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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

Schlaefli's polynomial, expression by means of Bessel functions      287
Schlaefli's polynomial, expression in terms of Lommel's function      350
Schlaefli's polynomial, integrals evaluated in terms of      350
Schlaefli's polynomial, recurrence formulae for      285
Schlaefli's solution of Riccatti's equation      90
Schlaefli, L.      10 14 27 28 30 32 33 63 64 65 67 72 79 90 91 143 145 151 160 171 174 175 176 178 179 181 185 195 196 215 216 228 253 274 276 278 284 285 286 288 289 290 341 342 344 345 508 577 579 581 582 583 585
Schloemilch series      618—649 (Chapter XIX)
Schloemilch series, definition of      621
Schloemilch series, definition of generalised      623
Schloemilch series, expansion of au arbitrary function of a real variable into      619 623 629
Schloemilch series, nature of convergence of      637 645
Schloemilch series, null-functions expressed by      634
Schloemilch series, Riemann's theorem on trigonometrical series (analogue of)      642 647
Schloemilch series, special cases of      632
Schloemilch series, symbolic operators in the theory of      626 627
Schloemilch series, theory of functions of complex variables connected with      623
Schloemilch series, uniqueness of      643 647
Schloemilch, O.X.      14 18 33 34 35 36 153 173 183 617 618 619 621 622 628 655 656
Schonholzer, J.J.      145
Schott, G.A.      551 556 572 573
Schwarz, K.H.A.      358 643
Schwarzschild, K.      361
Schwerd, F.M.      477 654
Scrret, J.A.      171 173 188
Segar, H.W.      483
Series containing Bessel functions      see Dini series Fourier-BeBsel Kapteyn Neumann
Series of Bessel functions, definition of      580
Series of positive terms, approximation to the sum of (greatest term method)      8
Sharpe's differential equation      105
Sharpe's solution by generalised hypergeometric functions      105
Sharpe, H.J.      105 157
Sheppard, W.F.      199 454 579 595 615
Siacoi, F.      92
Siegel, C.L.      485
Siemon, P.      328 398
Sign of remainders in asymptotic expansions      206 207 209 215 315 333 449
Sign of Struve's function      337 417
Sine-integral expressed as a series of squares of Bessel coefficients      152
Singularities of functions defined by Neumann serios (Pincherle's theorem)      526
Singularities of the generalised Riccati equation      94
Smallest zeros of Bessel functions      5 500 516
Smith, Bernard A.      655
Smith, Clara E.      621
Smith, Otto Andreas      50
Sommerfeld's expansion      see Kneser — Sommerfeld expansion
Sommerfeld, A.J.W.      56 57 178 361 389 395 406 417 464 499
Sonine — Mehler integrals representing Bessel functions      100 170
Sonine's definite integral      373
Sonine's definite integral, generalised      382
Sonine's discontinuous infinite integrals      115
Sonine's infinite integrals      432
Sonine, N.J.      82 83 132 137 139 143 169 170 171 175 176 177 180 279 280 281 290 353 354 362 363 367 373 374 375 376 377 378 383 386 391 394 395 398 401 411 115 417 418 431 432 433 434 439 454
Sound, Sharpe's differential equation in the theory of      105
Spherical geometry used to express Bessel functions as limits of Legendre functions      155
Spherical geometry used to obtain transformations of integrals      51 374 376 378
Spitzer, S.      68 71 153
Squares of Bessel functions      see Products of Bessel functions
Stability of a vertical pole associated with Bessel functions of order one-third      96
Stationary phase, applied to Bessel functions      231 238
Stationary phase, method of      225 229
Stearn, H.      482 621
Steepest descents, applied to Bessel functions      237 241 244 245 262
Steepest descents, applied to functions of Anger and Weber      310
Steepest descents, applied to Struve's function      333
Steepest descents, connexion with Laplace's method of approximation      421
Steepest descents, method of      235
Steiner, L.      655 661
Steinthal, A.K.      171 387
Stephenson, A.      579
Stern, M. A,      500
Stieltjes, T.J.      195 196 207 208 209 213 214 464
STIRLING, JAMES      7 8 214
Stokes' method of computing zeros of Bessel functions and cylinder functions      503 505 507
Stokes' phenomenon of the discontinuity of arbitrary constants      201 203 238 336
Stokes, Sir George Gabriel      8 12 53 55 68 69 70 80 95 97 188 189 190 201 202 225 229 238 320 324 336 391 405 503 505 507 605 659
Strutt, J.W.      see Rayleigh
Struve's function $H_\nu(z)$       328
Struve's function $H_\nu(z)$ of order $\pm(n+\frac12)$       333
Struve's function $H_\nu(z)$ , connexion with Weber's function      336
Struve's function $H_\nu(z)$ , differential equation satisfied by      329
Struve's function $H_\nu(z)$ , inequalities connected with      328
Struve's function $H_\nu(z)$ , infinite integrals containing      392 397 417 425 436
Struve's function $H_\nu(z)$ , integral representations of      328 330
Struve's function $H_\nu(z)$ , occurrence in generalised Schloemilch series      622 623 631 645 646 647
Struve's function $H_\nu(z)$ , recurrence, formulae for      829
Struve's function $H_\nu(z)$ , sign of      337 417
Struve's function $H_\nu(z)$ , tables of      668 666
Struve's function $H_\nu(z)$ , Theisinger's integral for      338
Struve's function $H_\nu(z)$ , with imaginary argument      329 832
Struve's function $H_\nu(z)$ , with large argument and order, asymptotic expansions of      333
Struve's function $H_\nu(z)$ , with large argument, asymptotic expansions of      332
Struve's function $H_\nu(z)$ , zeros of      479
Struve's infinite integrals      390 397 421
Struve, U.      328 329 333 337 392 396 397 417 661
Sturm's methods applied to determine the reality of zeros of Bessel functions      483
Sturm's methods applied to determine the reality of zeros of Lommel's polynomials      30—1 305 306
Sturm's methods applied to estimate the value of the smallest zero of Bessel functions and cylinder functions      517 518
Sturm, J.C.F.      304 477 479 517 518 521
Suchar, P.J.      90
Svanberg, A.F.      173
Symbolic operators in expressions representing Bessel functions      50 170
Symbolic operators in expressions representing solutions of various differential equations      41 51 108
Symbolic operators in the theory of Schloemilch series      627
Tables (references to) of Airy's integral      659
Tables of       663 698—713
Tables of Bessel coefficients (of order n)      664 730—732
Tables of Bessel coefficients (of orders 0 and 1)      662 666—697
Tables of Bessel coefficients (with equal order and argument)      664 746
Tables of Bessel coefficients and functions derivable from them      654 655 656 658
Tables of Bessel functions (of orders $n+\frac12$ , $-n-\frac12$ )      658 659
Tables of Bessel functions (of orders $\pm\frac13$ , $\pm\frac23$ )      659
Tables of Bessel functions (of orders $\pm\frac14$ , $\pm\frac34$ )      659
Tables of Bessel functions of the first kind (of order $n+\frac12$ , $-n-\frac12$ )      664 740—741
Tables of Bessel functions of the first kind (of order $\frac13$ )      664 714—729
Tables of Bessel functions of the second kind      655 656 658
Tables of Bessel functions of the second kind (of order $\frac13$ )      664 714—729
Tables of Bessel functions of the second kind (of order n)      664 732^735
Tables of Bessel functions of the second kind (of orders 0 and 1)      662 666—697

Tables of Bessel functions of the second kind (with equal order and argument)      664 747
Tables of Bessel functions of the third kind      657
Tables of Bessel functions of the third kind (of order n)      664 714—729
Tables of Bessel functions of the third kind (of orders 0 and 1)      662 666—697
Tables of Bessel functions with imaginary argument      657 658
Tables of Bessel functions with imaginary argument (of order $\frac13$ )      664 714—729
Tables of Bessel functions with imaginary argument (of order 0 and 1)      663 698—713
Tables of Bessel functions with imaginary argument (of order n)      664 736 737—739
Tables of Fresnel's integrals      664 744—740
Tables of integrals of Bessel functions and Struve's functions      661
Tables of integrals of Bessel functions of order zero      664 702
Tables of Lommel's functions of two variables      660
Tables of Struve's functions (of orders 0 and 1)      663 666—697
Tables of Thomson's functions ber(x) and bei(x), etc.      658
Tables of zeros of Bessel confidents and functions of integral order n and of order $\frac13$       664 748—751
Tables ot zeros of Bessel coefficients, functions and associated functions      659 660
Takeuchi, T.      313
Tannery, J.      11 156 302
Theisinger's integral representation of Bessel functions      184
Theisinger's integral representation of Struve's and Weber's functions      338
Theisinger, L.      184 185 338
Thomson's (Sir William) functions, ber(z), bei(z)      81
Thomson's (Sir William) functions, connexion with Bessel functions      81
Thomson's (Sir William) functions, generalisations      81
Thomson's (Sir William) functions, references to tables of      658
Thomson's (Sir William) functions, squares and products of      82 148
Thomson's (Sir William) functions, with large argument, asymptotic expansions of      203
Thomson, Sir Joseph John      65 173
Thomson, Sir William      see Kelvin
Tisserand, F.      371
Tnree-term relations connecting Bessel functions, cylinder functions and Lommel's polynomials      300 301
Todhunter, I.      27 157 199
Transcendants, elementary, definition of      111
Transcendants, order of      111
Transcendants, solutions of differential equations by      112
Transitional regions associated with Bessel functions of large order      248
Turriere, K      15
Unferdinger, F.      310
Uniformity of convergence of Dini series      601
Uniformity of Fourier — Bessel series      598 594
Uniformity of Kupteyn series      575
Uniformity of Schloemilch series      632
Uniqueness of Fourier — Bessel and Dini series      616 649 651
Uniqueness of Schloemilch serios      643 647
Upper bounds      see Inequalities
Valewink, G.C.A.      196
Van Vleck, E.B.      480
Vandermondc, A.      102
Verdet, E.      477
Vessiot, E.      94
Viscous fluid, motion of, associated with Airy's integral      189
Volterra, V.      579 621
Voronoi, G.      200
Voss, A.      406
Wagner, C      13 142
Walker, Gilbert Thomas      360 361
Walker, James      328 331 333 537 544
Wallenburg, G.      94
Waring, E.      503
Watson, G.N.      11 105 125 158 226 231 249 268 355 444 483 485 513 519 566 575
Wave-motions, equation of, general solutions      125
Wave-motions, equation of, generalised to p dimensions      12K
Wave-motions, equation of, used to obtain addition theorems for Bessel functions      129
Waves on water, and the method of stationary phase      229
Waves, electric      56 226 446
Webb, H.A.      351 523 533 536
Weber's (H. F.) function $E_\nu(z)$       308
Weber's (H. F.) function $E_\nu(z)$ , connexion with Anger's function      310
Weber's (H. F.) function $E_\nu(z)$ , connexion with Struve's function      336
Weber's (H. F.) function $E_\nu(z)$ , differential equation satisfied by      312
Weber's (H. F.) function $E_\nu(z)$ , integrals expressed in terms of      312
Weber's (H. F.) function $E_\nu(z)$ , recurrence formulae for      311
Weber's (H. F.) function $E_\nu(z)$ , representation of Airy's integral (generalised) by      321
Weber's (H. F.) function $E_\nu(z)$ , tables of      see Struve's function
Weber's (H. F.) function $E_\nu(z)$ , Theisinger's integral for      338
Weber's (H. F.) function $E_\nu(z)$ , with large argument and order, asymptotic expansion of      310
Weber's (H. F.) function $E_\nu(z)$ , with large argument, asymptotic expansion or      313
Weber's (H.) discontinuous factor      405
Weber's (H.) infinite integrals      391 393 395 396
Weber's (H.) infinite integrals (discontinuous typos of)      398 402 405 406 408 411
Weber, Heinrich      63 64 67 75 165 167 195 196 210 211 212 386 391 392 393 394 395 396 398 402 405 406 408 421 450 451 452 453 454 455 408 469 470 495
Weber, Heinrich Friedrich      308 309 310 311 312 3L5 320
Weierstrass, C.T.W.      358
Weierstrassian products, expression for Bessel functions as      497
Wendt, Caecilie      363
Weyl, H.      189 454
Weyr, IS.      93 94
Whewell, W.      479
Whipple's transformation of Legendre functions      387
Whipple, F.J.W.      177 313 387
Whitehead, C.S.      81 82 132 148 203
Whittaker's function $W-\nu(z)$       339
Whittaker's function $W-\nu(z)$ , differential equation satisfied by      339
Whittaker's function $W-\nu(z)$ , recurrence formulae for      339
Whittaker's function $W-\nu(z)$ , with large argument, asymptotic expansion of      340
Whittaker, E.T.      44 50 124 126 173 197 339 503 632 633
Wigert, C.S.      200
Williamson      13. 110
Willson, R.W.      501 660 664
Wilton, J.R.      154
Wirtinger, W.      189 190
Wronskian determinant      42 76 77
Young, W.H.      10 351 406 578 579 580 582 586 596 616 617
Zeros of Bessel functions      477—521 (Chapter XV)
Zeros of Bessel functions, computation of (various methods of)      142 500 502 503 516
Zeros of Bessel functions, inequalities connected with, limits of, rates of growth of      485 489 490 491 494 507 513 516 518
Zeros of Bessel functions, infinity of      4 478
Zeros of Bessel functions, interlacing of      479 480 481
Zeros of Bessel functions, non-coincidence of (Bourget's hypothesis)      484
Zeros of Bessel functions, non-repetition of      479
Zeros of Bessel functions, number of, in a strip of arbitrary width      495
Zeros of Bessel functions, reality of      482 483
Zeros of Bessel functions, tables of      664 748—751
Zeros of Bessel functions, tables of (references to)      659
Zeros of Bessel functions, values of      4 5 512 516
Zeros of Bessel functions, with imaginary argument      511
Zeros of Bessel functions, with unrestrictedly large order      513 516
Zeros of Lommel's polynomials (reality of)      304 305 306
Zeros of Struve's function      479

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