Complex analysis: definition of cluster point

by Szeto   Last Updated August 10, 2018 11:20 AM

Cluster point refers to a point where poles become ‘infinitely dense’. For example, $0$ is a cluster point of $\csc \frac1z$.

Recently I invented a definition of cluster point (because I don’t see any rigorous definition on the internet):

Suppose $f$ admits an analytic continuation to $\mathbb C\setminus\{s_k\}$. $c$ is a cluster point iff $$\{z~|~0<|z-c|<r\}\cap\{s_k\}\ne\emptyset\qquad{\forall r>0}$$

I think this definition is not rigorous enough as it does not rule out the case of ‘natural boundaries’, as mentioned in Wikipedia.

So, what is the true definition of cluster point? And how can one rigorously use the true definition to prove $0$ is a cluster point of $\csc\frac1z$?



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