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:

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

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

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

Thus, we get:

$\operatorname {PV} \int _{-\infty }^{\infty }\int f(x)e^{iax}\,dz=2\pi i\sum \operatorname {res} (f;z_{j})$

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