I have trouble evaluating the integral:
$$\int_{B(0,\frac{3R}{|h|})} \frac{1}{(r e^{2i a}-e^{i a})}dr da$$
In fact I just need to estimate it from above in terms of $|h|log (\frac{1}{|h|})$, where $R\geq 1$ and $0 < |h| \leq \frac{1}{2}.$
I don't know how to compute an integral about an area with a pole in $re^{ia}=1,$ applying the residue formula does not seem to work.
I got to the point where I need to know now how exactly and if I can apply the Cauchy principal value to:
$$\int_0^\frac{3R}{|h|} \frac{1}{(r-\frac{1}{2})}dr$$
Can somebody tell me if and how that works here? I have trouble finding literature on CPV for the complex case.
Best regards!