<s>[INST] Finding the residue at $z=0$ for complex function $1/(z^2\sin z$) Find the residue at $z=0$ for complex function $1/(z^2\sin z$)
I know $z=0$ is a pole of order 3 but can't seem to calculate the residue value for it.
 [/INST] Try some power series expansion around zero:
$$\sin z=z-\frac{z^3}{6}+\ldots\implies\frac{1}{\sin z}=\frac{1}{z\left(1-\frac{z^2}{6}+\ldots\right)}=\frac{1}{z}\left(1+\frac{z^2}{6}+\frac{z^4}{120}+\ldots\right)\implies$$
$$\frac{1}{z^2\sin z}=\frac{1}{z^3}\left(1+\frac{z^2}{6}+\ldots\right)=\frac{1}{z^3}+\frac{1}{6z}+\ldots$$
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