14 
IN ECI E 
(«+ m + 1) dn? 
pee 2 slog n 
n 
that is <{e, since (logn):n? is zero for n= oo. By this the conver- 
gence of the first part of the series of derivatives (47) has been 
proved, together with the wniform convergence in S. The second 
part need not be examined more closely, since this part forms a 
series which, save as to the factor We) not depending on n, is 
equal to the series of factorials itself. 
On account of its uniform convergence in the domain S and the 
continuity of its individual terms the series of derivatives in question 
represents in S the derivative of the function determined by the 
original series of factorials. The differential coefficient of this function 
is therefore determined at any point of S, and is independent of the 
direction in which it is taken: thus the series of factoruuls represents 
an analytic function in the domain S. 
14. We now proceed to settle the question indicated at the 
beginning of the preceding section, and we first consider the case 
4.=0, in which the upper limit of a, for =o is equivalent to 
ne. We further assume the limit of convergence and divergence of 
the series (6) to be given by R(x) = 9, 9 being a positive number 
less than unity. Since, as we showed, the number 4 is in any case 
not greater than 0, we have 
lim(l—i)*g()=90 for R(x) >. 
i=1 
From this it follows that the integral (1) converges absolutely 
for R(«) > 6. At the same time this convergence is uniform in any 
domain S wholly included in the finite part of the half-plane 
R(«z) > 6+ 4, where d is a certain fixed, but for the rest arbitrarily 
small positive number. The same holds for the integral 
1 
fo (t) (1—t)?—! log (1—+) dt, 
0 
which may be deduced from the integral (1) by differentiating the 
latter under the integration-sign with regard to x. Hence this integral 
represents in S the derivative with regard to 2 of the integral (1), 
so that the latter is an analytic function of # in the domain S.*) 
1) The statements mentioned here briefly have been established by PINCHERLE 
in his paper “Sur les fonctions déterminantes”’, Ann. de l’Ec. Norm. (3) XXII, 1905, 
p. 13—17. Their analogy with those we proved in the preceding section is 
manifest. 
