I'm having trouble figuring out how to evaluate the integral $\int_{|z|=\rho} \frac{|dz|}{|z-a|^2}$ where $|a| \neq \rho$. This is a problem in Ahlfors in the section on Cauchy's Integral Formula, and I think by convention when he says $|z|=\rho$, he means the parametrization $z=\rho e^{it}, \; 0 \leq t \leq \pi$ (so that the winding number of a point inside this circle would be 1).
I'm guessing there is some smart way to apply the integral formula (since it's in this section), and I naively tried to expand the integrand. However, you end up with $\frac{1}{(z-a)(\bar{z}-\bar{a})}$, and I don't believe $\bar{z}-\bar{a}$ is an analytic function, so I'm not sure how to proceed from here.