INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
287 
where £<2, the asymptotic expansion (B) lias only been proved to be valid over the 
area given by | arg x | < n—^kn. 
§ 45. It is at once evident from the expansion (B) that, if 2>a>0, we have, when 
I ar g x I = l a7r > 
1 i * 
E a (x) — - exp { + tr a } — 2 - 
W a 1 ; n =iX n T 
when | x | = r. 
Therefore | E a ( x ) | behaves like - . 
CL 
In thus finding the complete asymptotic expansions for E a (x), when 0<a<2, we 
have incidentally verified all Mittag-Leffler’s results. 
§ 46. We proceed next to consider the asymptotic value of E a (x) when a > 2. 
In this case Mittag-Leffler shows that:— 
1. If — 7r<arg &’<7r, 
1 . 2iJ.n +arg x 
E a ( x ) — 2, - exp { | x \ 1,a e a } 
/1 \ ' ° ki 
(1 — an) 
diminishes indefinitely as \x\ increases, the summation embracing all real numbers y, 
such that 
2. If arg x = ± 7r, 
2y.Tr + arg x 
a. 
r7T. 
m — 1 o 
E a ( x) — % - exp 
p= 0 Cl 
i ij 2 p +1 ] 
X ' COS —- TT 1 COS 
a J 
X 
1/a 
9/, 
sin 
)p +1 
a 
(wherein a = 2 m + &, 0 > 3- > — 1 and m = 1 , 2, 3, ...) diminishes indefinitely as \x\ 
increases. 
§ 47. To obtain the complete asymptotic expansions which correspond to these 
results, we consider the contour integral 
1 
2ttl 
J Sill 7 TS 
x s ds, 
wherein q is an odd positive integer equal to 2p +1 (say), and the contour of the 
integral embraces the positive half of the real axis, and encloses the poles of r( — as) 
and the points s = 0, 1, 2, ... go, but no other poles of the subject of integration. 
By Cauchy’s Theorem of Residues the integral is equal to 
v (— )“ l ~ l x ml ° 1 sin tt (qmjci — m) ^ (— ) m T (— am) sin 7 t (q — a.) ?nx m 
am! sin 7rm/a m=0 tt 
- _ ^ xm ' a s i n Trgm/ot + ^ x m 
m=o a.m ! sin Trmja m=0 T (1 + am) 
m —0 
