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

798 THEORY OF BESSEL FUNCTIONS
algebraic integral, 117; soluble in finite terms when and only when the functions satisfying it
are of order n +, 52, 119; solution of, in ascending series, 39, 40, 57, 59-61; solution of, in
descending series, see Asymptotic expansions; symbolic solution of, 41; transformations of, 94,
97. See also Bessel coefficients and Bessel functions
Bessel's integral representing Bessel coefficients, 19, 21; generalisations and extensions of, see
Anger's function, Bourget's function, Bruns' function and Weber's function; modifications of,
to represent Bessel functions of arbitrary order, 175, 176, 177, 178, 181; Theisinger's transformation of, 184; used in theory of diffraction, 177; used to obtain asymptotic expansions, 215. See
also Parseval's integral
Bounds, upper, see Inequalities
Bourget's function J, k (z), 326; differential equation satisfied by, 327; recurrence formulae for,
326
Bruns' function J (z; s, k), 327
Carlini's approximation for Bessel functions of large order, 6, 7; extended by Meissel, 226, 227
Cauchy's numbers N_,,, 324; recurrence formulae for, 325
Cayley's solution of Riccati's equation, 88
Chain, oscillations of a uniform heavy, 3, 4, 576
Cognate Riccati equations, 91
Complex variables, expansions of arbitrary functions of, see Kapteyn series and Neumann series
Complex zeros of Bessel functions, 483; of Bessel functions with imaginary argument, 511; of
Lommel's polynomials, 306
Composition of Bessel functions of the second kind of integral order, 340
Computation of zeros of Bessel functions by various methods (Graeffe's), 500, 502; (Stokes'), 503;
(Sturm's, for the smallest zero), 516. See also Zeros of Bessel functions
Constant phase, Schlafli's method of, 216
Constants, discontinuity of arbitrary (Stokes' phenomenon), 201, 203, 238, 336
Continuants, connected with Schlafli's polynomial, 288
Continued fractions representing quotients of Bessel functions, 153; convergence of, 154, 303
Convergent series, Hadamard's conversion of asymptotic expansions into, 204
Crelier's integral for Schlafli's polynomial, 288. See also Neumann's integral for Neumann's
polynomial
Cross-ratio of solutions of Riccati's equation, 94
Cube of a Bessel function, expansion of, 149
Cut necessary for definition of Bessel functions, 45, 77
Cylinder (circular), motion of heat in, 9, 10, 576, 577
Cylinder functions, v (z), 4, 82, 480; addition theorems, 143, 361, 365; connexion with Bessel
functions, 83; origin of the name, 83; rank of, 129; solutions of differential equations of order
higher than the second by, 106; three-term relations connecting, 300. See also Bessel functions
and Hemi-cylindrical functions
Darboux' method of approximating to functions of large numbers, 233
Definite integrals, containing Bessel functions under the integral sign, 373-382 (Chapter xnI);
evaluated by geometrical methods, 374, 376, 378; the Ramanujan-Hardy method of evaluation,
382. See also Infinite integrals
Definite integrals representing special functions, see Bessel functions and Integrals
Determinants, representing Lommel's polynomials, 294; Wronskian, 42, 76, 77
Difference equations (linear with linear coefficients) solved by means of Bessel functions, 83. See
also Functional equations and Recurrence formulae
Differentiability of Fourier-Bessel expansions, 605; of special Schlomilch series, 635
Differential coefficients, fractional, 107, 125
Differential equations (ordinary), linear of the second order, equivalent to the generalised Riccati
equation, 92; of order higher than the second solved by Bessel functions, 106; oscillation of
solutions of, 518; satisfied by the product of two Bessel functions, 145, 146; solved by elementary transcendants, 112; symbolic solutions of, 41, 108. See also under the names of special
equations, such as Bessel's differential equation, and under the names of various functions and
polynomials satisfying differential equations, such as Anger's function
Differential equations (partial), solution of by an integral containing Bessel functions, 99; see also
Laplace's equation and Wave-motions, equation of
Diffraction, theory of, connected with Airy's integral, 188; with Bessel's type of integral, 177; with
Schlomilch series, 633; with Struve's functions, 417
Diffusion of salts in a liquid, and infinite integrals containing Bessel functions, 437
Dini expansion, 580. See also Dini series