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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Olver F.W.J. — Asymptotics and Special Functions |
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| Ïðåäìåòíûé óêàçàòåëü |
Exponential integral, asymptotic expansion of 1—3 112 227
Exponential integral, complementary 40
Exponential integral, converging factors for 523—531 543—544
Exponential integral, generalized 43 72 112 527
Exponential integral, relation to incomplete Gamma functions 45
Fabry’s transformation 231
Faxen, H. 332
Faxen’s integral 332
Fedoryuk, M. V. 105 361 479 517 518 N.
Ferrers functions 185—189 (see also “Legendre functions Legendre
Ferrers functions of large degree 467—469 472—473
Ferrers functions, zeros 469
Feshbach, H. 501
Fields, J. L. 321
Fix, G. 228
Ford, W. B. 321
Fourier integrals 75
Fourier integrals, asymptotic expansion of 75—79
Frank, W. M. 479
Franklin, J. 68 544
Fresnel integrals 44 60
Fresnel integrals, asymptotic expansion of 67
Friedman, B. 351 361 544
Frobenius, G. 149 152
Frobenius’ method 152 163 243
Froese, C 317 320
Froman, N. 517 518
Froman, P. O. 517 518
Fubini, S. 479
Fuchs, L. 145
Functions defined parametrically by integrals 31—32
Furlan, G. 479
Furry, W. H. 517
Gamma function 31 (see also “Incomplete Gamma Psi
Gamma function, asymptotic expansion of 87—88 129—130 293—295
Gamma function, asymptotic expansion of ratio 118—119
Gamma function, Binet’s formula 295
Gamma function, canonical product 34
Gamma function, duplication formula 35
Gamma function, Euler’s integrals 31
Gamma function, expansion at z = 1 64
Gamma function, Gamma function, Euler’s limit formula 33
Gamma function, Hankel’s loop integral 38
Gamma function, history 64
Gamma function, minimum of 39
Gamma function, multiplication formula 35 295
Gamma function, Pochhammer’s loop integral 38
Gamma function, recurrence formula 32
Gamma function, reflection formula 35
Gamma function, singularities 32
Gans — Jeffreys formulas 493—494 510—513
Gans, R. 228 516
Gauss, C. F. 161
Geller, M. 64
Generating function 51
Goodwin — Staton integral 44 115 543
Goodwin, E. T. 44
Green, G. 228
Haar, A. 316 317 318
Haar’s method 315—320 321
Handelsman, R. A. 104 105 332 361
Hankel functions 238
Hankel functions of half-odd-integer order 241
Hankel functions of large argument 238 240 266—270
Hankel functions of large order 424 434
Hankel functions, analytic continuation 239
Hankel functions, bounds for 270
Hankel functions, Hankel’s integrals 241
Hankel functions, recurrence relations 241
Hankel functions, relation to Bessel functions 239 242
Hankel functions, Sommerfeld’s integrals 241
Hankel functions, Wronskians 241
Hankel functions, zeros 434
Hankel, H. 37 133
Hankel’s expansions 238
Hankel’s expansions, error bounds for 266—269
Hanson, R. J. 433
Hardy, G. H. 321
Harris, B. 329
Hartman, P. 143 189
Heading, J. 517
Hermite polynomials 48—50 52
Hermite polynomials of large degree 315 403 412
Hermite polynomials, relation to confluent hypergeometric functions 259
Hermite polynomials, zeros 408
Herstein, I. N. 336
Hethcote, H. W. 228 278 433 434
Hobson, E. W. 184 189
Hochstadt, H. 65
Holomorphic function 9
Horn, J. 277 391
Hsieh, P.-F. 277
Hsu, L. C 361
Hukuhara, M. 391
Hull, T. E. 317 320
Hypergeometric equation 156
Hypergeometric equation, connection formulas for solutions 164—167
Hypergeometric equation, generalized 168
Hypergeometric equation, second solution when c is a nonpositive integer 168
Hypergeometric function 159 (see also “Hypergeometric equation”)
Hypergeometric function as function of its parameters 160
Hypergeometric function of large argument 167 302
Hypergeometric function, asymptotic expansions for large parameters 162
Hypergeometric function, Barnes’ integral 301 321
Hypergeometric function, behavior at z= 1 161 165—166
Hypergeometric function, contiguous 162 177
Hypergeometric function, derivatives 162
Hypergeometric function, generalized 168 189 309
Hypergeometric function, history 189
Hypergeometric function, integral representations 161 162 301
Hypergeometric function, Pochhammer’s loop integral 162
Hypergeometric function, quadratic transformation 167
Hypergeometric function, relation to elementary functions 160 161
Hypergeometric function, singularities 159
Hypergeometric series 159
Improper integrals 74 104
Ince, E. L. 189 231
Incomplete gamma functions 45 64 100
Incomplete Gamma functions, asymptotic expansion of 66—67 69—70 109—112 137
Incomplete Gamma functions, bounds for 67 70 135
Incomplete Gamma functions, complementary 45
Incomplete Gamma functions, relation to confluent hypergeometric functions 259
Integral equations 217—220 228
Integration of series, asymptotic 20—21
Integration of series, convergent 54
Inverse Laplace transforms 315—316
Inverse Laplace transforms, asymptotic expansions of 316—317
Irregular singularities 148 (see also “Singularities of differential equations”)
Irregular singularities at infinity 154
Irregular singularities of inhomogeneous equation 270—274
Irregular singularities, asymptotic solutions at 197—202 223—224 232—236
Irregular singularities, characteristic equation 231
Irregular singularities, characteristic values 231
Irregular singularities, circuit exponents 482
Irregular singularities, circuit roots 482
Irregular singularities, connection formulas at 481—482
Irregular singularities, error bounds for asymptotic solutions 193 196 222 265—266 278
Irregular singularities, formal solutions at 230 231
Irregular singularities, history 277—278
Irregular singularities, LG approximation at 230
Irregular singularities, rank 148 154
Ismail, M. E. 321
Jacobi polynomials 48—49 52 167
Jacobi’s lemma 185
Jeffreys, B. S. 192
Jeffreys, H. 192 206 228 433 495 501 516 517 544
Join 15
Jones, A. L. 251
Jones, D. S. 105 136 361
Jordan’s inequality 42
| Jorna, S. 433 434
Karmazina, L. N. 479
Kazarinoff, N. D. 278 479
Kelvin (Lord) 105
Kelvin functions 61
Kelvin’s method of stationary phase see “Method of stationary phase”
Kemble, E. C 497 517
Kiyek, K. H. 391 433
Klein, F. 159
Knopp, K. 283
Kostomarov, D. P. 478
Kramers, H. A. 228
Kronecker’s delta symbol 46
Kummer’s function 255 (see also “Confluent hypergeometric functions”)
Laguerre polynomials 48—50 52
Laguerre polynomials, relation to confluent hypergeometric functions 259
Landau, E. 4
Langer, R. E. 228 433 478 479 517
Laplace (Le Marquis de) 80
Laplace integrals 67
Laplace integrals, analytic continuation 107—108
Laplace integrals, asymptotic expansion of 67—70 106—109
Laplace transform 112
Laplace transform, abscissas of convergence 113
Laplace transform, inversion of 315—316
Laplace’s method 80—81 85—86
Laplace’s method for contour integrals 121—127
Laplace’s method, error bounds for 90—96 105 135—136
Laplace’s method, examples 82—84 87—88 127—130
Laplace’s method, generalizations 325—329 331—333 336—339 361
Laplace’s method, history 105
Laplace’s method, relation to method of stationary phase 98
Laurent coefficients, asymptotic expansion of 309—310 321
Lauwerier, H. A. 130
Legendre functions 170 178 Conical Ferrers Legendre
Legendre functions of degree or order ±i 173—174
Legendre functions of integer degree 179 180—185
Legendre functions of large degree 206 311—313 463—473 479
Legendre functions on cut 185
Legendre functions, addition theorem 183
Legendre functions, analytic continuation 179
Legendre functions, behavior at singularities 171 173—174 181
Legendre functions, bound for 185
Legendre functions, connection formulas 171 179 490—491
Legendre functions, degree 174
Legendre functions, generating function 185
Legendre functions, Heine’s formula 473
Legendre functions, Heine’s integral 185
Legendre functions, history 189 479
Legendre functions, integral representations 174 177—183 185 189
Legendre functions, Neumann’s integral 183
Legendre functions, order 174
Legendre functions, recurrence relations 176—177 179
Legendre functions, Whipple’s formula 174
Legendre functions, Wronskians 172
Legendre polynomials 48 (see also “Ferrers functions Legendre
Legendre polynomials of large degree 84 127—129 130 311—313 463—473 479
Legendre polynomials, bound for 52
Legendre polynomials, differential equation for 50
Legendre polynomials, generating function 51
Legendre polynomials, Laplace’s integral 52
Legendre polynomials, Neumann’s expansion theorem 473
Legendre polynomials, recurrence relations 50 52
Legendre polynomials, Rodrigues’ formula 49
Legendre polynomials, Schlafli’s integral 51
Legendre polynomials, zeros 469
Legendre, A. M. 291
Legendre’s equation 169
Lerch, M. 299
Levinson, N. 31 123 128 146 247 479
Lew, J. S. 104 332 361
LG approximation 191
LG approximation at singularities 200—202 204—206 208
LG approximation with complex variables 222
LG approximation, asymptotic properties with respect to independent variable 197—202 223—224
LG approximation, asymptotic properties with respect to parameters 203—206 224
LG approximation, condition for exactness 193
LG approximation, derivative of error term with respect to parameter 216
LG approximation, error bounds for 193 196 203—204 206 222
LG approximation, history 228
LG approximation, special extension 208—211
LG functions 191
Lindelof, E. 294 321
Liouville transformation 191 228 363
Liouville — Green see “LG approximation LG
Liouville — Neumann expansion 146 218 364
Liouville, J. 228
Logarithmic integral 41 64
Lommel, E. C. J. von 245
Lommel’s method for locating zeros 245 414
Loop contour 37
Lorch, L. 434
Ludwig, D. 342
Luke, Y. L. 64 162 189 278 321 344 434
Lyness.J. N. 104 306 321
Macdonald, H. M. 252
Macdonald’s function 250 (see also “Modified Bessel functions”)
MacFarlane, G. G. 321 544
Maclaurin coefficients, asymptotic expansion of 309—315 329—331
MacRobert, T. M. 189
Massey, H. S. W. 433
Mathieu equation 496
Maximon, L. 78
McHugh, J. A. M. 228 433 517
McKenna, J. 361
McLeod, J. B. 479 518
Medhurst, R. G. 94
Mellin transform 336
Method of Chester, Friedman, and Ursell 351 361
Method of Chester, Friedman, and Ursell, examples 352—357 425
Method of Chester, Friedman, and Ursell, extension of the region of validity 358
Method of stationary phase 96 100—102 104
Method of stationary phase, examples 102—104
Method of stationary phase, generalizations 342 361
Method of stationary phase, history 105
Method of stationary phase, relation to other methods 98 105 138
Method of steepest descents 136—138 (see also “Laplace’s method”)
Method of successive approximations 141
Miller, G. F. 346 527
Miller, J. C. P. 154 228 433 524 530 540 543
Milne — Thomson, L. M. 321
Modified Bessel equation 60
Modified Bessel equation, inhomogeneous form 386
Modified Bessel equation, numerically satisfactory solutions 251
Modified Bessel functions 60 250
Modified Bessel functions of large argument 82—83 92—94 250 251 269
Modified Bessel functions of large complex order 381—382
Modified Bessel functions of large imaginary order 382 425
Modified Bessel functions of large positive order 328 374—381
Modified Bessel functions, analytic continuation 253 381
Modified Bessel functions, ascending series 60 253
Modified Bessel functions, asymptotic expansion of v-derivatives 325
Modified Bessel functions, auxiliary functions for 454—457
Modified Bessel functions, Bassett’s integral 254
Modified Bessel functions, collected properties 435—436 453—454
Modified Bessel functions, converging factor for 536
Modified Bessel functions, generating function 61
Modified Bessel functions, graphs 251
Modified Bessel functions, infinite integral of 254
Modified Bessel functions, integral representations 60 250 254
Modified Bessel functions, monotonicity properties 251
Modified Bessel functions, recurrence relations 60 253
Modified Bessel functions, relation to confluent hypergeometric functions 255 259
Modified Bessel functions, Wronskians 251
Modified Bessel functions, zeros 252 254
Modulus function 394 436 456
Moments 70
Monodromy theorem 146
Monotonicily condition 222
Monotonicily condition, necessity of 236
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