430 
MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
co 
“ sum.” Thus the divergent series t represents the non-uniform function 
z~ l log (1 — z), which becomes uniform when a cross-cut is made along the real axis 
from 1 to + 00 . 
§ 27. Suppose now that we have any function with singularities lying outside 
a circle of radius p, within which the function is represented by the convergent series 
a o “h a i z d - • • • T" a n% n + • • • 
We may join the singularities by straight lines to infinity, each line being the 
continuation of the direction from the origin to its initial point. Then within the 
simply connected area thus formed we may replace the function .by a set of integrals 
of the type 
_ ■ / ^ ( G (xz) e~ x (— a-) 0-1 dx. 
2 sin l 6 J v ' v ' 
Which we can therefore regard as the “ sum ” of the divergent series within the 
region in question, whenever this set of integrals has a meaning. 
Although in general this will not be the case, we can nevertheless, if the function 
represented by the series has only a finite number of poles outside its circle of 
convergence and within a circle of finite radius cr, greater than the radius of conver¬ 
gence p, split up the given series into a sum of others each of which, except the last, 
will be divergent, but capable of being represented by an integral of the foregoing- 
type, while the last series is convergent within this circle of radius cr. The circum¬ 
stances under which the whole series can be represented by a definite integral over 
the region of its existence I hope to discuss elsewhere. The problem is bound up 
with the determination of the number and nature of the singularities of a Taylor’s 
series and is, therefore, connected naturally with the researches of Darboux, # 
Hadamard,! Borel,! Fabry, § Le Roy,|| Lindelof,1F and LealW* * * § ** 
8 28. So far we have been concerned with the summation of divergent series of 
ascending powers of 2 which are convergent for sufficiently small values of |z|. We 
will now define asymptotic series as those which are divergent, however small \z\ 
may be, and we proceed to consider their summation. 
At the outset we can see that the problem is essentially different from the one 
* Darboux, ‘Liouville’ (1878), 3 ser., t. 4, pp. 5-56, 377-416. 
t Hadamard, ‘Liouville’ (1892), 4 ser., t. 8, pp. 101-186. 
\ Borel, ‘ Comptes Rendus,’ October 5 and December 14, 1896 ; December 12, 1898; ‘ Acta Mathe¬ 
matical 21; ‘ Liouville’ (1896), 5 ser., t. 2. 
§ Fabry, ‘Ann. de l’Ec. Nor. Sup.’ (1896), 3 ser., t. 13, pp. 367-399 ; ‘Acta Mathematica’ (1 899), 
t. 22, pp. 65-87; ‘Liouville ’ (1898), 5 ser., t. 4, pp. 317-358. 
|| Le Roy, ‘ Comptes Rendus,’ October 21, 1898, and February 20, 1899. 
IT Linbelof, ‘Ac-ta Societatis Fennicse,’ 1898. 
** Leau ‘Liouville’ (1899), 5 ser., t. 5, pp. 365-425. 
