A few questions on an application of the residue theorem I have got the following closed curve $\gamma$

and I am supposed to calculate
\begin{equation}
\int_\gamma\frac{\cos(z)}{z^3(z^2 +1)}dz
\end{equation}
with the help of the residue theorem.
Now, I've got a few questions

*

*$0$, i and -i are the isolated singularities of our given function. As far as I understand it, the winding number of $0$ is $-2$, the winding number of $-$i is $-1$ and the winding number of i is $0$. Is this correct?

*How can I determine the residue in this case? For a start, I tried finding the Laurent expansion for $z_0=0$, but I have no idea how to rearrange the equation further. I only got as far as
\begin{equation}
\frac{\cos(z)}{z^3(z^2 +1)}=\frac{1}{z^2+1}\sum\limits_{n=0}^{\infty}(-1)^n\cdot\frac{z^{2n-3}}{(2n)!}.
\end{equation}
Could someone give me a hint on how to find the Laurent expansion?

Thanks in advance!

*

*You are right.

*Let $f(z)=\frac{\cos(z)}{z^2+1}$. Then\begin{align}\operatorname{res}_{z=0}\left(\frac{\cos(z)}{z^3(z^2+1)}\right)&=\operatorname{res}_{z=0}\left(\frac{f(z)}{z^3}\right)\\&=\frac{f''(0)}{2!}\\&=-\frac32.\end{align}
