[Math] Infinite number of poles in a domain of a meromorphic function

Question

Let $U\in \mathbb{C}$ be an open set and $f$ be meromorphic. Let $\gamma$ be a simple closed path in $U$ and let $D$ be the interior of $\gamma$ in $U$. Also suppose there are no zeros on the trace of the path. Using this I can prove that there can be only finitely many zeros of $f$ in $D$ which is clearly a bounded domain. Now what if the set $P=${$z\in D|z$ is a pole of $f$} was infinite? Then clearly $P$ would have an accumulation point in $\overline D$. Can this be possible for a meromorphic function? I feel like it cannot since it would not be possible to find any disk in $D$ that does not contain a pole so it cannot be analytic in $D$. Can someone give me a proper answer if possible it would be really helpful. Thanks in advance

Answer 1

Consider the function $$f(z) = {\sin (1/z)}^{-1}$$ on the positive real half plane $\mathbf H$, say. The set of poles of $f$ is indeed discrete in $\mathbf H$, although the poles accumulate at $0$. By definition, $f(z)$ is thus meromorphic.