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Watson G.N. — Treatise on the Theory of Bessel Functions
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: en

Рубрика: Математика/Анализ/Специальные функции/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-е издание

Год издания: 1944

Количество страниц: 804

Добавлена в каталог: 18.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Abel, N.H.      68 616 621
Adamoff, A.      196
Addition theorems      358—372 (Chapter XI)
Addition theorems for Bessel coefficients of order n      29
Addition theorems for Bessel coefficients of order zero      128 359
Addition theorems for Bessel functions of the first kind (Gegenbauer's type)      362 367
Addition theorems for Bessel functions of the first kind (Graf's type)      130 143 359
Addition theorems for Bessel functions or cylinder functions of any kind (Gegenbauer's type)      363
Addition theorems for Bessel functions or cylinder functions of any kind (Graf's type)      143 361
Addition theorems for hemi-cylindrical functions      354
Addition theorems for Lommel's functions of two variables      543
Addition theorems for Schlaefli's function fn{z)      344
Addition theorems for Schlaefli's polynomial      289
Addition theorems physical significance of      128 130 361 363 366
Addition theorems special and degenerate forms of      366 368
Addition theorems, integrals derived from      367
Aichi, K.      80
Airey, J.R.      63 142 214 247 319 502 505 516 659 660
Airy's integral      188
Airy's integral, expressed in terms of Bessel functions of order one-third      192
Airy's integral, generalised by Hardy      320
Airy's integral, Hardy's expressions for the generalised integral in terms of the functions of Bessel, Anger and Weber      321
Airy's integral, references to tables of      659
Airy, Sir George B.      188 189 229 249 320 321 322 654 659
Aldis, W.S.      65 655 656 657 658 662 663 664
Alexander, P.      579
Analytic theory of numbers associated with asymptotic expansions of Bessel functions      200
Anding, E.      313 657
Anger's function $J_\nu(z)$      308
Anger's function $J_\nu(z)$, connexion with Weber's function      310
Anger's function $J_\nu(z)$, differential equation satisfied by      312
Anger's function $J_\nu(z)$, integrals expressed in terms of      312
Anger's function $J_\nu(z)$, recurrence formulae for      311
Anger's function $J_\nu(z)$, representation of Airy's integral (generalised) by      321
Anger's function $J_\nu(z)$, with large argument and order, asymptotic expansion of      316
Anger's function $J_\nu(z)$, with large argument, asymptotic expansion of      313
Anger, C.T.      21 22 308 309 310 311 312
Anisimov, V.A.      92
Appell, P.E.      146 371
Approximations of large order (extensions due to Meissel)      226 227 232 247 521
Approximations of large order (in transitional regions)      248
Approximations to Bessel coefficients of order zero with large argument      10 12
Approximations to Bessel functions of large order (Carlini)      6 7
Approximations to functions of large numbers (Darboux)      233
Approximations to functions of large numbers (Laplace)      421
Approximations to Legendre functions or large degree      65 155 157 158
Approximations to remainders in asymptotic expansions      213
Approximations to the sum of a series of positive terms      8. See also Asymptotic expansions Method Arbitrary expansions see Dini Fourier Neumann
Argument of a Bessel function defined      40
Asymptotic expansions for Bessel coefficients of order zero with large argument      10 12 194
Asymptotic expansions for Bessel functions of arbitrary order with large argument      194—224 (Chapter VII)
Asymptotic expansions for Bessel functions of arbitrary order with large argument (functions of the first and second kinds)      199
Asymptotic expansions for Bessel functions of arbitrary order with large argument (functions of the third kind by Barnes' methods)      220
Asymptotic expansions for Bessel functions of arbitrary order with large argument (functions of the third kind by Schlaefli's methods)      215
Asymptotic expansions for Bessel functions of arbitrary order with large argument (functions of the third kind)      196
Asymptotic expansions for Bessel functions of arbitrary order with large argument(functions with imaginary argument)      202
Asymptotic expansions for Bessel functions with order and argument both large      225—270 (Chapter VIII)
Asymptotic expansions for Bessel functions with order and argument both large (order greater than argument)      241
Asymptotic expansions for Bessel functions with order and argument both large (order less than argument)      244
Asymptotic expansions for Bessel functions with order and argument both large (order nearly equal to argument)      245
Asymptotic expansions for Bessel functions with order and argument both large (order not nearly equal to argument, both being complex)      262
Asymptotic expansions for combinations of squares and products of Bessel functions of large argument      221 448
Asymptotic expansions for Fresnel's integrals      545
Asymptotic expansions for functions of Anger and Weber (of arbitrary order with large argument)      313
Asymptotic expansions for functions of Anger and Weber (with order and argument both large)      316
Asymptotic expansions for Lommel's functions      351
Asymptotic expansions for Lommel's functions of two variables      549
Asymptotic expansions for Struve's function (of arbitrary order with large argument)      332
Asymptotic expansions for Struve's function (with order and argument both large)      333
Asymptotic expansions for Thomson's functions, ber(z) and bei(z)      203
Asymptotic expansions for Whittaker's function      340
Asymptotic expansions, approximations to remainders in      213
Asymptotic expansions, conversion into convergent series      204
Asymptotic expansions, magnitude of remainders in      206 211 213 236 314 332 352 449
Asymptotic expansions, sign of remainders in      206 207 209 215 315 333 449.
Autonne, L.      94
Bach, D.      96 102 110
Baehmann, P.      197
Baehr, G.F.W.      479
Ball, L. de      157
Barnes, E.W.      83 100 102 104 105 150 190 192 195 196 220 221 340 351 357 307 383 387 402 409
Basic numbers applied to Bessel functions      43
Basset, A.B.      76 77 78 80 172 173 180 385 388 395 425 426 454
Bateman's type of definite integral      379 382
Baterman, H.      130 131 367 370 372 373 376 379 380 389 406 417 437 456 530 533 535
Bauer, G.      50 128 368 370
Bei(z), Ber(z)      see Thomson's functions
Beltrami, E.      51 358 361 374 386 389 390 391 500 579 621
Bernoulli's (Daniel) solution of Ricatti's equation      85 89 123
Bernoulli, Daniel (1700—1782)      2 3 4 9 85 86 87 88 111 123 478 576
Bernoulli, James (1654—1705)      1 2 3 88 90 92
Bernoulli, John (1667—1748)      1 2 3
Bernoulli, Nicholas, (1687—1759)      2
Bernoulli, Nicholas, (1695—1726)      2
Bernoullian polynomials associated with Poisson's integral      49
Bessel coefficient of order zero, $J_0(z)$      3 4
Bessel coefficient of order zero, differential equation satisfied by      4 5
Bessel coefficient of order zero, differential equation satisfied by (general solution of)      5 12 59 60
Bessel coefficient of order zero, expressed as limit of a Legendre function      65 155 157
Bessel coefficient of order zero, oscillations of a uniform heavy chain and      3 4
Bessel coefficient of order zero, Parseval's integral representing      9
Bessel coefficient of order zero, with large argument, asymptotic expansion of      10 12 194
Bessel coefficient of order zero, zeros of      4 5. Bessel
Bessel coefficients $J_n(z)$      5 6 13 14—37
Bessel coefficients $J_n(z)$, addition theorem for      29
Bessel coefficients $J_n(z)$, Bessel's integral for      19
Bessel coefficients $J_n(z)$, expansion in power series of      15
Bessel coefficients $J_n(z)$, generating function of      14 22 23
Bessel coefficients $J_n(z)$, inequalities satisfied by      16 31 268
Bessel coefficients $J_n(z)$, notations for      13 14
Bessel coefficients $J_n(z)$, order of      14
Bessel coefficients $J_n(z)$, order of (negative)      16
Bessel coefficients $J_n(z)$, recurrence formulae for      17
Bessel coefficients $J_n(z)$, square of      32
Bessel coefficients $J_n(z)$, tables of (of order n)      664 730—732
Bessel coefficients $J_n(z)$, tables of (of orders 0 and 1)      662 666—697
Bessel coefficients $J_n(z)$, tables of (references to)      654 655 656 658. Bessel's
Bessel coefficients $J_n(z)$, tables of (with equal order and argument)      664 746
Bessel functions      38—84 (Chapter III)
Bessel functions of the first kind, $J_\nu(z)$      38
Bessel functions of the first kind, addition theorems for      143 359 362 363 367 868
Bessel functions of the first kind, Barnes' type of integral representing      190
Bessel functions of the first kind, Bessel's type of integral representing      176
Bessel functions of the first kind, cut in plane to render uniform      46
Bessel functions of the first kind, differential equation (Bessel's) satisfied by      38
Bessel functions of the first kind, expansion of, in ascending series      40
Bessel functions of the first kind, expansion of, in descending series      see Asymptotic expansions expressed as a generalised hypergeometric function 100 101
Bessel functions of the first kind, expressed as the limit of a hypergeometric function      154
Bessel functions of the first kind, expressed as the limit of a Legendre function      156
Bessel functions of the first kind, expressed as the limit of a Legendre function (physical significance of)      155
Bessel functions of the first kind, expressed as the limit of a Lommel polynomial      302
Bessel functions of the first kind, functional properties of      45
Bessel functions of the first kind, generalisations of      43 44 308—357
Bessel functions of the first kind, inequalities satisfied by      49 255 259 270 406
Bessel functions of the first kind, infinite integrals containing      Chapter XIII passim
Bessel functions of the first kind, of complex order      46
Bessel functions of the first kind, of order $n+\frac12$      41
Bessel functions of the first kind, of order $n+\frac12$ (expressed in finite terms)      52 55
Bessel functions of the first kind, of order $n+\frac12$ (notations for)      55 80
Bessel functions of the first kind, Poisson's integral representing      47 48
Bessel functions of the first kind, Poisson's integral representing, (modifications of)      161 163 164 169 170
Bessel functions of the first kind, products of      see Products of Bessel functions
Bessel functions of the first kind, quotient of two, expressed as a continued fraction      153 154 303
Bessel functions of the first kind, recurrence formulae for      45 294
Bessel functions of the first kind, relations with Lommel's polynomial      297
Bessel functions of the first kind, represented by integrals containing Legendre functions      173 174
Bessel functions of the first kind, symbolic formulae for      170
Bessel functions of the first kind, tables of (of order $-\frac13$, method of computing)      664
Bessel functions of the first kind, tables of (of order $n+\frac12$)      664 740—745
Bessel functions of the first kind, tables of (of order $\frac13$)      664 714—729
Bessel functions of the first kind, tables of (of order n)      664 730—782
Bessel functions of the first kind, tables of (of orders 0 and 1)      662 666—697
Bessel functions of the first kind, tables of (with equal order and argument)      664 746
Bessel functions of the first kind, tables of (zeros of)      664 748—751
Bessel functions of the first kind, tables of tables of (references to)      654 655 656 658 659 660
Bessel functions of the first kind, Weierstrassian product representing      497
Bessel functions of the first kind, with large argument      see Asymptotic expansions
Bessel functions of the first kind, zeros of      see Zeros of Bessel functions
Bessel functions of the second kind, $F^n(z)$ (after Neumann)      67
Bessel functions of the second kind, $G_n(z)$ (after Heine)      65
Bessel functions of the second kind, $Y_\nu(z)$ (after Hankel)      57 63
Bessel functions of the second kind, $Y_\nu(z)$ (after Weber — Schlaefli, the canonical form)      63
Bessel functions of the second kind, addition theorems for      144 361 365 368
Bessel functions of the second kind, Bessel's type of integral representing      177
Bessel functions of the second kind, component parts of      71 72 340
Bessel functions of the second kind, continuity of (qua function of their order)      63
Bessel functions of the second kind, differential equation (Bessel's) satisfied by      59 63
Bessel functions of the second kind, expansion of, in ascending series      59 60 61 69 72
Bessel functions of the second kind, expansion of, in descending series      see Asymptotic expansions
Bessel functions of the second kind, expressed as an integral containing functions of the first kind      5 133 382 433
Bessel functions of the second kind, infinite integrals containing      385 387 393 394 424 425 426 428 429 430 433
Bessel functions of the second kind, Poisson's typo of integral representing      68 73 165
Bessel functions of the second kind, Poisson's typo of integral representing (modifications of)      169 170
Bessel functions of the second kind, products of      149
Bessel functions of the second kind, products of (asymptotic expansions of)      221 448
Bessel functions of the second kind, products of (represented by infinite integrals)      221 441 446
Bessel functions of the second kind, recurrence formulae for      66 71
Bessel functions of the second kind, represented by integrals containing Legendro functions      174
Bessel functions of the second kind, symbolic formulae for      170
Bessel functions of the second kind, tables of (of order $\frac13$)      664 714—729
Bessel functions of the second kind, tables of (of order n)      664 732—735
Bessel functions of the second kind, tables of (of orders 0 and 1)      662 666—697
Bessel functions of the second kind, tables of (ot order $-\frac13$, method of computing)      064
Bessel functions of the second kind, tables of (with equal order and argument)      664 747
Bessel functions of the second kind, tables of (zeros of)      748—751
Bessel functions of the second kind, tables of, references to      665 656 658
Bessel functions of the second kind, with large argument      see Asymptotic expansions with negative 75
Bessel functions of the second kind, zeros of      see Zeros of Bessel functions. See also Neumann's polynomial
Bessel functions of the third kind, $H_\nu^{(1)}(z)$, $H_\nu^{(2)}(z)$      73
Bessel functions of the third kind, Barnes' integrals representing      192
Bessel functions of the third kind, Bessel's type of integral representing      178
Bessel functions of the third kind, Poisson's type of integral representing      166
Bessel functions of the third kind, Poisson's type of integral representing (modifications of)      108 169 170
Bessel functions of the third kind, represented by integrals containing Legendre functions      174
Bessel functions of the third kind, symbolic formulae for      170
Bessel functions of the third kind, tables of (of order ft      604 714—729
Bessel functions of the third kind, tables of (of orders 0 and 1)      662 666—697
Bessel functions of the third kind, tables of (references to)      657
Bessel functions of the third kind, with large argument and order, asymptotic expansions of      244 245 262
Bessel functions of the third kind, with large argument, asymptotic expansions of      199 210 216
Bessel functions of the third kind, with negative argument      75
Bessel functions whose order and argument are equal, approximations to      229 281 232 259 260 448 515
Bessel functions whose order and argument are equal, asymptotic expansions of      245
Bessel functions whose order and argument are equal, integrals representing      268
Bessel functions whose order and argument are equal, tables of      658 664 746 747
Bessel functions whose order and argument are equal, tables of (references to)      658
Bessel functions whose order is $\pm(n+\frac12)$      10 62 80
Bessel functions whose order is $\pm(n+\frac12)$, expressible in finite terms      52
Bessel functions whose order is $\pm(n+\frac12)$, notations for      55 80
Bessel functions whose order is $\pm(n+\frac12)$, tables of      664 740—745
Bessel functions whose order is $\pm(n+\frac12)$, tables of (references to)      658 659
Bessel functions whose order is a fraction, of orders $\pm\frac13$ (and Airy's integral)      190
Bessel functions whose order is a fraction, of orders $\pm\frac13$ (and the stability of a vertical pole)      96
Bessel functions whose order is a fraction, of orders $\pm\frac14, \pm\frac34$, tables of (references to)      669
Bessel functions whose order is a fraction, of orders $\pm\frac23$, tables of (references to)      659
Bessel functions whose order is a fraction, of small fractional orders, tables of zeros of (references to)      502 660.
Bessel functions whose order is a fraction, tables of      664 714—729
Bessel functions whose order is a fraction, tables of (references to)      659
Bessel functions whose order is a fraction, zeros of      751
Bessel functions whose order is large      225—270 (Chapter VIII)
Bessel functions whose order is large, asymptotic expansions of      241 244 245 262
Bessel functions whose order is large, Carlini's approximation to      6 7
Bessel functions whose order is large, Carlini's approximation to (extended by Meissel)      226 227
Bessel functions whose order is large, Horn's (elementary) approximation to      225
Bessel functions whose order is large, Laplace's approximation to      7 8 9
Bessel functions whose order is large, method of stationary phase applied to      232
Bessel functions whose order is large, method of steepest descents applied to      237
Bessel functions whose order is large, miscellaneous properties of      252—261
Bessel functions whose order is large, tables of (reference to)      658
Bessel functions whose order is large, transitional formulae for      248
Bessel functions whose order is large, zeros of      513 516 617 518.
Bessel functions with imaginary argument, $I_\nu(z), K_\nu(z), \mathbf{K}_\nu(z)$      77 78
Bessel functions with imaginary argument, differential equation satisfied by      77
Bessel functions with imaginary argument, integrals representing (of Bessel's type)      181
Bessel functions with imaginary argument, integrals representing (of Poisson's'type)      79 171 172
Bessel functions with imaginary argument, integrals representing (proof of equivalence of various types)      185—188
Bessel functions with imaginary argument, monotonic property of      446
Bessel functions with imaginary argument, of order $\pm(n+\frac12)$      80
Bessel functions with imaginary argument, recurrence formulae      79
Bessel functions with imaginary argument, tables of (of order $\frac13$)      604 714—729
Bessel functions with imaginary argument, tables of (of orders 0 and 1)      603 698—713
Bessel functions with imaginary argument, tables of (of various integral orders)      064 736 737—739
Bessel functions with imaginary argument, tables of (references to)      657 668
Bessel functions with imaginary argument, with large argument, asymptotic expansions of      202
Bessel functions with imaginary argument, zeros of      611
Bessel functions with imaginary argument, zeros of (computation of)      512
Bessel functions with imaginary argument, zeros of (references to)      660
Bessel functions, (compiled by Maggi and by Wagner)      13
Bessel functions, argument of, denned      40
Bessel functions, differential equations of order higher than the second satisfied by      106
Bessel functions, expressed as limits of Lame functions      159
Bessel functions, expressed as limits of P-functions      158
Bessel functions, history of      1—13 (Chapter I)
Bessel functions, indefinite integrals containing      132—138
Bessel functions, order of, defined      38 58 63 67 70
Bessel functions, rank of, defined      129
Bessel functions, relations between the various kinds of      74
Bessel functions, representation of cylinder functions in terms of      82
Bessel functions, solutions of difference equations in terms of      83 355
Bessel functions, solutions of Laplace's equation containing      83 124
Bessel functions, solutions of the equation of wave motions containing      125
Bessel functions, three-term relations connecting      300
Bessel functions, with negative argument      75. See also the two preceding and ten following entries and
Bessel's differential equation      1 19
Bessel's differential equation (generalised)      38
Bessel's differential equation for functions of order zero      5 12 59 60
Bessel's differential equation for functions with imaginary argument      77
Bessel's differential equation, fundamental system of solutions of      42 75
Bessel, F.W.      1 3 4 5 9 13 14 15 18 19 21 24 25 38 84 140 148 153 100 295 308 478 551 554 654
Binet, J.P.M.      183
Bocher, M.      46 57 64 376 494 495 517
Boehmer, P.E.      142
Boole, G.      27 47 110 627
Borel, E.      8 281 536
Bourget, J.      6 324 325 326 484 485 517 659 660
Brajtzew, J.R.      169
Brasainne, E.      91
Brenke, W. C      23
Bridgenmn, P.W.      597
Brims, H.      327
Bromwich, T.J.I'A.      8 11 44 68 156 187 189 191 202 203 214 230 231 234 279 302 349 360 385 391 393 399 574 575 601
Bryan, G.H.      127 480
Burkhardt, H.V.K.L.      236 280
Burnside, W.S.      305
Cailler, C.      44 149 386 395 415 437 455 536 537
Callandrean, O.      190 208 387
Cantor, G.K.L.P.      037
Carlini, F.      177 194 225 226 227 249 255 268 572
Carslaw, H.S.      177 366 395 499 500 507 509 553
Catalan, E.C.      21 22 27 96 173 188
Cauchy, (Baron) A.L.      7 15 16 21 150 183 230 231 232 233 217 249 259 309 319 324 449 545 554 557 579
Cayley, A.      88 90 96 102 103 109 188 502
Challis, H.W.      91
Chapman, S.      621
Chessiit, A.S.      135 175 325 346 382
Chree, C.      6 597
Christoffel, E.B.      154
Chrystal, G.      102 288 295
Cinelli, M.      633
Clebach, K.F.A.      359 363
1 2 3 4
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