# Meromorphic function with infinitely many poles must have essential singularity at point of infinity?

Question: Suppose $$f: \mathbb{C}\setminus P \to \mathbb{C}$$ is meromorphic on the complex plane, where $$P:=\{z_n: n\in\mathbb{N}\}$$ is a set of infinitely many isolated poles with $$\infty$$ as their only limit point on the extended complex plane. With this condition, can we conclude $$f$$ has essential sigularity at $$\infty$$?

I'm know if we require $$f$$ to be meromorphic on the extended complex plane, then there are at most finitely many poles, where $$\infty$$ can be either a removeable singularity or a pole. Since then $$f$$ must be holomorphic on $$|z|>R$$ for some $$R>0$$ and its complement is a compact set. But for a function to be meromorphic on $$\mathbb{C}$$, it doesn't matter how it behaves at $$\infty$$, and nothing prevents it from having a sequence of poles diverging to $$\infty$$. Hence the question. Thanks.

• it really depends on the meaning of singularity - if poles are considered singularities then $\infty$ is not an isolated singularity as the poles accumulate there, but if poles are considered regular (maybe in the extended sense of allowing infinity as a value) points, then yes infinity is essential with the proof mentioned in the OP Commented Jan 1 at 17:51

If a meromorphic function $$f(z)$$ does not have essential singularity at $$z=\infty$$, then the principal part of the Laurent expansion of $$f(w^{-1})$$ at $$w=0$$ only contains finitely many terms. That is, there is some $$R>0$$ such that when $$|w|, there is some $$N\in\mathbb N$$ such that

$$f(w^{-1})=\sum_{n\ge1}b_nw^n+a_0+a_1w^{-1}+\dots+a_Nw^{-N}.$$

In other words, for $$|z|>R$$, there is

$$f(z)=\sum_{n\ge1}b_nz^{-n}+a_0+a_1z+\dots+a_Nz^N.$$

This indicates that $$f$$ has no poles other than $$z=\infty$$ for $$|z|>R$$. Furthermore, because $$f$$ is meromorphic, it can only possess finitely many poles in $$|z|, so we conclude that

Proposition 1: If $$f$$ is meromorphic in $$\mathbb C$$ and at $$\infty$$, then it can only possess finitely many poles.

Because essential singularity needs to be isolated, the contrapositive of this proposition is

Proposition 2: If $$f$$ is meromorphic in $$\mathbb C$$ and has infinitely many poles, then $$z=\infty$$ is a singularity of $$f$$ that is not isolated.