vn 
865 
in which remains indefinite for the moment; this function has 
the radius of convergence 7, and therefore belongs to (9). Since a is 
the upper limit of the numbers a,, for the domain («), there is a 
point P, on the circumference of («), for which «a, is greater than 
a — (8—r), or, according to (7), than 7—a; let us suppose 
dy —r—at Jd. 
Then there are an unlimited number of m-values, for which, 
if ¢ has a given value < J, we have in the point ? 
lam (@)| > (r — a + Ee)”. 
If p is the argument of w—z, for P, we choose y=g, and then 
have in the same point 
ey un) ein 
Trott 
= > 7 
r—a ml! 
For the above m-values, we have therefore, in /, 
lam (#) ul, “al e Nm 
| et >—| 1+ —], 
| mm. | rt r—O 
and these 7-values being unlimited in number, the condition, neces- 
sary for convergence that the limit of the terms is zero, has not 
been satisfied in this case for the series (1); the series therefore 
diverges in P. 
2°. The condition is sufficient. The quantity a@,, being in an 
arbitrary point 2 of the domain (we) not greater than 3— a, there 
is corresponding to any arbitrarily chosen number € an integer 7, 
such that in the point « | 
beam (a) (B — a He", for m5 me 
If further the function « belongs to (9), it belongs also to a 
somewhat greater circle (@); let us suppose 
o=g Hd, (J > 9). 
Let Moe) be the maximum modulus of w on the circumference 
of (o); from the theory of functions it is known that in the whole 
domain (a) the condition 
um) oM(o) 
Ie A B 5 
is satisfied. 
We have thus in w, for m > mt: 
an UD) oM(o) /B—-ate” 
m! | p—at+d\B-atd 
If we now suppose that for ¢ a number smaller than 0 has been 
