Meromorphic functions on $\hat{\mathbb{C}}$ are rational functions

Question

I was reading through a proof of the fact that all meromorphic functions on $\hat {\mathbb{C}}$ are rational functions found here

http://math.haifa.ac.il/hinich/RSlec/lec1.pdf

and I didn't understand the justification behind the statement

"The set of poles of a meromorphic function is discrete, therefore, finite since $\hat{\mathbb{C}}$ is compact"

I understand that $\hat{\mathbb{C}}$ is compact, but I don't see how that gives us that the number of poles is finite.

for example, the analytic continuation of the gamma function has an infinite number of poles, and I thought that was meromorphic.

Thanks in advance for the help!

Answer 1

If there are infinitely many poles, they have a limit point in $\widehat{\mathbb C}$. In your example, that limit point is $\infty$. The limit point is a singularity of the function, but not an isolated singularity, so the function is not meromorphic on $\widehat{\mathbb C}$.