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

396 THEORY OF BESSEL FUNCTIONS [CHAP. XIII
13'32. Generalisations of Weber's second exponential integral.
When the Bessel functions in integrals of the type just considered are
not of the same order, it is usually impossible to express the result in any
simple form. The only method of dealing with the most general integral
f J, (at) Ja (bt) exp (- p2t) tl,- dt
0o
is to substitute the series of ~ 116 for the product of Bessel functions and
integrate term-by-term, but it seems unnecessary to give the result here.
In the special case in which X = v - /u, Macdonald* has shewn that the integral
is equal to
(-a)- fc 0' ' (.-ab sin 0 b + sin,
p2 r( ) cos2-2vl 0 sinv+1IV h ) exp (- 4 s )
by a transformation based on the results of ~~ 12'11, 1387.
An exceptional case occurs when a = b; if R (X 4+ + v) > 0, we then have
F,X + u + v)
J, (at) J, (at) exp (- p2t2) t'l- dt = 2+ ++ r (/- + ) P( + 1)
J/ l-0^+, 2bp,- b, Ff +. a I) v~1
x F (/+v4__ /z+v+2 X+A+a2
x 33 2 2 _L + l, v + 1, I. + v + 1 p2)
by using the expansion of ~ 5'41. Some special cases of this formula have
been investigated by Gegenbauert.
13'33. Strrve's integral involving products of Bessel functions.
It will now be shewn that, when R (Lt + v) > 0, then
(1) f J,(t)J(t) t (Fu + v) 1 (i-)
f \ )t = 2
J tJo +v 2L+ rF ( +~, ) + )r(t + )r( )'
This result was obtained by Struve, gemn. de 'Acacd. Imp. des Sci. de St Petersbozrg, (7)
xxx. (1882), p. 91, in the special case = v = 1; the expression on the right is then equal
to 4/(37r).
In evaluating the integral it is first convenient to suppose that P (tu) and
R (i) both exceed ~. It then follows from ~ 3-3 (7) that
J J (t)J(t) i t ( _ (2A - 1) (27 - 1)
0 + - 2f+V7r-r ( + 2) p (t + 2)
x f 2 f ' sin (t sin O)sin (t s in c) o s- 2 0 2osin 0 sin Oi dOddt.
o 0 0 t
* Proc. London Math. Soc. xxxv. (1903), p. 440.
t Iiener Sitzungsberichte, LXXXVIII. (1884), pp. 999-1000.