436 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Now 
Go ( 2 ) = 
-f • • • + 
yji _|_ 
r (0) 1 ' ' 1 r (n + 0) 
= a 0 " + . . . fi- ci,”z n + . . . (say). 
If the series G! ( 2 ) had a finite radius of convergence, or zero radius of the first order, 
the function Go ( 2 ) will be an integral function, and by the process just sketched, 
a definite meaning has been assigned to G 1 ( 2 ) and the original series. 
When, however, G x ( 2 ) has zero radius of convergency of the second or higher 
order, Go ( 2 ) will not be an integral function, but we must regard the series which it 
denotes as determining an analytic function 
2 sin 7 t 0 
|G, (xz) e - r (— x) (J 1 clx, 
whenever this integral has a meaning, that is, as a preliminary condition, whenever 
°R) = i% + ...+ 
ci n 
r (n + 6) 
z n fi- . 
is convergent over the whole plane. 
The procedure may be repeated indefinitely. If we have started with an asymptotic 
series which does not ultimately give rise to a function G„ ( 2 ) whose finite radius of 
convergency is a line of essential singularity, we shall ultimately get an analytic 
function of which the original series is the asymptotic expansion in the vicinity of its 
essential singularity 2 = 0. 
§ 34. The extension which we have just indicated is in harmony with the general 
theory, but we have still to determine the important point as to whether the 
asymptotic equality of series and functions satisfies Poincare’s arithmetic definition. 
Take for simplicity the series of the second order « 0 -f a 1 z fi- . . . fi- a u z n fi- . . . , 
for which the associated series 
u 0 1 tc l 
r (6) ^ r (i + 6) 
+ ... + 
r 0 n + 6) 
U fi¬ 
lms finite radius of convergency and represents the function 
iG (z). 
The given series gives rise to the function 
G (4 = j-bs jp ^ O- 1 dx. 
G„ ( 2 ) — a 0 fi- . . . fi- a,'Z”, 
Let 
