( 303 ) 
With this assumption about W we have immediately 
aaa 1 ri 
1 1 Qi a 
yr ee =a) eA ak 
r (= +1) Ww 
Pp 
nus iB ‘ 
en era? 1 aus 5 > 
A er en 5 49 ene ze Prdv, 
7 t £ 
r(2+41) WV 
Pp 
and finally, provided || remains small enough, 
a F 
ay Ee = = 
oh Er [ E Sen (ve? ze P rar, 
Te ce : 
or 
a 
1 a Sih dek 
ER | : Renee? hyde. 
W 
The latter equation however conveys no definite meaning, unless 
1 mt 
during the integration the value of f(#” ze ” ) can be fixed without 
ambiguity. This requires that the region bounded by W contains 
l rt 
none of the singular points ¢ of f(z’ ze ”). 
We will assume that in the z- plane f(z) can be continued across 
oo 
the circle of convergence of the series Sc, 2” and that it has outside 
0 
this circle a set of singular points e= Ac. Then, taking 2= ge}, 
1 zi 
the function f(#” ze ") has in the z-plane a similar set of singular 
points #, given by 
1 jp Pp 
zin en cos p(O—/) —t = sinp(O—/?) . 
In order to exclude the points ¢ from the region delimited by 
the cardioid, we confine the point - in the z-plane to an area, con- 
22 
Proceedings Royal Acad. Amsterdam. Vol. IL, 
