How to show that a function is meromorphic?

Question

Is the function $\frac{1}{\sin(z)}$ meromorphic on $\mathbb{C}$?

In general, how to show that a function is meromorphic or not meromorphic?

Answer 1

A meromorphic function $f$ on $\mathbb C$ is defined as a function which is holomorphic on $\mathbb C \setminus A$ where the set $A$ has no limit points and, at each point of $A$ it has a pole. A pole $z_0$ can be characterized as an isolated singularity such that $|f(z)| \to \infty$ as $z \to z_0$. The zeros of $\sin (z)$ are the points $n\pi$ with $n \in \mathbb Z$. These points form a set with no limit points. So all you have to do is show that $|f(z)| \to \infty$ as $z \to n\pi$ (for each fixed $n$). In our case this is true because $|sin (z)| \to 0$ as $z \to n\pi$