$f(z)$ is a function which has a simple pole at $z=0$, has finite amount of poles in the upper halfplane (not on the real axis) and for which holds $\lim \limits _{|z| \to \infty, \Im(z) \ge 0}|zf(z)|=0 $. We want to show by choosing a path, that : $\int_{-\infty}^{\infty} f(z)dz = 2\pi i( \sum \operatorname{Res} z_{i}) + (\operatorname{Res} 0)\pi i .$
Planned was to cut out spectacles, that means cut out two arcs with lower arc from $(-\epsilon, \epsilon)$ and upper arc of $(-R,R)$. Then we can pick $\epsilon$ so that $0$ doesn't lie in it but all the other polish points do. So from all those points we get : $2\pi i (\sum \operatorname{Res} z_{i})$.
Then for the $0$ I don't see how to continue.
Does anybody see a way ? Please do tell.