Jordan's lemma

Statement

Suppose $f$ is a function (possibly with isolated singularities) on an open subset $U$ of $\mathbb{C}$, that contains the real axis and upper half-plane $\mathcal{H}$, such that $f$ has no essential singularities in the strict upper half-plane, and only finitely many poles on the real axis and in the upper half-plane. Suppose further that:

$\underset{r\to \mathrm{\infty }}{lim}{sup}_{\left|z\right|=r,z\in \mathcal{H}}|f(z)|=0$

Then, if ${\gamma }_r$ denotes the semicircle of radius $r$ centered at the origin, and if $a>0$, we have:

$\underset{r\to \mathrm{\infty }}{lim}{\int }_{{\gamma }_r}f(z)e^{iaz}dz=0$

Thus, we get:

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

where the sum is taken over all poles in the upper half-plane.