I am having trouble categorizing the singularities of the following complex valued function:
$$f(z) = \frac{z^2}{\sin(z)}$$
It seems like the isolated singularities are $2n\pi$ where $n\in\{0,\pm1,\pm2,\cdots\}$ but I am having difficulty determining whether we have removable or poles or essential singularities? In the book it says we have a removable singularities if given an isolated singularities:
$$|f(z)| \mbox{ is bounded as } z\to z_0$$
Which I think means that the limit as $z\to z_0$ exists, which means that we ought to be able to calculate $f(z_0)$ to determine its value but for instance $f(0)$ does not exist but the limit does for its modulus. Does that mean it is not a removable singularity and possibly a pole or even an essential singularity (truth be told I am not fully sure what that means). Any help would be really great!