A complex integral (explicit solution?)

Question

Let $r<1$ be a positive number close to 1. How can I compute $$ \int_0^{2\pi}\frac 1 {e^{i\theta}-r}d\theta? $$ Can this be computed in a closed form? Thanks!

Answer 1

The answer is $0$. By Cauchy's Theorem $\int_C \frac 1 {1-rz}=0$. This gives $\int_0^{2\pi} \frac {ie^{i\theta}} {1-re^{i}\theta} d\theta=0$. Hence $\int_0^{2\pi} \frac 1 {e^{-i\theta}-r} d\theta=0$. Can you take it from here?