What results are known If $f,g$ are both analytic in $\mathbb{C}$, having infinitely many poles or zeros for the same $z$'s?

Question

What results are known If $f,g$ are both analytic in $\mathbb{C}$, having infinitely many poles (or zeros) that all coincide? Where each pole from $f$ has has the same order pole of $g$ for the same $z$.

Answer 1

Hint:

What happens if you multiply by an entire function without zeros - such as $h(z)=e^z$?

Do you know of any factorization theorems?

Answer 2

Theorem: Let $U\subseteq \mathbb{C}$ be open and connected, let $f:U\rightarrow \mathbb{C}$ be holomorphic, and let $A\subseteq U$ have an accumulation point in $U$. Then, if $f(A)=\{ 0\}$, then $f$ is identically $0$ on all of $U$.

In other words, the values a holomorphic function takes on an inifnite set with an accumulation point uniquely determine the function (on a certain connected component).

As for when $f$ and $g$ both have the same poles, I don't think you can say much. For example, if $h$ is holomorphic, then $f+h$ has exactly the same poles as $f$ and $g$. I guess you could try to apply the above theorem to $1/f$; however, if the set of points where $f$ has a pole has an accumulation point, then $f$ is identically equal to $\infty$ (follows from the above theorem). . .probably isn't a particularly useful fact.