well, could any one give me a hint for this one? $f$ is meromorphic and defined on $\mathbb{C}\setminus F$ where $F$ is a finite set. Assume $f$ has no essesntial singularity, then we need to show $f$ is a rational function.
a question on meromorphic function defined on $\mathbb{C}\setminus F$
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complex-analysis
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1Dear @ Thomas, please undele$t$e it (click on "undelete" right under your answer or write a new answer) after a little modification: conclude your correct argument at the beginning with a reminder that an entire function meromorphic at infinity is a polynomial, because if its Taylor series were infinite it would yield a singularity at infinity. Just do it, Thomas! – 2012-06-24
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Each point $a\in F$ is either removable singularity or a pole. The latter is removable for $(z-a)^nf(z)$ where $n$ is large enough. In this way we get a polynomial $q$ such that $qf$ is entire. Since there is no essential singularity at $\infty$ either, $qf$ is a polynomial. $\Box$