INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
293 
An independent solution of this equation is 
(«—p + 1 ; 2— p ; x). 
II. By considering the contour integral - ~ | r (n-s) x s ds, we may show that 
iFi {«; p ; x} = e x 1 F 1 {p-«; p ; -x}. 
This result is valid for all values of arg x. It has been otherwise obtained by Ore. 
III. By considering the contour integral 
- J_ f r(-s)r(q + s)(-a;) s J 
2m) r (p + s) 
we may show that, when (x) <0, JA {a ; p; x) admits the asymptotic expansion 
[{r (p)/r (,>-«)}](-aJ-.Foja, l-p+« ; -1}. 
IV. Combining II. and III, we show that, when E(«)>0, ^{aipix} admits 
the asymptotic expansion 
e * r (P ) p 1 \ 0 — a 
V. The combination of III. and IV. gives us the complete asymptotic expansion of 
iFi{« ;p;x\. 
This I have verified by means of integrals taken round double loop contours of 
Pochhammer’s type. 
VI. It is possible to take such a linear combination of the two solutions 
iFi{a ;p;x} and ^{a—p+1 ; 2-p ;x} x x ~ 9 
as will admit all round x — oo the single asymptotic expansion 
(—.r) _a 2 F 0 {a, 1— p + a; — 1 /x}, 
By considering the contour integral 
U f r (- s) r (l - P - S ) r («+ s ) * * 
we can, in fact, prove that, when | arg x | < 37 t/ 2 , 
r (“) r (I — p) iFi {a.; p ;x} + T (a + 1 — p) T (p — l) a; 1_p 1 F 1 {a—p+ 1 ;2 -p;x} 
= X~ a T (a) T (1 + a— p) 2 F 0 {«, 1 +a —p ; — 1 /x}. 
This theorem is equivalent to two different results when & (x) < 0. By this means 
we can obtain III. anew. 
