( 536 ) 
As second application we take the function fun‘. Again the 
function is not uniform in the whole plane; now there are two 
singular points + 4 and — d at final distance. So convergence 
is again only possible for |a@| > 1 and the mark of the series 
can fall at most to 1:|a |. The region of convergence is found 
by applying to circle G, of fig. 1 successively the transformations 
(x, ix) and (x —i«). The new region G is the part of the z-plane 
common to the two transformed circles. It is a double segment, 
bounded by the circles with the radius , having the centres 
: 5 and — — (fig. 3). If we describe out of these centres 
a— a— 
+ 
circles with the radius 
a 
these circles touch each other externally. So here is no region Gi 
within which the mark of the series is constantly equal tol: | @|- 
For a given point z inside G, the mark 1: #, is found by solving 
according to u the two equations 
(a) a 
x == eN 
a—wu 
The smallest value of |u| is R. To obtain the lowest mark for 
the series we must bring a circle through #(P),-+-7(A;) and —#(Q) 
and determine the midpoint J, of the are PQ. If we now describe 
he tal 
