1) Why do the domain of a complex function has to be a disk (circular neighborhood of Zo)? $$ |z-z_0|<p $$ 2) Domain is an open connected set. An open set D is said to be connected if every pair of points $z_1$,$z_2$ in S can be joined by a polygonal path that lies entirely in S.
so why are the following sets are not domains?
a) $-1<Im( z )<= 1$
b) $ (Re(z))^2 >1$