Improper integrals with residue theorem

Question

How to prove that when solving improper integrals with residue theorem, we should only include residues in the upper complex plane? I've tried working my way through with Jordan's lemma, but I got stuck.

Answer 1

Consider real integrals of the form $\int_{-\infty}^\infty f(x)\,dx$.

In order to apply contour techniques this integral must be transformed into a complex contour integral.

Take a symmetric interval, $[-R,R]$ and adding in a upper semi-circle of radius $R$ (centered at the origin). Then as $R \to \infty$ the semi-circle part should tend to zero so that in the limit one gets the (PV) integral over $(-\infty,\infty)$

As the half circle goes bigger and bigger it gathers more and more of the upper half plane, so in the limit you have essentially captured the entire upper half plane. Thus when you go to compute the contour integral using residues, you should only consider poles in the upper half plane.

Answer 2

You cannot prove that because it is false. For instance, if we compute$$\int_{-\infty}^{+\infty}e^{iax}f(x)\,\mathrm dx$$with $a\in(-\infty,0)$ and where $f$ is an analytic function such that $\lim_{z\to\infty}f(z)=0$, then we use the residues of the lower halfplane.