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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.

Аннотация:

To incorporate in this work the discoveries of the last twenty years would
necessitate the rewriting of at least Chapters XII—XIX; my interest in
Bessel functions, however, has waned since 1922, and I am consequently not
prepared to undertake such a task to the detriment of tny other activities.
In the preparation of this new edition 1 have therefore limited myself to the
correction of minor errors and misprints and to the emendation of a few
assertions (such as those about the unproven character of Bourget's hypo-
hypothesis) which, though they may have been true in 1922, would have been
definitely false had they been, made in 1941.
My thanks are due to many friends for their kindness in informing me of
errors which they had noticed; in.particular, I cannot miss this opportunity
of expressing my gratitude to Professor J. R. Wilton for the vigilance which
he must have exercised in the compilation of his list of corrigenda.

Язык:

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

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Издание: 2-е издание

Год издания: 1944

Количество страниц: 804

Добавлена в каталог: 18.04.2005

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Предметный указатель

Le Paige, C      96
Lebedeff, Wera Myller      99
Lebesgue's lemma      see Riemann — Lebesgue lemma
Lebesgue, Henri      457
Lebesgue, Victor Amedee      110 123
Lefort, P.      6
Legendre functions, Barnes' notation for      156
Legendre functions, integrals obtainhig      50 173 174 339 475
Legendre functions, limits of (physical significance of)      155
Legendre functions, limits of, expressed as Bessel functions      65 155 157
Legendre functions, of large degree, approximations to      158
Legendre functions, relation between two kinds of      174
Legendre functions, Whipple's transformation of      387. See also Gegenbauer's function $C_n^\nu(z)$
Legendre, A.M.      52 90 183 204 485 557
Leibniz, G.W.      1 2 3
Leroh, M.      382 393 433 434 617
Lindner, P.      484
Lindstedt, A.      545 661
Lipschitz — Hankel infinite integral      384
Lipschitz — Hankel infinite integral, generalised      389
Lipschitz, R.O.S.      11 12 195 200 206 331 339 3H 386 390 633
Lipuville, J.      27 28 87 111 112 116 117 119 120 123
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, $U_\nu(w, z)$ and $V_\nu(w, z)$       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 of two variables, with large argument, asymptotic expansions of      549
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 $\mu\pm\nu$ an odd negative integer      348
Lommel's functions, with large argument, asymptotic expansion of      351
Lommel's polynomial $R_{m, v}(z)$       294 295
Lommel's polynomial, differential equation satisfied by      297
Lommel's polynomial, Hurwitz' notation $g_{m, v}(z)$       303
Lommel's polynomial, limit of, expressed as a Bessel function      302
Lommel's polynomial, of negative order, $R_{-m, v}(z)$       299
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
Lommel, E.C.J. von      13 21 23 25 30 34 38 43 45 40 47 49 53 59 62 65 66 71 73 76 77 96 97 98 99 106 107 132 133 135 130 140 142 143 145 148 151 152 154 196 200 294 295 296 297 298 299 300 303 308 315 328 341 345 318 350 304 374 400 478 479 482 529 531 537 538 539 540 542 543 544 545 546 548 549 550 576 654 655 656 658 660 661 664
Lorenz, L.      57 96 224 229 500
Love, A.E. IL      56 226 417 449 654 662
Mac Robert, T.M.      197
Macdonald, H.M.      78 79 80 158 170 171 225 229 233 365 377 385 386 389 395 396 112 413 439 440 482 509 511
Maclean, M.      658
Maggi, G.A.      13
Magnitudes of remainders in asymptotic expansions      206 211 213 286 314 332 852 449
Malmsten, C.J.      99 173 185 187 L88 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.U.      550
McMahon, J.      61 195 200 505 507 581
Mean anomaly, expansions of elements of an orbit in trigonometrical series of      6 13 552 554 556
Mehler — Bonine integrals representing Bessel functions      169 170
Mehler — Dirichlet integral representing Legendre functions, limiting form expressed as Poisson's integral      157
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 types of Kapteyn series      557 561 664 566
Meissel, D.P.E.      7 145 204 226 227 229 232 233 234 247 391 521 557 558 561 564 672 655 656 658 660 662
Mellin, R. Hj.      190 196
Membrane, vibrations of a circular      5 576 GIB
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 23B
Method of steepest descents      235
Method of steepest descents, applied to Bessel functions      237 241 241 245 262
Method of steepest descents, applied to functions of Anger and Weber      816
Method of steepest descents, applied to Struve's function      838
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 $J_\nu(\nu x)/J_\nu(\nu)$       257
Monotonic properties of $J_\nu(\nu)$ and $J_\nu'(\nu)$       260
Monotonic properties of       446
Moore, C.N.      479 578 579 597 649
Morton, W.B.      65 66
Multiple infinite integrals      450—476 (Chapter XIV)
Multiple infinite integrals investigated by Neumann      453 470
Multiple infinite integrals investigated by Neumann (generalised by Hankel)      453 456 4(55
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 (Chapter XVI)
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'a theorem on the singularities of      526
Neumann series, special series      18 23 25 33 34 35 128 130 138 189 140 527 581
Neumann series, Webb-Kapteyn (real variable) theory of      533. See also Addition theorems and Lommel's functions of two variables
Neumann's factor $\epsilon_n$ (=1 or 2)      22
Neumann's integral for       82
Neumann's integral for Neumann's polynomial      278 280
Neumann's polynomial       271 272 273
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 formerly called a Bessel function of the second kind      67 273
Neumann's polynomial generalised by Gegenbauer      see Gegenbauer's polynomial $$A_{n</a></span> <span class=subjpages><a href=$ v}{t)$"/>
Neumann's polynomial of negative order defined      276
Neumann's polynomial recurrence formulae for      274
Neumann's polynomial , contour integrals containing      277
Neumann's polynomial , differential equation satisfied by      276
Neumann's polynomial , expressed in terms of Lommel's functions      850
Neumann's polynomial , generating function of      281 282
Neumann's polynomial , inequalities satisfied by      273 282
Neumann's polynomial , infinite integrals containing      488
Neumann's polynomial , Neumann's integral representing      278 280
Neumann's polynomial $\Omega_n(t)$       290 291
Neumann's polynomial $\Omega_n(t)$ , expressed as integral containing Neumann's polynomial       292
Neumann's polynomial $\Omega_n(t)$ , Gegenbauer's generalisation of      see Gegenbauer's polynomial $$B_{H</a></span> <span class=subjpages><a href=$ \mu \nu}(t)$"/>
Neumann's polynomial $\Omega_n(t)$ , recurrence formula for      292
Neumann, Carl Gottfried      16 19 22 23 30 31 32 33 34 36 37 46 59 60 65 66 67 68 69 70 71 73 128 143 150 161 155 271 273 274 276 277 278 280 281 284 286 290 291 292 345 358 359 361 363 365 386 418 424 440 441 453 455 456 470 471 473 474 475 476 522 523 524 525
Neumann, Friedrich E.      154
Newman, F.W.      663 664
Newton, Sir 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, Lereh'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, defined      38 58 63 67 70
Order of a Bessel function, integrals with regard to      4—19
Ordinary differential equations      see Differential equations
Orr, W. McF.      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 representing       9 21
Parseval's integral, modifications of      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
Perks, 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, K      358
Picard, C.E.      93 94
Pincherle's theorem on singularities of functions defined 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 for Bessel coefficients      12 24 25
Poisson's integral for Bessel functions      47 48 49
Poisson's integral for Bessel functions (generalised by Gegenbauer)      50
Poisson's integral for Bessel functions (symbolic form of)      50
Poisson's integral for Bessel functions of imaginary argument      80
Poisson's integral for Bessel functions of the wond kind      68 73
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      108 169 170.
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      51 374 370 378
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, Bateman's expansion of      130 370
Products of Bessel functions, expansions of arbitrary functions into series of      525 572
Products of Bessel functions, integrals representing      31 150 221 438 439 440 441 445 446 448
Products of Bessel functions, series of      30 151 152
Products of Bessel functions, with large argument, asymptotic expansions of      221 448
Products of Weierstrassian factors, Bessel functions expressed as      497
Puiseux, V.      559
Quotient of Bessel functions expressed as a continued fraction      153 154 303
Radius vector of an orbit, expansionos trigonometrical series of the mean anomaly      6 13 552 553 554
Raffy, L.      94
Ramanujan'B method of evaluating definite integrals      382
Ramanujan's integrals of Bessel functions with respect to their order      449
Ramanujan, S.      382 449
Random flights, problem of      419
Rank of Bessel functions and cylinder functions      129
Rawson, R.      91
Rayleigh (J.W. Strutt), Lord      50 55 56 74 95 137 155 157 189 230 231 233 331 333 374 389 395 419 421 477 502 510 511 616 618 660
Real variables, expansions of arbitrary functions of      see Dini series Fourier Neumann and
Reality of zeros of Bessel functions      482 483 511
Reciprocation formulae for Lommel's functions of two variables      542
Recurrence formulae for Anger's functions      311
Recurrence formulae for Bessel coefficients      17
Recurrence formulae for Bessel functions of the first kind      45
Recurrence formulae for Bessel functions of the second kind      66 71
Recurrence formulae for Bessel functions of the third kind      74
Recurrence formulae for Bessel functions with imaginary argument      79
Recurrence formulae for Bourget's functions      326
Recurrence formulae for Cauchy's numbers      325
Recurrence formulae for cylinder functions      82
Recurrence formulae for Gegenbauer's polynomials      283
Recurrence formulae for Lommel's functions      348
Recurrence formulae for Lommel's functions of two variables      539
Recurrence formulae for Lommel's polynomials      298 303
Recurrence formulae for Neumann's polynomials       274
Recurrence formulae for Neumann's polynomials $\Omega_n(t)$       283
Recurrence formulae for Schlaefli's functions      71 342 343
Recurrence formulae for Schlaefli's polynomials      285
Recurrence formulae for Struve's functions      329
Recurrence formulae for Weber's functions      311
Recurrence formulae for Whittaker's functions      339. See also Functional equations Hemi-cylindrical
Reduced functions, Gailler's      536
Remainders in asymptotic expansions, magnitudes of      206 211 236 314 332 352
Remainders in asymptotic expansions, signs of      200 207 209 215 315 333
Remainders in asymptotic expansions, Stieltjes' approximations to      213
Repetition of zeros of Bessel functions and cylinder functions, impossibility of,      479
Riccati's differential equation      1 2 85—94
Riccati's differential equation, connexion with Bessel's equation      1 90
Riccati's differential equation, cross-ratio of solutions      94
Riccati's differential equation, equation cognate to      91
Riccati's differential equation, equivalence with the linear equation of the second order      92
Riccati's differential equation, generalised      3 92 94
Riccati's differential equation, limiting form of      86
Riccati's differential equation, singularities of      94
Riccati's differential equation, soluble by various numbers (two, one or none) of quadratures      3 98
Riccati's differential equation, soluble cases of (D. Bernoulli's)      85
Riccati's differential equation, soluble cases of, exhausted by D. Bernoulli's formula and its limit      123
Riccati's differential equation, solutions by various mathematicians (Cayley)      88
Riccati's differential equation, solutions by various mathematicians (D. Bernoulli)      2 85 89
Riccati's differential equation, solutions by various mathematicians (Euler)      87
Riccati's differential equation, solutions by various mathematicians (Schlaefli)      90
Riccati's differential equation, solved by means of infinite series by James Bernoulli      1
Riccati's differential equation, transformations of      86
Riccati, (Count) J.F.      1 2 3 85 86 87 88 94
Riemann — Lebesgue lemma, analogues of the      457 471 589 599
Riemann's theorem analogues for series of Fourier — Bessel and Dini      649
Riemann's theorem on trigonometrical series, analogues for Schloemilch series      642 647
Riemann, G.F.B.      80 158 172 203 229 235 427 457 486 623 637 649
Riesz, M.      606 614
Rodrigues' transformation      see Jacobi's transformation
Rodrigues, O.      27
Roehrs, J.H.      10
Rudski, P.      477 508
Russell, A.      81 82 204
Rutgers, J.G.      373 374 370 376 380 079
Rybczynski, W. von      56 225 449
Sasaki, S.      507
Savidge, H.G.      82 204 658
Schafheitlin's discontinuous infinite integral      398 402 405 406 408 411
Schafheitlin's integrals representing Bessel functions and cylinder functions      108 169 490 491 493
Schafheitlin, P.      64 137 142 108 169 207 215 373 391 392 398 401 402 400 408 421 447 477 479 482 485 487 489 490 491 492 493 494 508 510 543
Scheibner, W.      6
Schlaefli's functions and       71 340 348
Schlaefli's functions, addition theorems for      344 345
Schlaefli's functions, differential equations satisfied by      342 343
Schlaefli's functions, of negative order      343
Schlaefli's functions, recurrence formulae for      71 342 343
Schlaefli's hypergeometric function      90
Schlaefli's polynomial       284 286
Schlaefli's polynomial, addition theorem for      289
Schlaefli's polynomial, connexion with Neumann's polynomial       285 286
Schlaefli's polynomial, Crelier's integral representation of      288
Schlaefli's polynomial, differential equation satisfied by      285

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