Q:

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types
$$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$

I already showed, that $f$ has poles at points $z=\pi n$ where $n\in\mathbb{Z}\setminus \{0\} $.
I suspect it has essential singularity at $0$ but I can't show it.

A:

Hint. Recall that, for $z$ near $0$, you have an essential singularity at $0$ for
$$
e^{\frac{1}{z}}=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{1}{z^n}, \tag1
$$
now, with
$$
h(z):=\frac{e^{iz}-1}{\sin{z}} , \quad h(0)=i\neq 0,\tag2
$$ you have$$ f(z)=h(z)\:e^{\frac{1}{z}} $$  then $f(z)$ has an essential singularity at $0$ from $(1)$ and $(2)$.

