INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
295 
III. For all values of arg x we have the asymptotic expansion 
TT S n ■ r 'l - 2x^ r(2p-l) 1/2U/2-P -p J n _l 3_ . 1 
.tj- p/p_l\ V ' / 2 r 0|P 2’ 2 P » ^pl/2 
0>-*) 
-2i'>/2 r ( 2p — 1 ) / _ , 1 : 
, - y -r r i,2\X/2-p -p J 1 3 . 1 
r(p-i) 1 ' 2 0 r 2 ’ 2 4 x 1/2 
We take that value of (4a; 1/2 ) 1/2-p which is equal to exp {(|-—p) log (4a; 1/2 )}, the 
logarithm being real when x is real and positive and having a cross-cut along the 
negative half of the real axis. Similarly (— 4.r 1/2 ) 1/2_p is equal to exp {(^—p) log (. —4,r 1/2 )} 
when the logarithm is real, when x is real and negative and has a cross-cut along the 
positive half of the real axis. 
IV. We may deduce the asymptotic behaviour of Bessel’s function. 
The theorem, though its expression is more concise, is equivalent to the results 
obtained by Stokes. 
V. By considering the integral 
1 
2m 
j T (— s) r (1 — p — s) x s ds we may prove that, 
if | arg x | < tt, 
r (1 -p) 0 F X {p ; a;} + af- p T (p—l) 0 F 1 {2—p ; x) 
= -p; p-i; -i-x- 
1/2 \ 
Part XI. 
The Generalised Hypergeometric Functions. 
§ 52. When p < q, the generalised hypergeometric functions are integral functions. 
We limit ourselves to this case. 
I. By considering the contour integral 
_ L f r ( -s) r («,+ s) r (i - Pl ->■)... r ( i - P - s ) rf 
2ttl J r (1 — a 2 — s) ... r (1 — a. p —s) 
we may show that the linear combination of functions, 
r ( a i ) r(i pi )... r(i p g ) p < . i_\p-* x \ 
r(i-oa)...r(i-Op) q{ 1 ””’ p ’ P 1 ’”” Pg ’ 1 ’ ; 
+ 
r(«i-p 1 + i)r(p 1 -i)r(p 1 -p 8 )• r^-p,) ^. 
r(pi-a 3 )r(pi-a 3 ) ... V(p 1 -a p ) 
pF?{«i-pi+!»■••, a P -pi+ 1 ; 2 -pi> l ~pi+p2,---, i—pi+pg] 
