I hope to evaluate the contour integral of $\displaystyle\int_\gamma \frac{z}{\sin z}dz$
where $\gamma$ is circle of radius $\frac{3\pi}{2}$ centered at $z = 0$ and oriented clockwise.
I have trouble evaluating the residue at $z = 0$. Even if I can evaluate the residue I wonder if the answer is just $2\pi i$ times the residues. I also wonder why I have to evaluate with this fixed value of radius. Is the integral is zero even if clockwise?
Thanks in advance.