INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
297 
We can infer that asymptotically, when \x\ is large, 
I = exp {—^}(27r)^- 1)/ V- 1/3 x {2a ' 2P+(M " 1>/2}/M J(*) i 
where j J (ac) | tends to unity as | x | tends to infinity. 
The complete asymptotic expansion is thus made to depend upon the determination 
of the singularities of S ( s ). 
The relation obtained holds when | x v>l \ < n, and is thus equivalent to q + 1 — p 
independent relations. 
In this way the (q + 1) asymptotic expansions of the differential equation for the 
generalised hypergeometric equation are connected with the solutions which are 
integral functions in the finite part of the plane. The results agree with those of 
Orr ; the methods, however, which have been suggested enable us to dispense 
entirely with his elaborate analysis. 
7 JULi&lo 
2 Q 
VOL. CCVI.—A. 
