INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
2G5 
For if 6 lie between — k and — (&+1), k being a positive integer, we consider 
(x ; 6 + k ). 
By the theorem just proved, G^ (.r; 6 + k) admits an asymptotic expansion. 
Also 
G^ (x; 6+k) - ~ 
G„ (x ; 6) - 
iAy 
n =o r(n+l) (n + 6y . 
Hence, since by a theorem due to Poincares, we may integrate an asymptotic 
series, the general result follows .* 
Finally we see that, if ll\ (x) > 0, and 6 be not zero or a negative integer, the 
function G^ (x ; 6) admits the asymptotic expansion 
C £ fi.r(l-j8) 
x p n= o T (1 —fi—n) x n ‘ 
3-1 
(1— y) e 1 = 2 c n (—y) n , valid where \y\ < 1. 
«=0 
The coefficients c n are determined by the expansion 
log (1—y) ~ 
- -y - 
§ 18. But when B (x) < 0, we have also obtained an asymptotic expansion, and 
the restriction that 6 shall not be real and negative can be replaced by the restric¬ 
tion that 6 shall not be zero or a negative integer. 
Combining the two results we see that, when \x\ is large and 6 not zero or 
a 
negative integer, the behaviour of G^ (x ; 6) is given by the sum of the two asymptotic 
expansions 
(-)T W ( 6 ) 
— t - C ” r ( 1 —^1— + ( x)~ S [~1 0 rr (- x l?" 1 V —_ 
x*n^Y(l-p-n)x n K ‘’ L ;J n =o T (n+1) r (/3—n){log (—x)}' 
(A). 
This double expansion is valid for all values of arg x between — tt and tt, except 
possibly those for which jarg x\ = —. It is, as we shall see later, true even in these 
2 
cases. 
x 
which was 
§ 19. When /3=1, the function reduces to G (x; 6) = 2 — 7 . . 
v ’ n=0 Y(n+l)(n+6y 
previously considered. 
oc 
The c’s are now determined by the expansion (1— y) e ~ l — 2 c„( — y) n . 
Therefore c n = (6—1) (6—2 )... (6—n)/n ! 
The asymptotic formula (A) therefore reduces to 
- 2 ••• ( 9 ~ n ) + (- x yr(6), 
00 ?i=o 00 
which was the result previously obtained (§ 6, III.). 
* The matter does not seem of sufficient importance for an elaborate proof. I may, however, refer the 
reader to Mr. Hardy’s paper, p. 419 (loc. cit., § 2). 
VOL. CCVI.—A. 2 M 
