# Meromorphic functions on unbounded domains

Suppose $D\subset\mathbb{C}$ is a bounded domain and $f$ is a meromorphic function on the exterior domain meaning on $D_+=\hat{\mathbb{C}}\setminus\overline{D}$. Moreover $f(\infty)=0$ and $f$ has only has poles which are of finite order. Does it follow that $f$ is a rational function on $D_+$?

Not at all. For instance, $f(z)=e^{1/z}-1$ is a counterexample for any $D$ containing $0$.