Show that a meromorphic function on \(\CP^1\) can have only finitely many poles. Show that moreover if \(f\) is meromorphic on \(\CC\) with infinitely many poles, then the poles must accumulate on an essential singularity at \(z=\infty\).
Solution
Since poles are isolated by definition, the set \(P_f\) of poles of \(f\) is a discrete subset of \(\CP^1\), which is compact. Note \(P_f\) is closed because being holomorphic is an open condition. Any discrete closed subset of a compact space is discrete and compact, thus necessarily finite.
Applying this to any bounded \(\Omega \subseteq \CC\), there can only be finitely many poles in any disc of radius \(R\). If there are infinitely many poles, this forces them to accumulate on \(z=0\infty\). A limit point of a sequence of poles of \(f\) is a limit point of a sequence of zeros of \(g\da 1/f\), making it an essential singularity for both.