# CVRC9

Evaluate

Consider the following four curves where :

where

where

and

where

We want to integrate over the closed contour .

By the residue theorem we know that

A very useful lemma is if are holomorphic functions defined on an open set that contains the point , and is a simple zero of then .

So we have

Now we integrate over each part of the contour, and then take the limit as goes to infinity.

The -test from single variable calculus tells us that this integral converges as goes to infinity. Therefore we simply have

.

Next we see that

For all large enough we have . Then for a large enough it's easy to see . Then getting back to out integral we have

Taking the limit as both as tend to infinity

Now we have just one more curve to integrate.

Again taking the limit as goes to infinity we get

.

i.e. times the original integral

Therefore

inserting the result of the integral on the left hand side we get

so that

Eventually,