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

13-4]

INFINITE INTEGRALS

401

Then in the last integral formula we may take z = b, and when this has been
done, if we use fonctions majorantes j ust as before, we find that the resulting
series has its terms less than the terms of an absolutely convergent series
(. (-)m (b)y+2m-I (IIa)- r (2a -- 2m () (1 - VX)-A-2m
m=o M! r (y + ) (a- + 02)+12 r (I - - /-m) F (a + m+)
I (- r ) (1 - VX)-2-n-1l
+ r( —m)r(+m)| j'
where X = C/(a2 + 02).
Hence, by the test of Weierstrass, the original series converges uniformly
with respect to c when 0 < c < C, and therefore the limit of the series when
c - 0 is the same as the value of the series when c = 0.
We have therefore proved that
lim m e-ct J'-a (at) Jv (bt) dt
C -^ o J0 ty — 3 dt
(-) ()(b)y+2-1 (a)a- P (2a+2 F) F(+, 2-m;-/3+q; 1),
m=o! r(y+ n) a - +
and therefore
o ttY- is-
(-)m bY+2m-l r (2a + 2m) r ()
~= mo! IF (y + in) 20-+Y+2'm-l a+P++2+ r (1 -/ ( - m) r (a + t t- 2)'
It has therefore been shewn that:L) f Ja-3(ato)J-Il(bt) b-lr(a) I b2
b ty d 2y —a- aa+-b r () nr (1- S2 1
(21 J J (at) JV (bt) dv a (k + 1 +
o tA 2A a2-+' 1+ (v ) r ( - + 2) F 2 l)V + )
provided that O < b < a, and that the integral is convergent. This is the result
obtained by Sonine and Schafheitlin.
If we interchange a and b, and also u, and v, throughout the work, we find
that, when O < ca < b and the integral is convergent, then
J, (Clat) Jy, (bt) dt _ ao-P F _ (a)
Jo t-a-j3 -27-a-p ba-r+l r (7-ce) r (a-a 1)
x r (a - 7-y + a-; -x a- + 1; ) -
W. B. F. 26