—389 396 398—399 450 485 511—512
Applied mathematics 14 277
Associated function 6 148—149
Associated Legendre polynomial 71 72
Astronomy 277
Asymmetrical saddle points 467
Asymptotic power series 2 26—55 100—109 275—284 371—381 431—451
Asymptotics v
Bailey, V. A. 328 330
Bailey, W. N. 55
Baker 403 429
Barnes 55 139 183 208 309 320 381
Basic terminants 198 406 415—429 484
Beer 55
Berghuis 215
Bernoulli numbers 75 91 92
Bernoulli polynomials 46 47 336
Berry 14 24
Bessel functions 36 50 51 54 62—65 78 82 86 95—97 101 104—105 108 166—174 200—202 240—241 248 250—251 281—282 283 286 303—307 334—339 349 440—441 449—451 467—472 482—485 495
Bessel functions of nearly equal order and argument 170—172 174 250—251 339 470—472 482—484
Bessel functions of the third kind 50 62 64 65 78 96 101 104—105 108 181—182 189 201 212 277 349 440—441 485
Bessel’s addition theorems 200
Bickley 468 488
Bijl 139
Binet 66 98 99 105 436
Binomial expansion 4 40 44 47
Blanch 54 130
Bleistein 215 254
Blumenthal 317 320
Borel summation 405—408 429
Bose — Einstein condensation 267—271
Bose — Einstein integral 53 69 73 98—99 109 449
Boundary layers 14
Bounds to remainders ix 401—403 405 488
Bowen 54 198 208
Braun 215
Bremmer 215
Brillouin 317 320
Buchholtz 54
Budden 320
Burmann series 405
Busbridge 54
Carleman 430
Carlini 316 320
Cauchy v 26 53 128 130 132 133 139
Chako 215
Chase 55
Chemistry 277
Cherry 349 403 430
Chester 247 254
Choquard 55
Circulis 139
Classification of series v 13—15 68—80 277
Code 408 410
Complete asymptotic expansions vi 8 19
Complete elliptic integrals 71
Conditionally convergent series 14 68—75
Confluent hypergeometric functions 24 33—35 46 50 60—62 94—95 103—104 175—180 203 249 275—277 279—281 286 356—362 366—369 370 438—440 486 504—505
Connection formulae 288—295 298 301 303 355—366 386 491—492 497 503 510—511
Continuation formulae, basic terminants 418
Continuation formulae, confluent hypergeometric function 33
Continuation formulae, error function 6 12
Continuation formulae, factorial 38
Continuation formulae, modified Bessel function 36
Continuation rules for asymptotic expansions 6—8 413—415
Continued fractions 403
Controversies about connection formulae 292—295
Convention adopted on fractional powers 12—13
Conventions adopted on cube roots of -1 and  147
Convergent power series listed 68—80
Convergent Series 1 26 53 403
Converging factor ix
Conversion by power identification 116 122 127 248—249 253
Conversion of power series into integral representations 56—99
Conwell 107 109
Copson x xi
Correspondences between late and early terms 148 151 300 303
Correspondences between quadratic and cubic contours 137 171 174 187 194 232
Cosine integral 77 447
Courant 140 154
Critical points 131
Crothers 295 320
Cube root conventions 147
Cube roots 123 136
Cubic contours 136—137
Cut in complex plane 13
Darboux method for calculating late terms in power series 4 24 140—142 152—154 157 409 411
Darwin 330
Darwin — Fowler method of selector variables 268
Davis 54 140 154
Dawson’s integral 224 263—264 273 274
de Haas-van Alphen effect 14
De Morgan 26 53
Debye 133 139
Defining asymptotic expansions v viii 16—21
defining functions 26—27 277
Derivatives, high 155—157
Difference equations viii 339
Difference expansions 433 451
Dingle vi vii xi 6 19 20 24 54 55 107 109 130 268 274 295 317 320 380 381 395 404 405 419 430 4A1 451 463 488
Dirichlet 26 53
Discontinuities in basic terminants 411—412 428 501
Dispersion relations 480 482 484
Distributions over geometrical progressions and basic terminants 408 415 431
Dixon 43 55
Doetsch 53 54
Double integrals 209—215
Dougall 277 284
Duhamel 53
Dunham 330
Duplication formula for factorials 42
D’Alembert 26 53
Earthquakes 14 318
eigenvalue problems vii 17 325 362 411 495
Elliptic integrals 71
Engineering 277
Erdelyi x xi 24 54 55 248 320 321 349
Error function 1—2 5—6 12—13 18 20 26—27 29 30 76 282 403—404 413
Euler v xi
Euler constant 30 37 66 73 83
Euler numbers 75 92
Euler polynomials 46
Euler transformation 15
Evgrafov 321
Expansion points, alternative 107 271—272
Exponential base 129
Exponential integral 31—33 54 59—60 76 94 102 103 162 434—435
Exponential type of series 14
Extinction point ix 354 500
Factorial function 10 37—38 65—66 105—106 157—160 435—436 448 461—462
Factorial function, incomplete 30—31 44 57—59 77 93 102 160—164 249 283 361 369—370 415 434—435 462—464 480—481 506
Factorial sequences 431—434
Fedorcuk 321
Fermi — Thomas calculations 14
Fermi-Dirac integral 20 38—39 55 66—67 69 74 106 164—166 436—437.
ferromagnetism 14
Flamme 140 154
Focke 215
Ford 49 55
Fractional powers 12—13
| Franklin 107 109
Fraser 268 274
Fresnel integrals 77 446 451
Friedman 107 709 247 254
Frobenius 26 53 275 284
Froman, N. and P. O. 294 295 317 321
Furry 321 330
GANs 292 316 321
Gegenbauer polynomials 45 47
General exponential integral 102 162
Geometrical progression 406
Goldstein 17 25 317 321
Goodwin 403 430 446 451
gray 54
Green v xi 285 287 288 292 294 316 321 323
Gregory 26 53 433 451
Gunson vii xi
Haar 140 154
Hamy 140 154
Handelsman 54 215
hankel 39 277 284
Heading 6 25 295 321
Hermite polynomials 46 47
High derivatives 155—157
hilbert 140 154
Historical remarks v-vii 13—14 16 26 107 132—134 140 216—218 241 247—248 275—278 292—295 316—318 327—329 401—405 408
Hobson 64 99
homogeneous differential equations 275—370 489—506
Horn 276 284
Hsu 215
Hydrodynamics 14 277
Hypergeomctric function 14 15 72 88
Ikaunieks 451
Incomplete factorial function 30—31 44 57—59 77 93 102 160—164 249 283 361 369—370 415 434—435 462—464 480—481 506
Incomplete gamma function 54
Indentations in contour 30 280
inhomogeneous differential equations 371—400 507—513
Integrals of Bessel functions 450—451
Integration by parts 21 112 404
Integration constants in recurrence relations 297 300 303 333 355—356 386
Interchanging order of summation and integration 410
Interdependence of late and early terms in asymptotic expansions 148—149 151 152 299—300 303
Interpretation of asymptotic expansions vi 401—513
Inverse factorial sequence 431—432
Ionosphere 14 318
Ising model 14 15
Ittman 321
Jacobi polynomials 45
Jacobian 211
Jeffreys x xi 292 295 317 321 404 430
Jones 215
JORDAN 156 199 208
Jorna x xi 317 318 321 346 350 362 473 488
Kato vii xi
Kazarinoff 350 362
Kelvin 134 139
Kelvin functions 78
Kemble 317 321
Kline 215
Kontorovitch 215
Kramers 292 317 321
Kreyszig 447 451
Kummer relation 33 50 54 61 175 357 447 451
Lagrange reversion theorem 112 127—130
Laguerre function 46 163 283
Landsberg 268 274
Lang 224 254
Laplace v xi 130 133 139
Laplace-type representations 27 100 276 409 410
Laplace’s method for expanding integrals 133
Late derivatives 155—157
Lauwerier xi 349 350
Le Caine 54
Legendre polynomials 45 47 71 72
Leibnitz 155 434
Lew 54
Liouville v xi 285 287 288 292 294 316 321 323
Liouville-Green method 285—321 323
Logarithm of factorial 37 54 105—106 435—436 448
Logarithmic derivative of factorial function 29 31 41—42 72—73 78—79 89—91 93—94 105—106 435^36 448
Lohmander 224 254 264 274
Lommel function of two variables 80 108 109 215 378—381 391—394 399 400 442—444 450—451
Lommel polynomials 46 47
Longer 286 294 295 317 321 349
Macdonald 224 254
MacFarlane 403 430
Maclaurin v xi
MacRobert 54
Magnus 54
Malaviya x
Malmsten 66 99 105 106
Many-valued functions 12 148
Mapping 247
Maslov 321
Mathews 54
Mathieu functions x 17 277 .117
McDougall 55
McGill x
Meijer 55
Meissel vi xi
Mellin transforms 27 30 48 53 223 231 239—240 278 331—334 363—366 371 428
Miller vi xi 404 408 430
Milne-Thomson 335 350
Mittag-Leffler 38 55
Modified Bessel function 36 50 51 54 62—65 78 86 95—96 104—105 108 181—182 189 201 212 240 281—282 440—441 451
Modified Lommel function of two variables 381 451
Modified Struve function 79 80 371—376 381 389—391 445—446
Modifying factor 431—434
mount 14 24
Multivalued functions 12 148
Muravev 215
Murnaghan 408 430
Mutter x xi
Neumann — Lommel polynomials 46
Neutralizer 132
Nevanlinna 430
Newton series 216 258 433 451
Ng vii xi
Nicholson 248
Nielsen 54
Nikiforov 321
Ninham 140 154
Non-linear phase-integral method 322—330 495—496
Non-numerical compliance definition of an asymptotic expansion 19—21 30 35
Norlund 335 336 350
Numerical analysis 411
Oberhettinger 54 55
Olver vi x xi 20 134 139 248 292 317 321 349
Optics 14
Ott 263 274
Overlapping contributions from expansion points ix 131—132 141
Pade sequences 15 403
Parabolic cylinder functions 8—12 203—204 224 258 283 319—320 329—330 349 449 486—487 498
Partial sum of asymptotic series 401
Pearson 54
Perturbation expansions vii
Phase convention for fractional powers 12—13
Phase-integral methods 8 10 14 285—350 362 489—498
Physics vii 13 14 277
Placzek 54 130
Poincare v xi 16 25
Poincare — Watson definition of an asymptotic power series 17—19 28 30 101
Poincare’s definition of an asymptotic power series v 16—17 294—295
Poiseuille functions 349
poisson 63
Polynomial form of recurrence relation 296 353 384
Principal value of an integral 412
Psi function 29 31 41—42 72—73 78—79 89—91 93—94 105—106 435—436 448
Quadratic contours 135—136
quantum mechanics 11 14 318
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