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

19-24, 19-3]

S-CHII LOMILCH SERIES

629

and divide it into two parts, namely the first N terms and the remainder of the terms,
where N is the integer such that
N <T I z I <V+1.
When n < N, the terms of n, (z) do not exceed rrn- 1 z l/n!, and therefore, when n I NV,,n (Z) t Ln z1}Z In/(n!).rn-l
+1 < Iz + lz- n+l.. z (-;l,)!'
When n N, we have () I < r -1 sinh r I z I, and therefore
N1 N |pn 7rn — - sinh 7r'lz 0 ItP,,+N+l
I, I () ^, (1-1)! 7r I Z jtN n=0 IIn
Since sinh rr T z
Since
I ZIN+ I
tends to zero as z - oo, it is evident that a sufficient condition for F(z) to be bounded
as [ z -- oo is that the series
E In l
~2=1
should be convergent; and this is the case if/(x) is such that
n q+ IIf(n) (O) I
6=l1
is convergent.
19'3. The expansion of an arbitrary function into a generalised Schlnmilch
series.
Now that the forms of the coefficients in the generalised Schlomilch
expansion have been ascertained by Filon's method, it is an easy matter to
specify sufficient conditions for the validity of the expansion and then to
establish it.
The theorem which we shall prove* is as follows:
Let v be a number such that - 1 < v < 1; and let f (x) be defined arbitrarily
in the interval (- r, wr), subjectt to the following conditions:
(I) The function h(x), defined by the equation
h (x) = 2v(x)+ xf' (x),
exists and is continuous in the closed interval (- r, 7r).
(II) The function h(x) has limited total fluctuzation in the interval (-r, 7r).
(III) If v is negative + the integral
fJ [ x 12 {f (x) -f(O)}] dx
is absolutely convergent when A is a (small) number either positive or negative.
* The expansion is stated by Nielsen, Handbuch der Theorie der Cylinderfunktionen (Leipzig,
1904), p. 348; but the formulae which he gives for the coefficients in the expansion seem to be
quite inconsistent with those given by equation (2).
f The effect of conditions (I) and (II) is merely to ensure the uniformity of 'the convergence
of a certain Fourier series connected with h (x).
+ If v is positive, this Lipschitz condition is 'satisfied by reason of (II).