I know that the Cauchy Principal value of
$\int_{-\infty}^{\infty}\frac{1}{x-i}dx=i\pi$
but I do not understand how to get to this. I understand how to use the principal value for functions with real poles, but with this integral I always end up struggling with the fact that going to the complex plane the integral over the half circle in the upper half plane does not go to zero as $R\rightarrow\infty$. Any suggestions?