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| Watson G.N. — Treatise on the theory of Bessel functions |
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Abel, N.H. 68 616 621
Adamoff, A. 196
Addition theorems 358—372
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  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,  308
Anger's function, , connexion with Weber's function 310
Anger's function, , differential equation satisfied by 312
Anger's function, , integrals expressed in terms of 312
Anger's function, , recurrence formulae for 311
Anger's function, , representation of Airy's integral (generalised) by 321
Anger's function, , with large argument and order, asymptotic expansion of 316
Anger's function, , 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 194—224
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 225—270
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,  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,  5 6 13 14—37
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 664 730—732
Bessel coefficients, tables of, of orders 0 and 1 662 666—697
Bessel coefficients, tables of, references to 654 655 656 658
Bessel coefficients, tables of, with equal order and argument 664 746
Bessel functions 38—84 (see also “Cylinder functions”)
Bessel functions of the first kind with large argument see “Asymptotic expansions”
Bessel functions of the first kind, 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 44 308—357
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 1/3 664 714—729
Bessel functions of the first kind, tables of, of order n 664 730—732
Bessel functions of the first kind, tables of, of order n+1/2 664 740—745
Bessel functions of the first kind, tables of, of orders 0 and 1 662 666—697
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 664 748—751
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,  , after Heine 65
Bessel functions of the second kind,  , after Neumann 67
Bessel functions of the second kind,  , after Weber — Schlaefli, the canonical form 63
Bessel functions of the second kind,  , 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 1/3 664 714—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, 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 748—751; 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,  ,  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 1/3 664 714—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 with imaginary argument of order ±(n+1/2) 80
Bessel functions with imaginary argument,  ,  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, recurrence formulae 79
Bessel functions with imaginary argument, tables of, of order 1/3 664 714—729
Bessel functions with imaginary argument, tables of, of orders 0 and 1 663 698—713
Bessel functions with imaginary argument, tables of, of various integral orders 664 736 737—739
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 1—13
Bessel functions, history of, compiled by Maggi and by Wagner 13
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, 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 664 714—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 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 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 513 516 517 518
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 664 740—745
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 59—61
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  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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