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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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: Treatise on the theory of Bessel functions

: Watson G.N.

: en

: // /

:

ed2k: ed2k stats

: 1922

: 812

: 06.11.2004

: | | ID
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Abel, N.H.      68 616 621
Adamoff, A.      196
Addition theorems      358372
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 $T_n (z)$      344
Addition theorems for Schlaefli's polynomial      289
Addition theorems, integrals derived from      367
Addition theorems, physical significance of      128 130 361 363 366
Addition theorems, special and degenerate forms of      366 368
Aichi, K.      80
Airey, J.R.      65 142 214 247 319 502 505 516 659 660
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, 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      see also Asymptotic expansions Method Method
Approximations to Bessel coefficients of order zero with large argument      10 12
Approximations to Bessel functions of large order (Carlini)      6 7;
Approximations to Bessel functions of large order (extensions due to Meissel)      226 227 232 247 521
Approximations to Bessel functions of large order in transitional regions      248
Approximations to functions of large numbers (Darboux)      233
Approximations to functions of large numbers (Laplace)      421
Approximations to Legendre functions of large degree      65 155 157 158
Approximations to remainders in asymptotic expansions      213
Approximations to the sum of a series of positive terms      8
Arbitrary functions, expansions of      see Neumann series Kapteyn Dini Fourier Neumann Schloemilch
Argument of a Bessel function defined      40
Asymptotic expansions      see also Approximations
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      194224
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      196
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 with imaginary argument      202
Asymptotic expansions for Bessel functions with order and argument both large      225270
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
Bachmann, P.      197
Baehr, G.F.W.      479
Ball, L. de      157
Barnes, E.W.      83 100 102 104 105 156 190 192 195 196 220 221 340 351 357 367 383 387 402 409
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
Bateman, 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 Riccati's equation      85 89 123
Bernoulli, Daniel      2 3 4 9 85 86 87 88 111 123 478 576
Bernoulli, James      1 2 3 88 90 92
Bernoulli, John      1 2 3
Bernoulli, Nicholas      2
Bernoullian polynomials, associated with Poisson's integral      49
Bessel coefficient of order zero      see also Bessel coefficients Bessel Bessel's
Bessel coefficient of order zero with large argument, asymptotic expansion of      10 12 194
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, zeros of      4 5
Bessel coefficients      see also Bessel coefficient of order zero Bessel's Bessel
Bessel coefficients, $J_n (z)$      5 6 13 1437
Bessel coefficients, addition theorem for      29
Bessel coefficients, Bessel's integral for      19
Bessel coefficients, expansion in power series of      15
Bessel coefficients, generating function of      14 22 23
Bessel coefficients, inequalities satisfied by      16 31 268
Bessel coefficients, notations for      13 14
Bessel coefficients, order of      14
Bessel coefficients, order of, negative      16
Bessel coefficients, recurrence formulae for      17
Bessel coefficients, square of      32
Bessel coefficients, tables of, of order n      664 730732
Bessel coefficients, tables of, of orders 0 and 1      662 666697
Bessel coefficients, tables of, references to      654 655 656 658
Bessel coefficients, tables of, with equal order and argument      664 746
Bessel functions      3884 (see also Cylinder functions)
Bessel functions of the first kind with large argument      see Asymptotic expansions
Bessel functions of the first kind, $J_{\nu} (z)$      38
Bessel functions of the first kind, addition theorems for      143 359 362 363 367 368
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      45
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
Bessel functions of the first kind, 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 308357
Bessel functions of the first kind, inequalities satisfied by      49 255 259 270 406
Bessel functions of the first kind, of complex order      46
Bessel functions of the first kind, of order n+1/2      41
Bessel functions of the first kind, of order n+1/2, expressed in finite terms      52 55
Bessel functions of the first kind, of order n+1/2, 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 -1/3, method of computing      664
Bessel functions of the first kind, tables of, of order 1/3      664 714729
Bessel functions of the first kind, tables of, of order n      664 730732
Bessel functions of the first kind, tables of, of order n+1/2      664 740745
Bessel functions of the first kind, tables of, of orders 0 and 1      662 666697
Bessel functions of the first kind, tables of, references to      654 655 656 658 659 660;
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 748751
Bessel functions of the first kind, Weierstrassian product representing      497
Bessel functions of the second kind      see also Neumann's polynomial
Bessel functions of the second kind with large argument      see Asymptotic expansions
Bessel functions of the second kind with negative argument      75
Bessel functions of the second kind, $G_n (z)$, after Heine      65
Bessel functions of the second kind, $Y^{(n)} (z)$, after Neumann      67
Bessel functions of the second kind, $Y_v (z)$, after Weber Schlaefli, the canonical form      63
Bessel functions of the second kind, $Y_{\nu} (z)$, after Hankel      57 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 type of integral representing      68 73 165
Bessel functions of the second kind, Poisson's type 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 Legendre functions      174
Bessel functions of the second kind, symbolic formulae for      170
Bessel functions of the second kind, tables of, of order -1/3, method of computing      664
Bessel functions of the second kind, tables of, of order 1/3      664 714729
Bessel functions of the second kind, tables of, of order n      664 732735
Bessel functions of the second kind, tables of, of orders 0 and 1      662 666697
Bessel functions of the second kind, tables of, references to      655 656 658
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      748751; see Zeros of Bessel functions
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      199 210 215
Bessel functions of the third kind with negative argument      75
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      168 169 170
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 1/3      664 714729
Bessel functions of the third kind, tables of, of orders 0 and 1      662 666697
Bessel functions of the third kind, tables of, references to      657
Bessel functions with imaginary argument of order (n+1/2)      80
Bessel functions with imaginary argument, $I_{\nu} (z)$, $K_{\nu} (z)$      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      185188
Bessel functions with imaginary argument, monotonic property of      446
Bessel functions with imaginary argument, recurrence formulae      79
Bessel functions with imaginary argument, tables of, of order 1/3      664 714729
Bessel functions with imaginary argument, tables of, of orders 0 and 1      663 698713
Bessel functions with imaginary argument, tables of, of various integral orders      664 736 737739
Bessel functions with imaginary argument, tables of, references to      657 658
Bessel functions with imaginary argument, with large argument, asymptotic expansions of      202
Bessel functions with imaginary argument, zeros of      511
Bessel functions with imaginary argument, zeros of, computation of      512
Bessel functions with imaginary argument, zeros of, references to      660
Bessel functions with negative argument      75
Bessel functions, argument of, defined      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      113
Bessel functions, history of, compiled by Maggi and by Wagner      13
Bessel functions, indefinite integrals containing      132138
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, whose order and argument are equal, approximations to      229 231 232 259 260 448 515
Bessel functions, whose order and argument are equal, asymptotic expansions of      245
Bessel functions, whose order and argument are equal, integrals representing      258
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 a fraction      see also Bessel functions whose
Bessel functions, whose order is a fraction, of orders 1/3, and Airy's integral      190
Bessel functions, whose order is a fraction, of orders 1/3, and the stability of a vertical pole      96
Bessel functions, whose order is a fraction, of orders 1/4, 3/4, tables of, references to      659
Bessel functions, whose order is a fraction, of orders 2/3, 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 714729
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      225270 see also (Bessel functions whose
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      252261
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      513 516 517 518
Bessel functions, whose order is (n+1/2)      10 52 80
Bessel functions, whose order is (n+1/2), expressible in finite terms      52
Bessel functions, whose order is (n+1/2), notations for      55 80
Bessel functions, whose order is (n+1/2), table's of, references to      658 659
Bessel functions, whose order is (n+1/2), tables of      664 740745
Bessel's differential equation      1 19 Bessel
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's differential equation, generalised      38
Bessel's differential equation, has no algebraic integral      117
Bessel's differential equation, soluble in finite terms when and only when the functions satisfying it are of order n+1/2      52 119
Bessel's differential equation, solution of, in ascending series      39 40 57 5961
Bessel's differential equation, solution of, in descending series      see Asymptotic expansions
Bessel's differential equation, symbolic solution of      41
Bessel's differential equation, transformations of      94 97
Bessel's integral      see also Parseval's integral
Bessel's integral, generalisations and extensions of      see Anger's function Bourget's Bruns' Weber's
Bessel's integral, modifications of, to represent Bessel functions of arbitrary order      175 176 177 178 181
Bessel's integral, representing Bessel coefficients      19 21
Bessel's integral, Theisinger's transformation of      184
Bessel's integral, used in theory of diffraction      177
Bessel's integral, used to obtain asymptotic expansions      215
Bessel, F.W.      1 3 4 5 9 13 14 15 18 19 21 24 25 38 84 140 148 153 160 295 308 478 551 554 654
Binet, J.P.M.      183
Boecher, M.      46 57 64 376 494 495 517
Boehmer, P.E.      142
Boole, G.      27 47 110 627
Borel, E.      8 281 536
Bounds, upper      see Inequalities
Bourget's function $J_{n,k} (z)$      326
Bourget's function, differential equation satisfied by      327
Bourget's function, recurrence formulae for      326
Bourget, J.      6 324 325 326 484 485 517 659 660
Brajtzew, J.B.      169
Brassinne, E.      91
Brenke, W.C.      23
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