15 
Now, the two analytic functions represented by the integral (1) 
and by the factorial series (6) are known to be equal in the domain 
of absolute convergence of that series, formed by the half-plane 
R(x) > 1. According to a well-known theorem of the theory of 
functions they cannot therefore differ in the domain of only con di- 
tional convergence 
Bake 
Thus the indicated proposition has been proved in the definite case 
2=0. Further, if 2 >0 and the limit of the domains of conver- 
gence and divergence of the series (6) be given by 
R(s)=i! +0, 
then, again, 2<A +60, as we proved as well in the preceding 
communication; and similar statements as before hold with regard 
to the integral (1), which therefore is again equal to the series of 
factorials in its domain of conditional convergence 
WOK) SN +1, 
as well as in its domain of absolute convergence. If, at last, A! and 
À!' + 0, the limit between convergence and divergence of the series 
of factorials, are both negative, then the integral (1) in general 
exists only for A(x) >0, but in this case the series affords the 
analytic continuation of the integral in the part of the plane given by 
MOK Ra) <0. 
We may also say that a certain remainder of the series (6) is 
represented by an integral of the form 
1 
fom (OPOE te se oh ea eee ACS) 
0 
in the whole domain of convergence of the series. 
We thus have proved the theorem: A series of factorials, whose 
domain of convergence is the half-plane on the right of the line 
R(x) =, is represented in a possibly existing domain of only 
conditional convergence by the same integral of the form (1) as 
in the domain of absolute convergence, if §>0. Lf, however, 
E<O, the same relation exists between a certain remainder of 
the series and an integral of the form (8). 
15. Finally we shall give a small correction and complement to 
the last two parts of N°. 6 in the second communication. The clause: 
“if it (the function) is continuous and “a écart fini’ on that circum- 
ference, or 7f a certain derivative of negative order —w has this 
property”, contains an in-correctness because the possibility might be 
