28 !) 
INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
Apparently, then, the expansion is not unique. But this indeterminateness is 
illusory, for it only corresponds to terms of the sum 
p 
2 exp {af /a e 2 ’ r ‘' t/a }, 
for which ar gx}/a does not lie between ±^ 7 r. 
If we neglect these terms, which-may be absorbed in J k , we may say that E a (x) 
admits the asymptotic expansion 
m =i T (1 — am) x m! 
" heiein p takes all integral values (positive, negative, or zero) such that 
27rp + arg x < jfav, 
ar g x having any value between + 7r, these limits included. 
This is equivalent to Mittag-Leffler’s results. 
§ 48. It is evident that the results just obtained admit of many extensions. As 
typical of these we may consider the function 
wherein a^> 0 and 0 is not equal to zero or a negative integer. 
We see at once that 
the integral being taken round a contour which embraces the positive half of the real 
axis and includes the points s — 0, 1, 2, ... oo, but no other poles of the subject of 
integration. 
We assume that x s = exp{s log a;}, and that the logarithm has its principal value 
whose argument lies between ±u. We also assume that (s-t-#) 8 = exp{/3 log(s + #)|, 
and that if 6 be not real, the logarithm has its principal value with respect to a 
cross-cut drawn from s = —6, parallel to the negative direction of the real axis. If 6 
veie real, we should slightly deform this cross-cut. We omit the consideration 
of this particular case in the following investigation. 
We assume in the first place that 2 > a > 0. Then if |arg x \<tt (1—^-a), the 
integral vanishes along that part of a great circle at infinity for which IX (s) > — 1\ 
YOL. CCVI.—A, 2 P 
