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

GENERAL INDEX

803

with the linear equation of the second order, 92; singularities of, 94; soluble by various numbers (two, one or none) of quadratures, 3, 93
Riemann-Lebesgue lemma, analogues of the, 457, 471, 589, 599
Riemann's theorem on trigonometrical series, analogues for Schlomilch series, 642, 647; analogues
for series of Fourier-Bessel and Dini, 649
Rodrigues' transformation, see Jacobi's transformation
Schafheitlin's discontinuous infinite integral, 398, 402, 405, 406, 408, 411
Schafheitlin's integrals representing Bessel functions and cylinder functions, 168,169, 490, 491, 493
Schlafli's functions T (z) and Un(z), 71, 340, 343; addition theorems for, 344, 345; differential
equations satisfied by, 342, 343; of negative order, 343; recurrence formulae for, 71, 342, 343
Schlafli's hypergeometric function, 90
Schlafli's polynomial S{ (t), 284, 286; addition theorem for, 289; connexion with Neumann's
polynomial 0, (t), 285, 286; Crelier's integral representation of, 288; differential equation
satisfied by, 285; expression by means of Bessel functions, 287; expression in terms of Lommel's
function, 350; integrals evaluated in terms of, 350; recurrence formulae for, 285
Schlafli's solution of Riccati's equation, 90
Schlomilch series, 618-649 (Chapter xix); definition of, 621; definition of generalised, 623; expansion of an arbitrary function of a real variable into, 619, 623, 629; nature of convergence of,
637, 645; null-functions expressed by, 634; Riemann's theorem on trigonometrical series (analogue of), 642, 647; special cases of, 632; symbolic operators in the theory of, 626, 627; theory
of functions of complex variables connected with, 623; uniqueness of, 643, 647
Series containing Bessel functions, see Dini series, Fourier-Bessel series, Kapteyn series, Neumann
series and Schlomilch series
Series of Bessel functions, definition of, 580
Series of positive terms, approximation to the sum of (greatest term method), 8
Sharpe's differential equation, 105; solution by generalised hypergeometric functions, 105
Sign of remainders in asymptotic expansions, 206, 207, 209, 215, 315, 333, 449; of Struve's function, 337, 417
Sine-integral expressed as a series of squares of Bessel coefficients, 152
Singularities of functions defined by Neumann series (Pincherle's theorem), 526; of the generalised
Riccati equation, 94
Smallest zeros of Bessel functions, 5, 500, 516
Sommerfeld's expansion, see Kneser-Sommerfeld expansion
Sonine-Mehler integrals representing Bessel functions, 169, 170
Sonine's definite integral, 373; generalised, 382
Sonine's discontinuous infinite integrals, 415
Sonine's infinite integrals, 432
Spherical geometry used to obtain transformations of integrals, 51, 374, 376, 378; used to express
Bessel functions as limits of Legendre functions, 155
Sound, Sharpe's differential equation in the theory of, 105
Squares of Bessel functions, see Products of Bessel functions
Stability of a vertical pole associated with Bessel functions of order one-third, 96
Stationary phase, method of, 225, 229; applied to Bessel functions, 231, 233
Steepest descents, method of, 235; applied to Bessel functions, 237, 241, 244, 245, 262; applied
to functions of Anger and Weber, 316; applied to Struve's function, 333; connexion with
Laplace's method of approximation, 421
Stokes' method of computing zeros of Bessel functions and cylinder functions, 503, 505, 507
Stokes' phenomenon of the discontinuity of arbitrary constants, 201, 203, 238, 336
Struve's function Hy (z),. 328; connexion with Weber's function, 336; differential equation satisfied
by, 329; inequalities connected with, 328; infinite integrals containing, 392, 397, 417, 425, 436;
integral representations of, 328, 330; occurrence in generalised Schlomilch series, 622, 623, 631,
645, 646, 647; of order 4 (n+), 333; recurrence formulae for, 329; sign of, 337, 417; tables of,
663, 666-697; Theisinger's integral for, 338; with imaginary argument, 329, 332; with large
argument, asymptotic expansions of, 332; with large argument and order, asymptotic expansions of, 333; zeros of, 479
Struve's infinite integrals, 396, 397, 421
Sturm's methodsIapplied to determine the reality of zeros of Bessel functions, 483; of Lommel's
polynomials, 304, 305, 306; applied to estimate the value of the smallest zero of Bessel functions
and cylinder functions, 517, 518
Symbolic operators in expressions representing Bessel functions, 50, 170; in expressions representing solutions of various differential equations, 41, 51, 108; in the theory of Schlomilch
series, 627