103] 
11 
103. 
ON CERTAIN DEFINITE INTEGRALS. 
[From the Cambridge and Dublin Mathematical Journal, vol. vi. (1851), pp. 136—140.J 
Suppose that for any positive or negative integral value of r, we have ^¡r(x + ra) 
= Ur'yjrx, U r being in general a function of x, and consider the definite integral 
I = I Tp'x'ty'xdx ; 
J — oo 
x Vx being any other function of x. In case of either of the functions y]rx, '\ r x becoming- 
infinite for any real value a of x, the principal value of the integral is to be taken, 
that is, I is to be considered as the limit of 
if + f j "ijrx "fyxdx, (e = 0), 
'wa + e J—oo/ 
and similarly, when one of the functions becomes infinite for several of such values 
of x. 
We have 
aJrx^xdx; 
or changing the variables in the different integrals so as to make the limits of each 
a, 0, we have 
I — I [%yjr(x + ra) 'P (x + ra)] dx, 
J 0 
2 extending to all positive or negative integer values of r, that is, 
I — I yjrx [2 CAP (# + ra)] dx, 
.(A) 
2—2