Watson G.N. — Treatise on the theory of Bessel functions :: Электронная библиотека попечительского совета мехмата МГУ

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Watson G.N. — Treatise on the theory of Bessel functions

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Название: Treatise on the theory of Bessel functions

Автор: Watson G.N.

Язык:

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1922

Количество страниц: 812

Добавлена в каталог: 06.11.2004

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Предметный указатель

Inequalities, by Struve's function      328 337 417
Inequalities, by zeros of Bessel functions      485 489 490 492 494 515 516 521
Inequalities, satisfied by Bessel functions      16 31 49 255 259 268 406
Infinite integrals      see also “Lipschitz — Hankel infinite integral”
Infinite integrals, containing Bessel functions under the integral sign      383—449
Infinite integrals, discontinuous      398 402 406 408 411 415 421
Infinite integrals, generalised      441
Infinite integrals, methods of evaluating, described      383
Infinite integrals, Ramanujan's type (integrals of Bessel functions with respect to their order)      449
Infinity of the number of zeros of Bessel functions and cylinder functions      4 478 481 494 495
Integrals      see also “Definite integrals” “Infinite
Integrals with oscillatory integrands      183
Integrals with the polynomials of Neumann and Gegenbauer under the integral sign      277 285
Integrals, expressed in terms of Lommel's functions of two variables      540
Integrals, expressed in terms of the functions of Anger and Weber      312
Integrals, Fresnel's      544 545 660 661 664 744 745
Integrals, Gilbert's      548 549
Integrals, values of, deduced from addition theorems      367
Interference      229
Interlacing of zeros of Bessel functions and of cylinder functions      479 480 481
Irrationality of $\pi$       90 485
Isherwood, J.G.      657 658 664
Jackson, Dunham      343
Jackson, Frank Hilton      43 44
Jackson, William Hartas      177
Jacobi's transformation, connecting $sin n{\theta}$ with the (n- l)th differential coefficient of $sin^{2n-1}{\theta}$ with respect to $cos {\theta}$       26
Jacobi's transformation, erroneously attributed to Rodrigues      27
Jacobi's transformation, various proofs of      27 28
Jacobi, C.G.J.      6 8 14 21 22 25 26 27 28 29 84 195 379 555 572
Jamet, E.V.      94
Johnson, W.W.      92
Jolliffe, A.E.      528 529
Julius, V.A.      65 200
Kalaehne, A.      505 507 660
Kapteyn series      6 13 551—575
Kapteyn series, connexion with Kepler's problem      551
Kapteyn series, diagram of      270
Kapteyn series, expansion of an arbitrary analytic function into      570
Kapteyn series, expansions into, derived from Kepler's problem      554 555
Kapteyn series, fundamental expansions into      557 559 561 564 566 568 571
Kapteyn series, Kapteyn's domain K, of convergence of      559
Kapteyn series, nature of convergence outside and on the boundary of K      574
Kapteyn series, second kind of      572
Kapteyn's domain K      559
Kapteyn's domain, diagram of      270
Kapteyn's polynomial       568
Kapteyn's polynomial , expressed in terms of Neumann's polynomial      569
Kapteyn's type of definite integral      380
Kapteyn, W.      35 183 200 268 279 281 282 292 351 370 373 376 380 386 404 413 498 499 531 532 533 535 536 538 551 559 560 562 565 566 568 569 570
Kelvin, Lord      81 124 203 225 229 230 233 248 654 658
Kepinski, S.      99
Kepler's problem      6 551 554
Kepler's problem, Bessel's solution of      13
Kepler's problem, Lagrange's solution of      6
Kepler, J.      551 552
Kinds of Bessel functions, first      40
Kinds of Bessel functions, second      58 63 64 65 67
Kinds of Bessel functions, third      73
Kirchhoff, G.      107 196 203 389 578 616
Kluyver, J.G.      367 419 420
Kneser — Sommerfeld expansion of a combination of Bessel functions as a Fourier — Bessel series      499
Kneser, J.C.C.A.      499 578 583
Knockenhauer, K.W.      545
Koenig, J.      354 523
Koppe, M.      247
Kummer's formulae, connecting generalised hypergeometric functions      101 102
Kummer, E.E.      49 90 101 102 104 105 148 185 190 191 196 203 394 409
Lacroix, S.F.      27
Lagrange, J.L. de      6 27 28 551
Lamb, H.      56 96 385 416 475 502
Lambert, J.H.      485
Lame functions, limiting forms expressed as Bessel functions      159
Lame, G.      96 159
Landau, E.G.H.      197
Laplace's equation, general solution due to Parseval      9
Laplace's equation, general solution due to Whittaker      124
Laplace's equation, solutions involving Bessel functions      83 124
Laplace's equation, used to obtain addition theorems for Bessel functions      127
Laplace's methods of approximating to functions of large numbers      8 421
Laplace's transformation      280 395
Laplace, P.S. de      6 7 8 9 53 280 395 421 450
Large numbers      see also “Approximations” “Asymptotic
Large numbers, methods of approximation to functions of (Darboux)      233
Large numbers, methods of approximation to functions of (Laplace)      8 421
Large order      see “Anger's function” “Bessel whose “Struve's “Weber's
Largeteau, C.L.      501
Laurent, Paul Mathieu Hermann      157
Laurent, Pierre Alphonse      100
Le Paige, G.      96
Lebedeff, Wera Myller-      99
Lebesgue's lemma      see “Riemann — Lebesgue lemma”
Lebesgue, Henri      457
Lebesgue, Victor Amedee      110 123
Lefort, F.      6
Legendre functions      see also “Gegenbauer's function $C_n^{\nu} (z)$ ”
Legendre functions of large degree, approximations to      158
Legendre functions, Barnes' notation for      156
Legendre functions, integrals containing      50 173 174 339 475
Legendre functions, limits of, expressed as Bessel functions      65 155 157
Legendre functions, limits of, expressed as Bessel functions, physical significance of      155
Legendre functions, relation between two kinds of      174
Legendre functions, Whipple's transformation of      387
Legendre, A.M.      52 90 183 204 485 557
Leibniz, G.W.      1 2 3
Lerch, M.      382 393 433 434 617
Lindner, P.      484
Lindstedt, A.      545 661
Liouville, J.      27 28 87 111 112 116 117 119 120 123
Lipschitz — Hankel infinite integral      384
Lipschitz — Hankel infinite integral, generalised      389
Lipschitz, R.O.S.      11 12 195 200 206 331 339 384 386 390 633
Lobatschevsky, N.      503
Lobatto, R.      49 90
Lodge, A.      224 229
Lommel's functions $s_{{\mu}, {\nu}} (z)$ , $S_{{\mu}, {\nu}} (z)$       345 347
Lommel's functions of two variables with large argument, asymptotic expansions of      549
Lommel's functions of two variables, $U_{\nu} ({\omega}, z)$ , $V_{\nu} ({\omega}, z)$       537 538
Lommel's functions of two variables, addition formulae for      543
Lommel's functions of two variables, integrals representing      540 546
Lommel's functions of two variables, reciprocation formulae      542
Lommel's functions of two variables, recurrence formulae      539
Lommel's functions of two variables, special case of      581 752
Lommel's functions of two variables, tables of      752
Lommel's functions of two variables, tables referred to      660
Lommel's functions with large argument, asymptotic expansion of      351
Lommel's functions, cases of expression in finite terms      350
Lommel's functions, integrals representing      346 350
Lommel's functions, recurrence formulae      348
Lommel's functions, special cases expressible by the polynomials of Gegenbauer, Neumann and Schlaefli      350
Lommel's functions, special cases with ${\mu}±{\nu}$ an odd negative integer      348
Lommel's polynomial $R_{m, {\nu}} (z)$       294 295
Lommel's polynomial of negative order, $R_{-m, {\nu}} (z)$       299
Lommel's polynomial, differential equation, satisfied by      297
Lommel's polynomial, Hurwitz' notation $g_{m, {\nu}} (z)$       303
Lommel's polynomial, limit of, expressed as a Bessel function      302
Lommel's polynomial, recurrence formulae      298
Lommel's polynomial, recurrence formulae in Hurwitz' notation      303
Lommel's polynomial, relations with Bessel functions      295 297 302
Lommel's polynomial, three-term relations connecting      300 301
Lommel's polynomial, zeros of      304 305 306
Lomrnel, E.C.J. von      13 21 23 25 30 34 38 43 45 46 47 49 53 59 62 65 66 71 73 76 77 96 97 98 99 106 107 132 133 135 136 140 142 143 145 148 151 152 154 196 200 294 295 296 297 298 299 300 303 308 315 328 341 345 348 350 364 374 406 478 479 482 529 531 537 538 539 540 542 543 544 545 546 548 549 550 576 654 655 656 658 660 661 664
Lorenz, L.      57 96 224 229 500
Love, A.E.H.      56 226 417 449 654 662
Macdonald, H.M.      78 79 80 158 170 171 225 229 233 365 377 385 386 389 395 396 412 413 439 440 482 509 511
Maclean, M.      658
MacRobert, T.M.      197
Maggi, G.A.      13
Magnitudes of remainders in asymptotic expansions      206 211 213 236 314 332 352 449
Malmsten, C.J.      99 173 185 187 188 196 203
Manfredius, G.      92

March, H.W.      56 225 449
Marcolongo, R.      135
Marshall, W.      505 506
Mathews, G.B.      64 65 78 194 206 454 480 655 656 657 658 660
Maxima of Bessel functions      488
Maxima of Fresnel's integrals, table of      745
Maxima of integrals of Bessel functions, table of      752
Maxwell, J. Clerk      125
Mayall, R.H.D.      550
McMahon, J.      64 195 200 505 507 581
Mean anomaly, expansions of elements of an orbit in trigonometrical series of      6 13 552 554 556
Mehler — Dirichlet integral, representing Legendre functions, limiting form expressed as Poisson's integral      157
Mehler — Sonine integrals, representing Bessel functions      169 170
Mehler, F.G.      65 155 157 169 170 180 183 425 431 455 475 476
Meissel's approximations to Bessel functions of large order      226 227 232 247 521
Meissel's approximations to Bessel functions of large order, types of Kapteyn series      557 561 564 566
Meissel, D.F.E.      7 145 204 226 227 229 232 233 234 247 391 521 557 558 561 564 572 655 656 658 660 662
Mellin, R.H.      190 196
Membrane, vibrations of a circular      5 576 618
Membrane, vibrations of a sectorial      510
Method of constant phase (Schlaefli's)      216
Method of stationary phase      225 229
Method of stationary phase, applied to Bessel functions      231 233
Method of steepest descents      235
Method of steepest descents, applied to Bessel functions      237 241 244 245 262
Method of steepest descents, applied to functions of Anger and Weber      316
Method of steepest descents, applied to Struve's function      333
Method of steepest descents, connexion with Laplace's method of approximation      421
Minima of Bessel functions      488
Minima of Fresnel's integrals, table of      745
Minima of integrals of Bessel functions      752
Mittag-Leffler, M.G.      83 497
Molins, H.      106
Monotonic properties of $J_{\nu} ({\nu})$ and $J'_{\nu} ({\nu})$       260
Monotonic properties of $J_{\nu} ({\nu}x)/J_{\nu} ({\nu})$       257
Monotonic properties of       446
Moore, C.N.      479 578 579 597 649
Morton, W.B.      65 66
Multiple infinite integrals      450—476
Multiple infinite integrals, investigated by Neumann      453 470
Multiple infinite integrals, investigated by Neumann, generalised by Hankel      453 456 465
Multiple infinite integrals, investigated by Neumann, generalised by Orr      455
Multiple infinite integrals, investigated by Neumann, modified by Weber      468
Multiple infinite integrals, Riemann — Lebesgue lemmas, analogues of      457 471
Multiple infinite integrals, Weber's type of      450
Murphy, R.      91 156 157
Myller-Lebedeff, W.      see “Lebedeff”
Nagaoka, H.      340 633 634
Neumann series      522—537 (see also “Addition theorems” “Lommel's
Neumann series, expansion of an arbitrary analytic function into      523
Neumann series, generalised      525
Neumann series, generalised, special series      30 31 36 69 71 151
Neumann series, Laurent's expansion, analogue of      524
Neumann series, Pincherle's theorem on the singularities of      526
Neumann series, special series      18 23 25 33 34 35 128 130 138 139 140 527 581
Neumann series, Webb — Kapteyn (real variable) theory of      533
Neumann's factor ${\epsilon}_n$ (=1 or 2)      22
Neumann's integral for       32
Neumann's integral for Neumann's polynomial      278 280
Neumann's polynomial       271 272 273
Neumann's polynomial ${\Omega}_n (t)$       290 291
Neumann's polynomial ${\Omega}_n (t)$ , expressed as integral containing Neumann's polynomial       292
Neumann's polynomial ${\Omega}_n (t)$ , Gegenbauer's generalisation of      see “Gegenbauer's polynomial $$B_{n</a></span> <span class=subjpages><a href=$ {\mu} {\nu}}
Neumann's polynomial ${\Omega}_n (t)$ , recurrence formula for      292
Neumann's polynomial of negative order defined      276
Neumann's polynomial, connected with Kapteyn's polynomial      569
Neumann's polynomial, connected with Neumann's polynomial ${\Omega}_n (t)      292
Neumann's polynomial, connected with Schlaefli's polynomial      285 286
Neumann's polynomial, contour integrals containing      277
Neumann's polynomial, differential equation satisfied by      276
Neumann's polynomial, expressed in terms of Lommel's functions      350
Neumann's polynomial, formerly called a Bessel function of the second kind      67 273
Neumann's polynomial, generalised by Gegenbauer      see “Gegenbauer's polynomial $$A_{n</a></span> <span class=subjpages><a href=$ {\nu}}
Neumann's polynomial, generating function of      281 282
Neumann's polynomial, inequalities satisfied by      273 282
Neumann's polynomial, infinite integrals containing      433
Neumann's polynomial, Neumann's integral representing      278 280
Neumann's polynomial, recurrence formulae for      274
Neumann, Carl Gottfried      16 19 22 23 30 31 32 33 34 36 37 46 59 60 65 66 67 68 69 70 71 73 128 143 150 151 155 271 273 274 276 277 278 280 281 284 286 290 291 292 345 358 359 361 363 365 386 418 424 440 441 453 455 456 470 471 473 474 475 476 522 523 524 525
Neumann, Friedrich E.      154
Newman, F.W.      663 664
Newton, Isaac      120
Nicholson's infinite integrals      431 441
Nicholson, J.W.      107 108 145 146 149 150 189 226 229 231 247 248 249 250 252 262 329 332 413 415 425 426 431 440 441 446 448 505 656
Nicolas, J.      77 84
Nielsen — Hankel functions      see “Bessel functions of the third kind”
Nielsen, N.      24 44 49 64 73 74 77 82 83 132 142 145 148 149 154 169 224 297 298 299 315 350 355 357 359 392 405 455 465 522 523 525 526 527 528 571 572 574 597 622 629 636
Niemoeller, F.      57 68 195
Null-functions, Lerch's theorem on integrals representing      382
Null-functions, represented by Schloemilch series      634 636 642 647
Numbers, analytic theory of, associated with asymptotic expansions of Bessel functions      200
Numbers, Cauchy's      324
Numbers, recurrence formulae for      325
Olbricht, R.      158 481
Oltramare, G.      173
Order of a Bessel function, defied      38 58 63 67 70
Order of a Bessel function, integrals with regard to      449
Ordinary differential equations      see “Differential equations”
Orr, W.M.      145 146 206 224 454 455 579
Oscillation of solutions of linear differential equations      518
Oscillations of membranes      5 510 576 618
Oscillations of uniform heavy chains      3 4 576
Oscillatory integrands, Du Bois Reymond's integrals with, expressed in terms of Bessel functions      183
Otti, H.      71 274 286 341
P-functions, limiting forms, expressed as Bessel functions      158
Panton, A.W.      305
Paoli, P.      53 95 186
Parseval's integral, modifications of      21
Parseval's integral, representing       9 21
Parseval, M.A.      9 21 24 68 105 229 358 359 384
Partial differential equations      see “Differential equations”
Pearson, Karl      98 99 419 421
Peirce, B.O.      501 660 664
Peres, J.      44
Perron, O.      154
Petzval, J.      49
Phase, applied to Bessel functions      231 233
Phase, method of stationary, general principles of      225 229
Phase, Schlaefli's method of constant      216
Phragmen, E.      358
Picard, C.E.      93 94
Pincherle's theorem on singularities of functions, denned by Neumann series      526
Pincherle, S.      190 196 271 274 386 526 528
Plana, G.A.A.      10 38 42 45 49 53 95 96 99 195 554
Plummer, H.C.      270 552 555
Pochhammer, L.      100 101 297 346 410
Pocklington, H.C.      537
Poincare, J. Henri      236
Poisson's integral      see also “Parseval's integral” “Struve's
Poisson's integral for Bessel coefficients      12 24 25
Poisson's integral for Bessel functions      47 48 49
Poisson's integral for Bessel functions of imaginary argument      80
Poisson's integral for Bessel functions of the second kind      68 73
Poisson's integral for Bessel functions, generalised by Gegenbauer      50
Poisson's integral for Bessel functions, symbolic form of      50
Poisson's integral, limit of the Mehler — Dirichlet integral for Legendre functions as      157
Poisson's integral, transformation into contour integrals to represent Bessel functions of any order (of the first kind)      161 163 164
Poisson's integral, transformation into contour integrals to represent Bessel functions of any order (of the second kind)      165
Poisson's integral, transformation into contour integrals to represent Bessel functions of any order (of the third kind)      166 167
Poisson's integral, transformation into contour integrals to represent Bessel functions of any order, with imaginary argument      171 172
Poisson's integral, transformations of the contour integrals      168 169 170
Poisson, S.D.      6 9 10 11 12 13 24 25 38 47 49 52 67 68 69 73 95 96 160 173 183 185 186 187 194 195 308 369 477 501
Polar coordinates, change of axes of, used to obtain transformations of integrals      51 374 376 378
Polar coordinates, used to express Bessel functions as limits of Legendre functions      155
Porter, M.B.      299 477 480 485 515 517
Preece, C.T.      27
Probleme de moments of Stieltjes      464
Products of Bessel functions      30 31 32 82 146 147 148 149
Products of Bessel functions with large argument, asymptotic expansions of      221 448
Products of Bessel functions, Bateman's expansion of      130 370

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