## A Big Delay for Me and a Fourier TransformSeptember 20, 2009

Posted by Phi. Isett in Uncategorized.
Tags: ,
So that the entry is not completely lame, a way to compute the Fourier transform of $f(x) = e^{-|x|}$ on the real line:
Differentiating in the sense of distributions, we have $f'' - f = -2 \delta$ where $\delta$ is the delta-function (the density function corresponding to a point mass at the origin).  By taking the Fourier transform of both sides, we conclude (depending on “where you put the $2 \pi$“)
$\hat{f}(\xi) = \frac{2}{1 + (2 \pi \xi)^2}$
(In particular, we’ve actually computed the integral of $f$ to be $2$ corresponding to $\xi = 0$)