MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
437 
then 
where 
G (z) — G„ (z) = * ([ X G (xz) — jG* (xz) }e x (- x) e 1 clx, 
bill 7T (7 J 
Q (G\ _ V _ -L _ 
1 r J o r ( r + 0 )- 
Now 2 - " -1 ( X G (xz) — jG,, (xz)} is an analytic function of x of the same character as 
jG (xz): hence the natures of the two functions G(z) and z~" _1 {G (z) — G n (z)\ near 
2 = 0 are substantially the same. And therefore, in general, if G (z) tends uniformly 
to a finite limit as z tends to zero in any direction, z~ n ~ l [G (z) — „G(z)} also tends 
uniformly to a finite limit as z tends to zero in the same direction. That is to say, 
| z~ n {G (z) — n G ( 2 ) } | tends to zero as z tends to zero, so that the divergent series is 
arithmetically asymptotic for the function G (z). 
It is evident that a repetition of the same argument will prove the arithmetic nature 
of the asymptotic dependence of a series of any order and the function to which it 
gives rise by successive applications of the exponential process. But one case of 
exception must be noticed. At each step the equivalence of the asymptotic series 
and the derived function fails along certain lines or within certain areas radiating 
from z = 0. And, since the effect of such failure is cumulative, it may happen that 
before the process is finished the equivalence has failed over the whole area around 
z=0. Either the series is hopeless—an artificial monstrosity that cannot arise in 
practice—or we need some other process by means of which it can be interpreted. 
§ 35. As an example of the process just sketched, consider the asymptotic 
expansion 
S n '(a) z"+ 1_1 
s + n — 1 ’ 
which, qud function of z, is an asymptotic series of the second order and wherein 
s and a are any complex or real parameters. 
Applying the integral process associated with the exponential function to the series, 
we obtain the integral 
~ | G, (zx) e~ x z dx, 
where 
G, (u) = — r (1 — S) (- 1 + 2 tyyg) 
and we have, for convenience, taken the auxiliary function to be 
i (~) n r (2 — ft - s)x n , ■ 
n =0 
so that 0 is absorbed in s. 
Now G x (u) is a series of finite radius of convergency, and the analytic function 
