Singularities of $\dfrac{z}{\sin\pi z^{2}}$?

Question

The function $\dfrac{z}{\sin\pi z^{2}}$ of complex variable $z$. It has a simple pole at $z=0$. There is also 4 poles at $z=\pm\sqrt{n}$ and $z=\pm i\sqrt{n}$ (Where $n\in Z^{+}$). I need to find the order of the pole.please help me on this.

Answer 1

All zeros of $\sin$ are of order 1 and located at $\pi\Bbb Z$.

The order of the zero $z=0$ of $\sin z^2$ is 2 because $z^2$ has a zero of order 2 at 0 (and $\sin$ has a simple zero there).

Hence, all poles of $z/\sin z$ are of order 1.