INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
285 
Part VIII. 
CO yi 
The Function E a (x) = 2 —— -r , a> 0. 
n=o r (1 + an) 
§ 43. After the previous investigations it is natural to consider the function 
00 
n=0 I (1+ an) 
has discussed its asymptotic behaviour. A reference to his papers will show how 
differently the methods of this memoir lead us to attack the problem which he has 
solved. 
We will first consider the results which Mittag-Leffler has obtained for the 
case 0 < a < 2. He shows that 
This function has. been denoted by E„ (x) by Mittag-Leffler, # who 
1. j E a ( x ) | approaches zero when 277 — \o.tt > </>>^a77, where arg x — <j>, 
2. When <j> = ± \a. 77, j E a (*) | approaches — . 
E ft (x) - - exp 
3. 
I <f) 
r « cos — 
a 
diminishes indefinitely when \a-rr > <f> > — ^an. 
It is evident that, where a = 0, E a (x) = 1/(1— x), and that, when a < 0, E a (x) is 
an asymptotic series. We assume then a > 0. 
It is evident that we may write 
7 TX S 
r(as+l) sin sit 
ds, 
where the integral is taken along a contour which encloses the points 5 = 0, 1, 2... oo, 
but no other poles of the subject of integration and which embraces the positive half 
of the real axis. 
Now when s = u + iv and | v | is large, r (s) behaves like exp{— Att | v j }. Therefore 
the integral vanishes when taken along any part of an infinite contour for which 
&(s) > —Tc, where h is a finite jiositive quantity, if | arg x | < (2 — a). In order 
that this equality may have a meaning, we assume 0 < a < 2. 
Hence, under these restrictions, 
k / \n-m-» 
E a (-x)= 2 a . +J to 
n= 1 r (1 — an) 
where I J* 1 is of order less than 
when 1*1 is larne. 
x 
Chan ging x into —x, we see that, if 0 < a < 2, we have the asymptotic equality 
when 277—> arg x > |-a 7 r. 
2 
n=i 
1 
PT(l-an) 
(A), 
* Mittag-Leffler, ‘Comptes Rendus,’ tome 137, pp. 554-558,1903. 
