Consider the following four curves 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
inserting the result of the integral on the left hand side we get