Semicircular contour theorem

Statement

Version where the integral over the contour tends to zero

Suppose $f$ is a meromorphic function on an open subset of $\mathbb{C}$ containing the closed upper half-plane. Further, suppose there exists $k>1$ and a constant $M$ such that for all sufficiently large $R>0$, we have:

$|f(z)|\le \frac{M}{|z{|}^k}$

for $z$ in the upper half-plane. Then, if ${\gamma }_R$ denotes the semicircular arc of radius $R$ centered at zero, then:

$\underset{R\to \mathrm{\infty }}{lim}|{\int }_{{\gamma }_R}f(z)dz|=0$

In particular, if $f$ has no poles on the real axis, and it has finitely many poles $z_1,z_2,\dots ,z_n$ in the upper half-plane, we get:

$\mathrm{P}\mathrm{V}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx=2\pi i{\sum }_j\mathrm{r}\mathrm{e}\mathrm{s}(f;z_j)$