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

GENERAL INDEX
[The numbers refer to the pages.]
Addition theorems, 358-372 (Chapter xi); for Bessel coefficients of order zero, 128, 359; for Bessel
coefficients of order n, 29; for Bessel functions of the first kind (Gegenbauer's type), 362, 367;
for Bessel functions of the first kind (Graf's type), 130, 143, 359; for Bessel functions or cylinder
functions of any kind (Gegenbauer's type), 363; for Bessel functions or cylinder functions of any
kind (Graf's type), 143, 361; for hemi-cylindrical functions, 354; for Lommel's functions of two
variables, 543; for Schlafli's function T, (z), 344; for Schlafli's polynomial, 289; integrals derived from, 367; physical significance of, 128, 130, 361, 363, 366; special and degenerate forms
of, 366, 368
Airy's integral, 188; expressed in terms of Bessel functions of order one-third, 192; generalised by
Hardy, 320; Hardy's expressions for the generalised integral in terms of the functions of Bessel,
Anger and Weber, 321; references to tables of, 659
Analytic theory of numbers associated with asymptotic expansions of Bessel functions, 200
Anger's function Jv (z), 308; connexion with Weber's function, 310; differential equation satisfied
by, 312; integrals expressed in terms of, 312; recurrence formulae for, 311; representation of
Airy's integral (generalised) by, 321; with large argument, asymptotic expansion of, 313; with
large argument and order, asymptotic expansion of, 316
Approximations to Bessel coefficients of order zero with large argument, 10, 12; to Bessel functions
of large order (Carlini), 6, 7; (extensions due to Meissel), 226, 227, 232, 247, 521; (in transitional
regions), 248; to functions of large numbers (Darboux), 233; (Laplace), 421; to Legendre functions of large degree, 65, 155, 157, 158; to remainders in asymptotic expansions, 213; to the sum
of a series of positive terms, 8. See also Asymptotic expansions, Method of stationary phase and
Method of steepest descents
Arbitrary functions, expansions of, see Neumann series and Kapteyn series (for complex variables);
Dini series, Fourier-Bessel series, Neumann series and Schlomilch series (for real variables)
Argument of a Bessel function defined, 40
Asymptotic expansions, approximations to remainders in, 213; conversion into convergent series,
204; for Bessel coefficients of order zero with large argument, 10, 12, 194; for Bessel functions
of arbitrary order with large argument, 194-224 (Chapter vni); (functions of the first and second
kinds), 199; (functions of the third kind), 196; (functions of the third kind by Barnes' methods),
220; (functions of the third kind by Schlafli's methods), 215; (functions with imaginary argument), 202; for Bessel functions with order and argument both large, 225-270 (Chapter vIII);
(order greater than argument), 241; (order less than argument), 244; (order nearly equal to argument), 245; (order not nearly equal to argument, both being complex), 262; for combinations of
squares and products of Bessel functions of large argument, 221, 448; for Fresnel's integrals,
545; for functions of Anger and Weber (of arbitrary order with large argument), 313; (with order
and argument both large), 316; for Lommel's functions, 351; for Lommel's functions of two
variables, 549; for Struve's function (of arbitrary order with large argument), 332; (with order
and argument both large), 333; for Thomson's functions, ber (z) and bei (z), 203; for Whittaker's
function, 340; magnitude of remainders in, 206, 211, 213, 236, 314, 332, 352, 449; sign of
remainders in, 206, 207, 209, 215, 315, 333, 449. See also Approximations
Basic numbers applied to Bessel functions, 43
Bateman's type of definite integral, 379, 382
Bei (z), Ber (z). See Thomson's functions
Bernoullian polynomials associated with Poisson's integral, 49
Bernoulli's (Daniel) solution of Riccati's equation, 85, 89, 123
Bessel coefficient of order zero, Jo (z), 3, 4; differential equation satisfied by, 4, 5; (general solution of), 5, 12, 59, 60; expressed as limit of a Legendre function, 65, 155, 157; oscillations of a
uniform heavy chain and, 3, 4; Parseval's integral representing, 9; with large argument, asymptotic expansion of, 10, 12, 194; zeros of, 4, 5. See also Bessel coefficients, Bessel functions and
Bessel's differential equation
Bessel coefficients J, (z), 5, 6, 13, 14-37 (Chapter II); addition theorem for, 29; Bessel's integral
for, 19; expansion in power series of, 15; generating function of, 14, 22, 23; inequalities satisfied
by, 16, 31, 268; notations for, 13, 14; order of, 14; (negative), 16; recurrence formulae for, 17;
square of, 32; tables of (of orders 0 and 1), 662, 666-697; (of order n), 664, 730-732; (with equal
order and argument), 664, 746; tables of (references to), 654, 655, 656, 658. See also Bessel
coefficient of order zero, Bessel's differential equation and Bessel functions
Bessel functions, 38-84 (Chapter II); argument of, defined, 40; differential equations of order
higher than the second satisfied by, 106; expressed as limits of Lame functions, 159; expressed
as limits of P-functions, 158; history of, 1-13 (Chapter i); (compiled by Maggi and by Wagner),
13; indefinite integrals containing, 132-138; order of, defined, 38, 58, 63, 67, 70; rank of, defined, 129; relations between the various kinds of, 74; representation of cylinder functions in
terms of, 82; solutions of difference equations in terms of, 83, 355; solutions of Laplace's