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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 $D_+$?

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

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  • $\begingroup$ As a matter of fact the particular domain I had in mind does contain zero $\endgroup$ – BigM Feb 11 '16 at 3:36

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