Finding $\text{Res}(f,0)$ where $f(z)=\frac{1}{z^2\sin(z)}$

Question

I am trying to determine the residue of $z=0$ where $f(z)=\frac{1}{z^2\sin(z)}$.

I have determined that $z=0$ is a pole of order $3$. Hence to compute the residue, I use $$\text{Res}(f,0)=\frac{1}{2}\lim_{z\to 0}\frac{\partial^2}{\partial z^2}\left(\frac{z}{\sin(z)}\right).$$ I am convinced I am doing something incorrect, as the calculations become very convoluted. I have computed that $$\frac{\partial^2}{\partial z^2}=\frac{z\sin^3(z)-2\sin^2(z)\cos(z)+2z\sin(z)\cos^2(z)}{\sin^4(z)},$$ but am unsure of how to further compute the residue, as I'm unable to solve this limit.

edit

I have solved the problem via the repeated use of L'Hopitals rule, but this took the work of computing some ugly limits. Is there another way?

Answer 1

$$\sin z=z-\dfrac{1}{3!}z^3+\dfrac{1}{5!}z^5-\dfrac{1}{7!}z^7+\cdots$$ then \begin{align} \dfrac{1}{z^2\sin z} &= \dfrac{1}{z^3\left(1-\dfrac{1}{3!}z^2+\dfrac{1}{5!}z^4-\dfrac{1}{7!}z^6+\cdots\right)} \\ &= \dfrac{1}{z^3}\left(1+\dfrac{1}{6}z^2+(\dfrac{1}{36}-\dfrac{1}{120})z^4+\cdots\right) \\ &= \dfrac{1}{z^3}+\dfrac{1}{6z}+\dfrac{7z}{120}+\cdots \end{align} s0 $a_{-1}=\dfrac16$.

Answer 2

$\frac{\partial^2}{\partial z^2}(\frac z{\sin z})=\frac{\partial}{\partial z}\frac{\sin z-z\cos z}{\sin^2 z}=\frac{(\sin z-z\cos z)2\sin z\cos z+z\sin^3 z}{\sin^4 z}=\frac{2\cos z(\sin z-z\cos z)}{\sin^3 z}+\frac z{\sin z}=\frac{2\cos z}{\sin^2 z}-\frac{2z\cos^2 z}{\sin^3 z}+\frac z{\sin z}$.

Now for the middle term (as the other two are easier) , we use L'Hôpital's: $\frac{2\cos^2 z-4 z\cos z\sin z}{3\sin^2 z\cos z}=\frac{2\cos z}{3\sin^2 z}-\frac{4z}{3\sin z}$

Now for the first term (as the other is easy), L'Hôpital's again: $-\frac1{3\cos z}\to-\frac13$.

The limit of each term is now easy:

$-1-(-\frac13-\frac43)+1=\frac13$.

I get a limit of $\frac13$.