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

Posted by Phi. Isett in Uncategorized.
Tags: ,

I’m not exactly sure if anybody reads this blog (as nobody has commented yet).  Just in case anybody does, I should have said a long time ago: I probably won’t post here again until the end of October.  I am still preparing for my general exam and that sort of monopolizes my time.  After that I have maybe 10-13 entries planned, but right now I am resisting the urge to actually write them.

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 $\xi = 0$)

—-  It should be noted, of course, that there are more elementary ways to compute this Fourier transform………  Also note that the Fourier transform has a meromorphic continuation into the complex plane whose poles can be anticipated from the physical space representation.