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| Watson G.N. — Treatise on the theory of Bessel functions |
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| Предметный указатель |
Bridgeman, P.W. 597
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
Bruns' function J (z; v, k) 327
Bruns, H. 327
Bryan, G.H. 127 480
Burkhardt, H.F.K.L. 236 280
Burnside, W.S. 305
Cailler, C. 44 149 386 395 415 437 455 536 537
Callandreau, O. 196 208 387
Cantor, G.F.L.P. 637
Carlini's approximation for Bessel functions of large order 6 7
Carlini's approximation, extended by Meissel 226 227
Carlini, F. 6 7 194 225 226 227 249 255 268 572
Carslaw, H.S. 177 366 395 499 500 507 509 583
Catalan, E.C. 21 22 27 96 173 188
Cauchy's numbers  324
Cauchy's numbers, recurrence formulae for 325
Cauchy, A.L. 7 15 16 21 150 183 230 231 232 233 247 249 259 309 319 324 449 545 554 557 579
Cayley's solution of Riccati's equation 88
Cayley, A. 88 90 96 102 103 109 188 502
Chain, oscillations of a uniform heavy 3 4 576
Challis, H. W. 91
Chapman, S. 621
Chessin, A.S. 135 175 325 346 382
Chree, C. 6 597
Christoffel, E.B. 154
Chrystal, G. 102 288 295
Cinelli, M. 633
Clebsch, R.F.A. 359 363
Clifford, W.K. 90 91
Coates, C.V. 173 180 313 622
Cognate Riccati equations 91
Complex variables, expansions of arbitrary functions of see “Kapteyn series” “Neumann
Complex zeros of Bessel functions 483
Complex zeros of Bessel functions with imaginary argument 511
Complex zeros of Lommel's polynomials 306
Composition of Bessel functions of the second kind of integral order 340
Computation see also “Zeros of Bessel functions”
Computation of zeros of Bessel functions by various methods (Graeffe's) 500 502
Computation of zeros of Bessel functions by various methods (Stokes') 503
Computation of zeros of Bessel functions by various methods (Sturm's, for the smallest zero) 516
Constant phase, Schlaefli's method of 216
Constants, discontinuity of arbitrary, Stokes' phenomenon 201 203 238 336
Continuants, connected with Schlaefli's polynomial 288
Continued fractions, convergence of 154 303
Continued fractions, representing quotients of Bessel functions 153
Convergent series, Hadamard's conversion of asymptotic expansions into 204
Cotter, J.R. 41
Crawford, L. 27
Crelier's integral see also “Neumann's integral for Neumann's polynomial”
Crelier's integral for Schlaefli's polynomial 288
Crelier, L. 286 287 288 295 300 301 302
Cross-ratio of solutions of Riccati's equation 94
Cube of a Bessel function, expansion of 149
Curtis, A.H. 96 110
Curzon, H.E.J. 395
Cut, necessary for definition of Bessel functions 45 77
Cylinder (circular), motion of heat in 9 10 576 577
Cylinder functions see also “Bessel functions” “Hemi-cylindrical
Cylinder functions,  4 82 480
Cylinder functions, addition theorems 143 361 365
Cylinder functions, connexion with Bessel functions 83
Cylinder functions, origin of the name 83
Cylinder functions, rank of 129
Cylinder functions, solutions of differential equations of order higher than the second by 106
Cylinder functions, three-term relations connecting 300
D'Alembert, J. le Rond 3
Dandelin, G.P. 503
Darboux' method of approximating to functions of large numbers 233
Darboux, J.G. 233
Darwin, C.G. 437
De la Vallee Poussin, Ch. J. 53 160 189
De Morgan, A. 188 190
Debye, P. 225 235 237 240 241 247 249 250 251 255 262 263 268 513 516
Definite integrals see also “Infinite integrals”
Definite integrals, containing Bessel functions under the integral sign 373—382
Definite integrals, evaluated by geometrical methods 374 376 378
Definite integrals, representing special functions see “Bessel functions” “Integrals”
Definite integrals, the Ramanujan — Hardy method of evaluation 382
Dendy, A. 107
Determinants, representing Lommel's polynomials 294
Determinants, Wronskian 42 76 77
Difference equations see also “Functional equations” “Recurrence
Difference equations, linear with linear coefficients, solved by means of Bessel functions 83
Differentiability of Fourier — Bessel expansions 605
Differentiability of special Schloemilch series 635
Differential coefficients, fractional 107 125
Differential equations see also “Laplace's equation” “Wave-motions equation
Differential equations (ordinary), linear of the second order, equivalent to the generalised Riccati equation 92
Differential equations (partial), solution of by an integral containing Bessel functions 99
Differential equations of order higher than the second solved by Bessel functions 106
Differential equations, oscillation of solutions of 518
Differential equations, satisfied by the product of two Bessel functions 145 146
Differential equations, solved by elementary transcendants 112
Differential equations, symbolic solutions of 41 108
Diffraction with Bessel's type of integral 177
Diffraction with Schloemilch series 633
Diffraction with Struve's functions 417
Diffraction, theory of, connected with Airy's integral 188
Diffusion of salts in a liquid, and infinite integrals containing Bessel functions 437
Dini expansion 580 (see also “Dini series”)
Dini series 577 580 596—605 615—617
Dini series, expansion of an arbitrary function of a real variable into 580 600
Dini series, methods of theory of functions of complex variables applied to 596 602
Dini series, Riemann — Lebesgue lemma, analogue of 599
Dini series, Riemann's theorem, analogue of 649
Dini series, summability of 601 615
Dini series, uniformity of convergence of 601 604
Dini series, uniqueness of 616 651
Dini series, value at end of range 602
Dini, U. 10 577 578 597 600 616 651
Dinnik, A. 579 659 660
Dirichlet's discontinuous factor 406
Dirichlet, P.G.L. 157 230 406 581 623
Discontinuity of arbitrary constants, Stokes' phenomenon 201 203 238 336
Discontinuous factor, Dirichlet's 406
Discontinuous factor, Weber's 405
Discontinuous integrals 398 402 406 408 411 415 421
Dixon, A.C. 35 480 481 482
Domain K (Kapteyn's) 559
Domain K, diagram of 270
Donkin, W.F. 109
Dougall, J. 65 411
Du Bois Raymond, P.D.G. 183 455 470
Du Bois Reymond's integrals with oscillatory integrands expressed in terms of Bessel functions 183
Duhamel, J.M.C. 38 49 59 68 227
Earnshaw, S. 108
Electric waves 56 226 449
Electromagnetic radiation 551 556
Elementary transcendants, definition of 111
Elementary transcendants, order of 111
Elementary transcendants, solution of differential equations by 112
Ellis, R.L. 95 109 110 173
Emde, F. 248
Encke, J.F. 342
Enestrom, G. 92
Enneper, A. 173
Epstein, S.S. 145 290
Equal order and argument, Bessel functions with 231 232 258 260
Equal order and argument, tables of 746 747
Equal order and argument, tables of, references to 658 664
Ermakoff, W. 455
Escherich, G. von 165
Euler's solution of Riccati's equation 87
Euler, L. 3 4 5 6 24 49 53 60 62 87 88 92 93 123 133 183 410 498 500 501 503 576 659
Exponential function, tables of 698—713
Exponential function, tables referred to 663 664
Factors, discontinuous, Dirichlets 406
Factors, discontinuous, Weber's 405
Factors, expression of Bessel functions as products of Weierstrassian 497
| Factors, Neumann's  (= 1 or 2) 22
Falkenhagen, J.H.M. 94
Fejer's theorem, analogue of, for Fourier — Bessel expansions 610
Fejer, L. 610
Feldblum, M. 92
Fields, J.C. 110
Filon, L.N.G. 51 578 622 623 625 629
Finite terms, Bessel functions of order ±(n+1/2) expressed in 52
Finite terms, Bessel functions of other orders not so expressible 119
Finite terms, solutions of Riccati's equation in 85 86 89
Finite terms, the solution of Riccati's equation in, not possible except in Daniel Bernoulli's cases and their limit 123
Flights, problem of random 419
Ford, W.B. 578 605
Forsyth, A.R. 42 57 107 109 117 346 358 400 499
Fourier — Bessel expansion 580 (see also “Fourier — Bessel series”)
Fourier — Bessel functions 4 84
Fourier — Bessel integrals see “Multiple infinite integrals”
Fourier — Bessel series 576—617 649—651
Fourier — Bessel series, expansion of an arbitrary function of a real variable into 576 580
Fourier — Bessel series, Fejer's theorem, analogue of 610
Fourier — Bessel series, Kneser — Sommerfeld expansion of a combination of Bessel functions into 499
Fourier — Bessel series, methods of theory of functions of complex variables applied to 582 607
Fourier — Bessel series, order of magnitude of terms in, Sheppard's theorem 595
Fourier — Bessel series, Riemann — Lebesgue lemma, analogue of 589
Fourier — Bessel series, Riemann's theorem, analogue of 649
Fourier — Bessel series, summability of 578 606 613
Fourier — Bessel series, term-by-term differentiation of 578 605
Fourier — Bessel series, uniformity of convergence of 593 594
Fourier — Bessel series, uniformity of convergence of, near origin 615
Fourier — Bessel series, uniformity of summability of 612
Fourier — Bessel series, uniqueness of 616 649
Fourier — Bessel series, value at end of range 594 603
Fourier, J.B.J. 4 9 10 13 22 84 135 449 450 454 455 456 478 482 483 501 576 577 578 616
Fractional differential coefficients 107 125
Freeman, A. 501
Frenet, F. 27
Fresnel's integrals 544
Fresnel's integrals, asymptotic expansion of 545
Fresnel's integrals, tables of 744 745
Fresnel's integrals, tables of maxima and minima of 745
Fresnel's integrals, tables of, references to 660 661 664
Fresnel, A.J. 544 545
Frobenius, F.G. 57
Frullani, G. 14 19
Functional equations, defining cylinder functions 82
Functional equations, generalised by Nielsen 355
Functions of large numbers see also “Approximations” “Asymptotic “Method “Method
Functions of large numbers, approximations due to Darboux 233
Functions of large numbers, approximations due to Laplace 8
Fundamental system of solutions of Bessel's differential equation 42 75 78
Gallop's discontinuous infinite integrals 421
Gallop, E.G. 405 421 422
Gamma functions, applications to determination of asymptotic expansions 220 223 352
Gamma functions, applications to evaluation of infinite integrals 383 434 436
Gamma functions, representation of Bessel functions by integrals containing 190 192 221
Gamma functions, representation of Lommel's functions by integrals containing 351
Gaskin, T. 109
Gasser, A. 509 517
Gauss, C.F. 191 506
Gegenbauer's addition theorem for Bessel functions 362 363 367
Gegenbauer's discontinuous infinite integrals 415 418
Gegenbauer's function  50 129 363 365 367 368 369 378 407
Gegenbauer's polynomial  283
Gegenbauer's polynomial  293 525
Gegenbauer's polynomial, contour integrals containing 284 524
Gegenbauer's polynomial, differential equation satisfied by 283
Gegenbauer's polynomial, equivalence with special forms of Lommel's function 351
Gegenbauer's polynomial, recurrence formulae for 283
Gegenbauer's representation of  by a double integral resembling Poisson's integral 51
Gegenbauer's type of definite integral 378
Gegenbauer, L. von 50 51 129 138 151 274 283 284 290 293 351 362 363 365 366 367 368 369 370 373 378 379 383 384 385 386 389 390 391 393 395 396 398 406 407 413 414 415 418 426 430 438 439 480 508 517 522 524 525 579
Generalised hypergeometric functions see “Hypergeometric functions (generalised)”
Generalised integrals, with implied exponential factor 188 441 463 464
Generating function of Bessel coefficients 14 22 23
Generating function of Neumann's polynomials 281 282
Genocchi, A. 119
Gibson, G.A. 197
Gifford, E. 662
Gilbert's integrals 548 549
Gilbert, L.P. 545 548 549
Giuliani's function see “Bourget's function”
Giuliani, G. 155 156 324 326 327
Glaisher, J.W.L. 89 96 102 103 108 109 140 171 173 183 664
Gordan, P.A. 55
Goursat, E.J.B. 120
Graeffe, C.H. 502 503
Graeife's method of calculating zeros 500 502
Graf's addition theorem for Bessel functions 359 361
Graf, J.H. 32 64 75 145 153 160 165 175 197 215 227 286 287 290 295 296 299 301 302 303 341 344 345 359 360 362 398 498 502 583
Gray, A. 64 65 78 194 206 454 480 655 657 658 660
Green, G. 124
Greenhill, A. George 91 96
Gregory, Duncan Farquharson 391
Gregory, Walter 224
Group velocity 229
Growth of zeros of Bessel functions 485
Grunert, J.A. 27
Gubler, E. 32 64 145 160 165 177 197 215 227 286 287 301 329 341 351 398 408 410 426 498 502 583
Gunther, S. 153
Gwyther, R.F. 621 636
Hadamard, J. 204 205 527
Haentzschel, E. 71 96 159
Hafen, M. 389
Hague, B. 656
Hall, A. 15
Hamilton, William Rowan 12 195 655
Hankel's infinite integrals 384 386 389 390 393 395 424 427 428 434
Hankel, H. 10 38 57 58 61 62 63 65 73 75 76 77 160 163 164 165 167 175 195 196 203 206 208 211 384 386 390 393 395 424 427 428 429 430 434 453 454 456 457 458 459 462' 464 465 471 488 513 514 577 579 581 582 633
Hansen's upper bound for  31
Hansen's upper bound, generalised 406
Hansen, P.A. 14 20 30 31 37 152 154 155 158 195 292 406 655 656
Hanumanta Rao, C.V. 437
Hardy's functions  ,  ,  , generalisations of Airy's integral 320
Hardy's functions, expressed in terms of functions of Bessel, Anger and Weber 321 322
Hardy's integrals, representing Lommel's functions of two variables 546
Hardy's method of evaluating definite integrals 382
Hardy, G.H. 8 111 180 183 188 189 200 309 320 321 322 324 373 382 386 395 406 421 422 437 441 442 463 464 542 546 547 573 575 579 606 615 621
Hargreave, C.J. 88 170 171
Hargreaves, R. 538
Harnack, A. 577
Harris, J.A. 15
Havelock, T.H. 125 171
Hayashi, T. 165
Heat, conduction of 9 10 450 576 577 616
Heaviside, O. 64 65 203 367 385 387 388 393 395 410 426
Heine, H.E. 4 56 65 66 84 154 155 156 157 181 358 363 365
Hemi-cylindrical functions  , addition theorem for 354
Hemi-cylindrical functions , defined 353
Hemi-cylindrical functions , expressed in terms of the function of order zero 353
Hermite, C. 55 477
Hertz, H. 80 81
Herz, N. 554 555
Hill, C.J.D. 94
Hobson, E.W. 10 33 54 58 125 128 129 149 172 174 280 353 363 369 385 386 387 480 485 578 586 591 602
Hopf, L. 178 406
Horn, J. 225 526
Hurwitz, A. 9 297 302 303 304 305 306 307 483 484
Hypergeometric functions (generalised) 90 100
Hypergeometric functions (generalised), Bessel functions expressed in terms of 100 101
Hypergeometric functions (generalised), notations for 100
Hypergeometric functions (generalised), relations between (Kummer's formulae) 101
Hypergeometric functions (generalised), Sharpe's differential equation solved by 105
Hypergeometric functions, limiting forms expressed as Bessel functions 154
I'Hospital, G.F.A. 134
Ignatowsky, W. von 365
Imaginary argument, Struve's functions with 329 332
Indefinite integrals, containing Bessel functions under the integral sign 132—138 350 581
Indefinite integrals, tables of 744 745 752
Indefinite integrals, tables of, references to 660 661 664
Inequalities, by Neumann's polynomial 273 282
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