Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to be
We shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.
(Received February 13 1919)
* Of. Maodonald, Proc. London Math. Soc., XXX.
* Cf. MacRobert's Functions of a Complex Variable, p. 239.
† Cf. Whittaker and Watson, Analysis, p. 376.
* Cf. Fourier, Théorie Analytique de la Chaleur, Ch. VI., Gray & Mathews, Ch. V., and Clifford, Mathematical Papers, p. 346.
* Cf. Macdonald, Proc. London Math. Soc., XXX.
* Kugelfunctionen, p. 237.
* Cf. Macdonald, Proc. London Math. Soc., XXX., p. 170, and Hardy, Quarterly Journal, XXXIX.
* Cf. Whittaker and Watson, Analysis, Chapter XVI.