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Let $f$ be holomorphic with an isolated singularity at $z_0$. Suppose that $\exists M,m,\epsilon$ positive numbers such that $|f(z)| \leq M|z-z_0|^{-m}$ for $0<|z-z_0|<\epsilon$. Prove that $z_0$ is either a removable singularity or a pole of order $m$.

How should I go about this? Any suggestions, hints, solutions will be deeply appreciated!

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That's not quite right: it is either removable or a pole of order $\le m$. And you want $m$ to be an integer. – Robert Israel Nov 20 '12 at 2:08

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up vote 3 down vote accepted

Hint: What kind of singularity does $(z - z_0)^m f(z)$ have at $z_0$?

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It seems to me that it should be removable. – user44069 Nov 20 '12 at 3:55

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