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

804

THEORY OF BESSEL FUNCTIONS

Tables of Bessel coefficients (of orders 0 and 1), 662, 666-697; (of order n), 664, 730-732; (with
equal order and argument), 664, 746; of Bessel functions of the first kind (of orders n + -, - n - ),
664, 740-741; (of order ~), 664, 714-729; of Bessel functions of the second kind (of orders 0 and
1), 662, 666-697; (of order n), 664, 732-735; (of order -), 664, 714-729; (with equal order and
argument), 664, 747; of Bessel functions of the third kind (of orders 0 and 1), 662, 666-697; (of
order ~), 664, 714-729; of Bessel functions with imaginary argument (of orders 0 and 1), 663,
698-713; (of order n,), 664,736,737-739; (of order ~), 664, 714-729; of ex, 663, 698-713; of Fresnel's
integrals, 664, 744-745; of integrals of Bessel functions of order zero, 664, 752; of Struve's functions (of orders 0 and 1), 663, 666-697; of zeros of Bessel coefficients and functions of integral
order n and of order 1, 664, 748-751
Tables (references to) of Airy's integral, 659; of Bessel coefficients and functions derivable from
them, 654, 655, 656, 658; of Bessel functions (of orders ni+ I, - n - ~), 658, 659; (of orders A-,
|-A), 659; (of orders a:, - +), 659; of Bessel functions of the second kind, 655, 656, 658; of
Bessel functions of the third kind, 657; of Bessel functions with imaginary argument, 657, 658; of
Fresnel's integrals, 661; of integrals of Bessel functions and Struve's functions, 661; of Lommel's
functions of two variables, 660; of Thomson's functions ber x and bei x, etc., 658; of zeros of
Bessel coefficients, functions and associated functions, 659, 660
Theisinger's integral representation of Bessel functions, 184; of Struve's and Weber's functions,
338
Thomson's (Sir William) functions, ber z, beiz, 81; connexion with Bessel functions, 81; generalisations, 81; references to tables of, 658; squares and products of, 82, 148; with large argument,
asymptotic expansions of, 203
Three-term relations connecting Bessel functions, cylinder functions and Lommel's polynomials,
300, 301
Transcendants, elementary, definition of, 111; order of, 111; solutions of differential equations
by, 112
Transitional regions associated with Bessel functions of large order, 248
Uniformity of convergence of Dini series, 601; of Fourier-Bessel series, 593, 594; of Kapteyn series,
575; of Schlomilch series, 632
Uniqueness of Fourier-Bessel and Dini series, 616, 649, 651; of Schlomilch series, 643, 647
Upper bounds, see Inequalities
Viscous fluid, motion of, associated with Airy's integral, 189
Wave-motions, equation of, general solutions, 125; generalised to p dimensions, 128; used to
obtain addition theorems for Bessel functions, 129
Waves, electric, 56, 226, 446; on water, and the method of stationary phase, 229
Weber's (H.) discontinuous factor, 405
Weber's (H.) infinite integrals, 391, 393, 395, 396; (discontinuous types of), 398, 402, 405, 406,
408, 411
Weber's (H. F.) function EB (z), 308; connexion with Anger's function, 310; connexion with Struve's
function, 336; differential equation satisfied by, 312; integrals expressed in terms of, 312; recurrence formulae for, 311; representation of Airy's integral (generalised) by, 321; tables of, see
Struve's function; Theisinger's integral for, 338; with large argument, asymptotic expansion of,
313; with large argument and order, asymptotic expansion of, 316
Weierstrassian products, expression for Bessel functions as, 497
Whipple's transformation of Legendre functions, 387
Whittaker's function Wv (z), 339; differential equation satisfied by, 339; recurrence formulae for,
339; with large argument, asymptotic expansion of, 340
Wronskian determinant, 42, 76, 77
Zeros of Bessel functions, 477-521 (Chapter xv); computation of (various methods of), 142, 500, 502,
503, 516; inequalities connected with, limits of, rates of growth of, 485, 489, 490, 491, 494, 507,
513, 516, 518; infinity of, 4, 478; interlacing of, 479, 480, 481; non-coincidence of (Bourget's
hypothesis), 484; non-repetition of, 479; number of, in a strip of arbitrary width, 495; reality of,
482, 483; tables of, 664, 748-751; tables of (references to), 659; values of, 4, 5, 512, 516; with
imaginary argument, 511; 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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