I am having trouble reconciling the following two questions:
- Can it be proved that a Meromorphic function only has a countable number of poles?
- $f$ meromorphic on $\mathbb{\hat{C}}$ $\implies$ $f$ has a finite number of poles
One seems to show that meromorphic functions have at most countably many poles while the other proofs that it is a finite number. My question is essentially whether the latter is generally true or whether there can be a meromorphic function with a countably infinite number of poles? If so, under what conditions is this possible?
My suspicion is that it is all about the domain, e.g. that functions that are meromorphic on $\mathbb{C}$ can have infinitely many poles while meromorphic functions on $\hat{\mathbb{C}}$ cannot. Then all functions which are meromorphic on $\mathbb{C}$ and have infinitely many poles are not meromorphic on $\hat{\mathbb{C}}$. Are these statements correct?
