A treatise on the theory of Bessel functions, by G. N. Watson.

GENERAL INDEX 799
Dini series, 577, 580, 596-605, 615-617 (Chapter xvIII), 651-653; expansion of an arbitrary function of a real variable into, 580, 600; methods of theory of functions of complex variables applied
to, 596, 602; Riemann-Lebesgue lemma, analogue of, 599; Riemann's theorem, analogue of,
649; summability of, 601, 615; uniformity of convergence of, 601, 604; uniqueness of, 616, 651;
value at end of range, 602
Dirichlet's discontinuous factor, 406
Discontinuity of arbitrary constants (Stokes' phenomenon), 201, 203, 238, 336
Discontinuous factor (Dirichlet's), 406; (Weber's), 405
Discontinuous integrals, 398, 402, 406, 408, 411, 415, 421
Domain K (Kapteyn's), 559; diagram of, 270
Du Bois Reymond's integrals with oscillatory integrands expressed in terms of Bessel functions, 183
Electric waves, 56, 226, 449
Electromagnetic radiation, 551, 556
Elementary transcendants, definition of, 111; order of, 111; solution of differential equations by,
112
Equal order and argument, Bessel functions with, 231, 232, 258, 260; tables of, 746, 747; tables
of (references to), 658, 664
Euler's solution of Riccati's equation, 87
Exponential function, tables of, 698-713; tables referred to, 663, 664
Factors, discontinuous (Dirichlet's), 406; (Weber's), 405; Neumann's e, (=1 or 2), 22; expression
of Bessel functions as products of Weierstrassian, 497
Fejer's theorem, analogue of, for Fourier-Bessel expansions, 610
Finite terms, Bessel functions of order - (n + -) expressed in, 52; Bessel functions of other orders
not so expressible, 119; solutions of Riccati's equation in, 85, 86, 89; the solution of Riccati's
equation in, not possible except in Daniel Bernoulli's cases and their limit, 123
Flights, problem of random, 419
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 (Chapter xvIII), 649-651; expansion of an arbitrary function of a
real variable into, 576, 580; Fejer's theorem, analogue of, 610; Kneser-Sommerfeld expansion
of a combination of Bessel functions into, 499; methods of theory of functions of complex variables applied to, 582, 607; order of magnitude of terms in (Sheppard's theorem), 595; RiemannLebesgue lemma, analogue of, 589; Riemann's theorem, analogue of, 649; summability of, 578,
606, 613; term-by-term differentiation of, 578, 605; uniformity of convergence of, 593, 594;
(near origin), 615; uniformity of summability of, 612; uniqueness of, 616,-649; value at end of
range, 594, 603
Fractional differential coefficients, 107, 125
Fresnel's integrals, 544; asymptotic expansion of, 545; tables of, 744, 745; tables of maxima
and minima of, 745; tables of (references to), 660, 661, 664
Functional equations defining cylinder functions, 82; generalised by Nielsen, 355
Functions of large numbers, approximations due to Darboux, 233; approximations due to Laplace,
8,421. See also Approximations, Asymptotic expansions, Method of stationary phase and Method
of steepest descents
Fundamental system of solutions of Bessel's differential equation, 42, 75, 78
Gallop's discontinuous infinite integrals, 421
Gamma functions, representation of Bessel functions by integrals containing, 190, 192, 221; applications to determination of asymptotic expansions, 220, 223; applications to evaluation of infinite
integrals, 383, 434, 436
Gamma functions, representation of Lommel's functions by integrals containing, 351; applications
to determination of asymptotic expansions, 352
Gegenbauer's addition theorem for Bessel functions, 362, 363, 367
Gegenbauer's discontinuous infinite integrals, 415, 418
Gegenbauer's function CY (z), 50, 129, 363, 365, 367, 368, 369, 378, 407
Gegenbauer's polynomial An, v(t), 283; contour integrals containing, 284, 524; differential equation satisfied by, 283; equivalence with special forms of Lommel's function, 351; recurrence
formulae for, 283
Gegenbauer's polynomial Bn, a, v (t), 293, 525
Gegenbauer's representation of Jv (z) by a double integral resembling Poisson's integral, 51
Gegenbauer's type of definite integral, 378
Generalised hypergeometric functions, see Hypergeometric functions (generalised)