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When reading Ahlfor's Complex Analysis book, I came across the notion of non essential singularity. I know that for a function $f(z)$ an element $a\in\mathbb C$ is a non essential singularity iff there exists $m\in\mathbb Z$ such that $g(z)=(z-a)^mf(z)$ has a removable singularity at $a$. The question is this: is a non essential singularity at $\infty$ removable or is it necessarily non removable, which would imply that it is a pole? Thanks.

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  • $\begingroup$ Isolated singularities are either removable singularities, poles, or essential singularities. By a non essential singularity Ahlfors just means either a removable singularity or a pole. It doesn't matter whether the point is infinity or an element in the complex plane. $\endgroup$ – Mehta Mar 27 '14 at 2:57
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It could be either. For example, $z^2$ has a pole at $\infty$, $1/z$ has a removable singularity there.

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