Contour integration of: $\int_C \frac{2}{z^2-1}\,dz$ I want to calculate this (for a homework problem, so understanding is the goal)
$$\int_C \frac{2}{z^2-1}\,dz$$ where $C$ is the circle of radius $\frac12$ centre $1$, positively oriented.

My thoughts:
$$z=-1+\frac12e^{i\theta},\,0\leq\theta\leq2\pi$$
$$z'=\frac{i}{2}e^{i\theta},f(z) = \frac{}{}$$, from here it is going to get really messy, so I doubt that is the direction I should go. Where should I go?
Or is it meant to get messy from here on out?

Use residue theorem to find integrals of this type. And observe that the only singular point for this analytic function in the disk above is at $z=1$.
