Write the Laurent series expansion of the function $f(z) = \frac {cos(z^2)}{z^7}$ in $0 < |z| < ∞$. Find the residue of this function at 0.
As in my previous question, I am not entirely sure how to find Laurent series expansions. My professor, while very intelligent, has a hard time teaching in a way that is understandable for students.
$$\frac{\cos z^2}{z^7}=\frac1{z^7}\sum_{n=0}^\infty(-1)^n\frac{z^{4n}}{(2n)!}=\frac1{z^7}\left(1-\frac{z^4}2+\frac{z^8}{24}-\ldots\right)$$
and with this you already have the order of the pole $\;z=0\;$ and the function's residue at it.