ME. E. W. BARNES ON INTEGRAL FUNCTIONS. 
435 
for the expansion 
* S„ ( a) 
n = l H ■ 
x n 
is only valid within a circle of radius 277. 
We see then that the asymptotic series is such that, if we denote the coefficient 
of b y 
Lt > 0, and U = 0. 
n=00 ^ n=CO ^ 
The series is thus of the second order, 
preceding paragraphs will be 
G (n) = 
And the associated function formed as in the 
" ( — ) n_L S„0) 
t -—-- ±n« 
which is not an integral function. 
Our analytical machinery therefore breaks down, and we must attempt to 
extend it. 
Just as the original problem admitted of an infinite number of solutions, so we 
may now proceed in an infinite number of ways to give an analytical meaning to 
asymptotic series of the second or higher orders. 
Of these two would appear to be most natural. We may either use some more 
powerful associated function than we used in the exponential process, or we may 
repeat the exponential process until we arrive at a finite analytical function. 
§ 33. Let us consider in the first place the second of these alternatives. 
If we have the asymptotic series 
a o + a i z + • • > + a n zn + • • • > 
we have agreed to say that this series is the expression of the analytic function 
—— [Gj ( xz ) e~ x (— x) d ~ l dx, whenever this integral has a meaning, that is, 
whenever G x (xz) is an integral function, and the integral is not infinite. 
Now 
G *W = r^ ) + r ( 1 V S) * + --- + 
= a 0 ' + a{z + . . . + a n 'z n -f . . . (say), 
and, if the series is not absolutely convergent over the whole jfiane, we shall be 
consistent with our former generalised point of view, if we regard it as determining 
an analytic function 
(Go (xz) e~ x (— x) B ~ l dx, 
2 sm 7T0 J ^ v 7 v 7 
whenever this integral has a meaning. 
3 K 2 
