( 625 j 
region of convergence G, is in this case easy to construct. The 
function tang-!« has in the finite part of the plane the two singular 
points +7 which are situated already symmetrically with respect to 
the origin. Therefore only ‘one condition of convergence has to be 
considered here, namely 
a 
goat | 
For our ease we shall consider only real values of a. The mark 
of the series 1: B, we can then find by solving / from the equation | 
u — : gef : . 
a—l a—l fk, 
According to the mark of the series the points « are arranged 
MO ae 
ER 
a—l 
1 
t 
in G, on confocal lemniscates with the foci + pA. py: Phe region 
a— 
G, is closed by the lemniscate 
1 a 
x? — ee 
reien ee 
passing through the singular points + 2. By taking 2, equal to a we 
find the lemniscate 
ar 
having a node in the origin. This lemniscate divides G, into two 
parts Gi and G,", and within the two loops the mark of the series 
is fixed and always equal to 1:a@. Outside them (fig. 6) the mark 
rises from 1:a to 1. 
a? — 
oel 
ee 
If the parameter a approaches 1 the lemniseate by which G, is 
bounded ‘opens, to pass finally into the equilateral hyperbola. 
