One definition I have seen is that a Meromorphic function has at most a countable number of poles. Another says that a function f is Meromorphic if every point is either a pole or the function is analytic there.
Now first an easy question, but I am unsure of it:
1. That poles are isolated, that is usually taken as a definition of the pole?
Now come the hard question:
2. Can it be proved that if we use definition 2 of a Meromorphic function it can at most have a coutnable number of poles? If the answer to question 1 is yes, I guess we can assume for contradiction that it has more than a countable number of poles, and then show that it must have a limit point, and hence not be isolated? Is this hard to prove?
PS: I think an equivalent question for 2 in terms of $\mathbb{R}^2$ is that lets say that the set A is bigger than just beeing countable(I don't know if this implies if it is uncountable?). Then the set A must have a sequence of distinct points, with a limit point in $\mathbb{R}^2$. But I don't think the limit point must be in A? Is this something we can prove?
Best Answer
Ad 1. Yes, it is part of the definition or a consequence of the definition that a pole is an isolated singularity of the function.
Ad 2. Yes, since poles are isolated singularities, a meromorphic function can have at most countably many poles. That is not hard to prove: Every compact subset $K$ of the domain of the function can only contain finitely many poles [otherwise the set of poles would have a limit point in $K$], and every open subset of $\mathbb{C}$ (or $\widehat{\mathbb{C}}$) is the union of countably many compact subsets (since it is locally compact and second countable), so the set of poles is a countable union of finite sets.