Let $f$ be a meromorphic function with $\text{dom} (f)=\mathcal{M}$, where $\mathcal{M}$ is a non-compact Riemann surface.
If $\mathcal{M}'= \mathcal{M} \cup \{\infty \}$ is the one-point compactification of $\mathcal{M}$, then is $f$ with domain $\mathcal{M}'$ still meromorphic? Does this depend on the compactification method?
Many thanks