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

408

THEORY OF BESSEL FUNCTIONS

[CHAP. XIII

13'44. Gubler's investigation of the Weber-Schafheitlint integral.
The integral
Jf J t(at) J (bt) dt
will now be investigated by the method due to Gubler*. It is convenient
first to consider the more general integral
tx
even though this integral cannot be evaluated in a simple manner by Gubler's
methods. It is first supposed that R (v) > 0, R (X) >, R (/t - X) > - 1; and,
as usual, a and b are positive, and a > b.
From the generalisation of Bessel's integral, given by ~ 6'2 (2), it is evident
that the integral is equal to
f Ja(Ctt)f(O+ z-v-l exp {2bt(-) dz dt.
27nio tA J-,
We take the contour as shewn in Fig. 29 to meet the circle z = 1 and the
0
Fig. 29.
line R (z) = 0 only at z = + i; and then, for all the values of z and t under
consideration,
R bt (z-l/z)^ (0;
and the repeated integral converges absolutely, since
I J. (at) dt + d
is convergent. The order of the integrations may therefore be changed, and
we have
f J, (at) J, (bt) 2 f (+) J, (at) ep dtd
dt =_ exp ~bt z - i -)f- dtdz.
If we write
b (z -l/ )=- (-/ 0,

* Math. Ann. XLVIII. (1897), pp. 37-48.