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

622

THEORY OF BESSEL FUNCTIONS

[CHAP. XIX

Bessel functions of the second kind and Struve's functions; and the types of
series to be considered may be written in the forms*:
Iao aqm J (nmx) + bm Yv (mx)
r ( +1) +i= (-x)
_ 0: a a,,n Jv (mx) + bm Hv (nx)
r( + 1) m=l (x mx)v
Series of the former type (with v = 0) have been considered by Coatest;
but his proof of the possibility of expanding an arbitrary function f(x) into
such a series seems to be invalid except in the trivial case in which f(x) is
defined to be periodic (with period 27r) and to tend to zero as x - o.
Series of the latter type are of much greater interest, and they form a
direct generalisation of trigonometrical series. They will be called generalised
Schlomilch series.
Two types of investigation suggest themselves in connexion with generalised Schlomilch series. The first is the problem of expanding an arbitrary
function into such a series; and the second is the problem of determining the
properties of such a series with given coefficients and, in particular, the
construction of analysis (resembling Riemann's analysis of trigonometrical
series) with the object of determining whether a generalised Schlomilch
series, in which the coefficients are not all zero, can represent a null-functioi.
Generalised SchlSmilch series have been discussed in a series of memoirs by Nielsen,
Math. Ann. LII. (1899), pp. 582-587; Nyt Tidsskrift, x. B (1899), pp. 73-81; Oversigt
K. Danske Videnskabernes Selskabs, 1899, pp. 661-665; 1900, pp. 55-60; 1901, pp.127-146; Ann. di Mat. (3) vi. (1901), pp. 301-329.
Nielsent has given the forms for the coefficients in the generalised
Schlomilch expansion of an arbitrary function and he has investigated with
great detail the actual construction of Schlomilch series which represent
null-functions, but his researches are of a distinctly different character from
those which will be given in this chapter.
The investigation which we shall now give of the possibility of expanding
an arbitrary function into a generalised Schlomilch series is based on the
investigation given by Filon~ for the case v = 0 in his memoir on applications
of the calculus of residues to the expansions of arbitrary functions in series of
functions of given form. It seems to be of some importance to give such an
investigationll because there is no obvious method of modifying the set of
* The reason for inserting the factor xv in the denominators is to make the terms of the second
series one-valued (cf. ~ 19'21).
t Quarterly Journal, xxi. (1886), pp. 189-190.
d See e.g. his Handbuch der Theorie der Cylinderfunktionen (Leipzig, 1904), p. 348.
~ Proc. London Math. Soc. (2) iv. (1906), pp. 396-430.
II It has to be assumed that - < v < 2. The results which will be proved in ~~ 19-41-19-62
suggest that it is only to be expected that difficulties should arise for other values of v.