Evaluating the squared sinc integral.
Posted by peeterjoot on December 10, 2011
In the Fermi’s golden rule lecture we used the result for the integral of the squared function. Here is a reminder of the contours required to perform this integral.
We want to evaluate
We make a few change of variables
Now we pick a contour that is distorted to one side of the origin as in fig. 1.1
We employ Jordan’s theorem (section 8.12 ) now to pick the contours for each of the integrals since we need to ensure the terms converges as for the part of the contour. We can write
The second two integrals both surround no poles, so we have only the first to deal with
Putting everything back together we have
On the cavalier choice of contours
The choice of which contours to pick above may seem pretty arbitrary, but they are for good reason. Suppose you picked for the first integral. On the big arc, then with a substitution we have
This clearly doesn’t have the zero convergence property that we desire. We need to pick the contour for the first (positive exponent) integral since in that range, is always negative. We can however, use the contour for the second (negative exponent) integral. Explicitly, again by example, using contour for the first integral, over that portion of the arc we have
 W.R. Le Page and W.R. LePage. Complex Variables and the Laplace Transform for Engineers. Courier Dover Publications, 1980.