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

464

THEORY OF BESSEL FUNCTIONS

[CHAP. XIV

But, if this limit exists, then, by ~ 1442,
f udu] F(R) J, (uR) J, (ur) RdR
also exists and is equal to it; and so we have proved Hankel's theorem, as
stated in. 14'4.
The use of generalised integrals in the proof of the theorem seems to be due to
Sommerfeld, in his K6nigsberg Dissertation, 1891. For some applications of such methods
combined with the general results of this chapter to the probliem des moments of Stieltjes,
see a recent paper by Hardy, Messenger, XLVII. (1918), pp. 81-88.
14'46. Note on Hankel's proof of his theorem.
The proof given by Hankel of his formula seems to discuss two points somewhat
inadequately. The first is in the discussion of
lim J F F(R) Jv (uR) Jv (ur) u dudR,
A-c- o 0o
which he replaces by
lim F (R) [RJ +1 (XR) J (Xr) - rJ + 1 (Xr) J, (XR)] &2 2
In order to approximate to this integral, he substitutes the first terms of the asymptotic
expansions of the Bessel functions without considering whether the integrals arising from
the second and following terms are negligible (which seems a fatal objection to the proof),
and without considering the consequences of XR vanishing at the lower limit of the path of
integration.
The second point, which is of a similar character, is in the discussion of
rr+S r\
lim J, (uR) J, (ur) uR dudR;
X-ce. Jr+t J
after proving by the method just explained that this is zero if: tends to a positive limit
and is ~ if 5=0, he takes it for granted that it must be bounded if — 0 as X-oo; and
this does not seem prima facie obvious.
14'5. Extensions of Hankel's theorem to any cylinder functions.
We shall now discuss integrals of the type
J udu F (R) V (uR) Y (ur) RdR,
in which the order v of the unrestricted cylinder function ', (z) is any real*
number. The lower limits of the integrals will be specified subsequently,
since it is convenient to give them values which depend on the value of v.
For definiteness we shall suppose that
^V (Z) - a {cos a. J (z) + sin a. Y (z)},
where oa and a are constants.
* The subsequent discussion is simplified and no generality is lost by assuming that v 0.