Many weeks ago, I wrote a post on how to compute the Fourier transform of the Gaussian function defined by using the methods of ordinary differential equations. Specifically, we showed that
In that post, I promised to later show how to prove the same result by means of complex analysis–Cauchy’s theorem for closed contours. As I keep my promises, though not necessarily with the promptness others might like, I will now give this second proof.
Consider the Fourier transform of the function , for fixed:
We can complete the square in the exponent to obtain
Consider the entire function , where is a fixed real number. Let denote the positively oriented rectangular contour with vertices , for . By Cauchy’s theorem,
By the dominated convergence theorem,
Let’s examine the growth of the second and third integral as .
It is well-known identity that , so by making the change of variable , we obtain
where we use the dominated convergence theorem to obtain that penultimate equality. We conclude that
In particular, if , then is a fixed point of the Fourier transform.