INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
III. If | arg as \ < 7r, we have the asymptotic expansion 
G(-x;0) = r(0)x- e -e~ x 2 . 
257 
n=0 
X 
When | arg x | is very small, we have the asymptotic expansion 
n=Q 
These expansions are truly asymptotic in the sense of § 3. 
IV. The large zeros of G (x ; 6) occur near the positive or negative directions of the 
imaginary axis. 
Part II. 
" x n 
The Function nix ; 6) defined , when lad <1, by the Taylor’s series 2 Q • 
»=o n + U 
§ 7. This function is, in Borel’s language, the function associated with C4 (as; 6). It 
is a particular case of the more general function considered in Part IV. The detailed 
analysis is given in the paper to which reference was made in Part I. 
I. The function g (x ; 6) can, for all values of as except those which lie on the real 
axis between 1 and + oo (the limits included), be represented by the system of 
integrals 
f G {xz ; 0) log {-%) e~ z dz , 
Z7U J D 
where D is a contour which encloses the origin and embraces some line in the positive 
half of the 2 -plane along which the integral is finite, and where log (— 2 ) is real when 
2 is real and negative and has a cross cut along this line. 
II. We deduce that the only finite singularities of g (as; 6) must lie on the real axis 
between as = 1 and as = + 00 (the limits included). 
III. By using I., coupled with the asymptotic expansion of G (as ;6), we can show 
that, if | arg as | <7r/2, and j (1 —as)/as | <1, 
-g(x;0)-xlj(0)x- e = 2 ^ ■ ~ 1 )-( (9 r n ) (lvg)- (log (l-as)-i//(n+l)}. 
71 = 0 lOl Jb 
This formula gives the nature of the singularity of g(x\6) at as = 1, and shows 
that g{x\0) has no other singularities in the finite part of the plane. 
IV. The function xg (as; 0) + x l ~ e log (1 —as) satisfies the differential equation 
1 —as 1 8 
as- 
1 —as 
2 L 
VOL. CCVI.-A. 
