The original question to me (from a friend) was stated as
Q:Find the first four Laurent series of $f(z) = \frac{\sin z - z}{z^2 \cos z}$ in the region $0 < |z| < 2 \pi$
I'm not sure how to do it, if possible I wish only to know this expansion about zero.
The coefficients are given by $ a_n = \frac1{2i\pi}\int _\gamma \frac{f(z)}{(z-0)^n} dz $ So I change $z = r e^{i \theta}$ and integrate from $0$ to $2\pi$ putting $r=1$ $ a_n = \frac1{2i\pi}\int _\gamma \frac{\sin (r {e^{i \theta}) - r {e^{i \theta}}}}{r^{n+2}e^{i\theta {(n+2)}} \cos (re^{i\theta})} r ie^{i \theta}d\theta $
Am I going in right direction?
EDIT:: Any similar solved problem link will be highly welcome as answer :D