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

802

THEORY OF BESSEL FUNCTIONS

Nielsen-Hankel functions, see Bessel functions of the third kind
Null-functions, Lerch's theorem on integrals representing, 382; represented by Schlomilch series,
634, 636, 642, 647
Numbers, analytic theory of, associated with asymptotic expansions of Bessel functions, 200
Numbers, Cauchy's, 324; recurrence formulae for, 325
Order of a Bessel function defined, 38, 58, 63, 67, 70; integrals with regard to, 449
Ordinary differential equations, see Differential equations
Oscillation of solutions of linear differential equations, 518
Oscillations of membranes, 5, 510, 576, 618; of uniform heavy chains, 3, 4, 576
Oscillatory integrands, Du Bois Reymond's integrals with, expressed in terms of Bessel functions,
183
P-functions, limiting forms expressed as Bessel functions, 158
Parseval's integral representing Jo (z), 9, 21; modifications of, 21
Partial differential equations, see Differential equations
Phase, method of stationary, general principles of, 225, 229; applied to Bessel functions, 231, 233
Phase, Schlafli's method of constant, 216
Pincherle's theorem on singularities of functions defined by Neumann series, 526
Poisson's integral for Bessel coefficients, 12, 24, 25; for Bessel functions, 47, 48, 49; (generalised by
Gegenbauer), 50; (symbolic form of), 50; for Bessel functions of imaginary argument, 80; for
Bessel functions of the second kind, 68, 73; limit of the Mehler-Dirichlet integral for Legendre
functions as, 157; transformation into contour integrals to represent Bessel functions of any order
(of the first kind), 161, 163, 164; (of the second kind), 165; (of the third kind), 166, 167; (with
imaginary argument), 171, 172; transformations of the contour integrals, 168, 169, 170. See
also Parseval's integral and Struve's function
Polar coordinates, change of axes of, used to obtain transformations of integrals, 51, 374, 376, 378;
used to express Bessel functions as limits of Legendre functions, 155
Probleme de moments of Stieltjes, 464
Products of Bessel functions, 30, 31, 32, 82, 146, 147, 148, 149; Bateman's expansion of, 130, 370;
expansions of arbitrary functions into series of, 525, 572; integrals representing, 31, 150, 221, 438,
439, 440, 441, 445, 446, 448; series of, 30, 151, 152; with large argument, asymptotic expansions
of, 221, 448
Products of Weierstrassian factors, Bessel functions expressed as, 497
Quotient of Bessel functions expressed as a continued fraction, 153, 154, 303
Radius vector of an orbit, expansionas trigonometrical series of the mean anomaly, 6, 13,552,553, 554
Ramanujan's integrals of Bessel functions with respect to their order, 449
Ramanujan's method of evaluating definite integrals, 382
Random flights, problem of, 419
Rank of Bessel functions and cylinder functions, 129
Real variables, expansions of arbitrary functions of, see Dini series, Fourier-Bessel series,
Neumann series (Webb-Kapteyn theory), and Schlomilch series
Reality of zeros of Bessel functions, 482, 483, 511
Reciprocation formulae for Lommel's functions of two variables, 542
Recurrence formulae for Anger's functions, 311; for Bessel coefficients, 17; for Bessel functions
of the first kind, 45; for Bessel functions of the second kind, 66, 71; for Bessel functions of the
third kind, 74; for Bessel functions with imaginary argument, 79; for Bourget's functions, 326;
for Cauchy's numbers, 325; for cylinder functions, 82; for Gegenbauer's polynomials, 283; for
Lommel's functions, 348; for Lommel's functions of two variables, 539; for Lommel's polynomials, 298, 303; for Neumann's polynomials 0O (t), 274; for Neumann's polynomials 0l, (t),
283; for Schlafli's functions, 71, 342, 343; for Schlafli's polynomials, 285; for Struve's functions, 329; for Weber's functions, 311; for Whittaker's functions, 339. See also Functional
equations, Hemi-cylindrical functions and Three-term relations
Reduced functions, Cailler's, 536
Remainders in asymptotic expansions, magnitudes of, 206, 211, 236, 314, 332, 352; signs of, 206,
207, 209, 215, 315, 333; Stieltjes' approximations to, 213
Repetition of zeros of Bessel functions and cylinder functions, impossibility of, 479
Riccati's differential equation, 1, 2, 85-94; connexion with Bessel's equation, 1, 90; equation
cognate to, 91; limiting form of, 86; soluble cases of (D. Bernoulli's), 85; soluble cases of,
exhausted by D. Bernoulli's formula and its limit, 123; solutions by various mathematicians
(D. Bernoulli), 2, 85, 89; (Cayley), 88; (Euler), 87; (Schlafli), 90; solved by means of infinite
series by James Bernoulli, 1; transformations of, 86
Riccati's differential equation generalised, 3, 92, 94; cross-ratio of solutions, 94; equivalence