Given an entire function $f(z)$, why can $f(1/z)$ have at worst an isolated singularity at $0$? Can a meromorphic function have non isolated singularities? What about other kinds of functions?
I'm probably missing something very simple given that there are no questions on this site about this..
By definition a meromorphic function is analytic except for isolated singularities which are poles. An example of a function with non-isolated singularity is $\frac 1 {\sin(\frac 1 z)}$. This function has singularites at each of the points $\frac 1 {n\pi}$ so $0$ is a non-isolated singularity.