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A Big Delay for Me and a Fourier Transform
*September 20, 2009*

*Posted by Phi. Isett in Uncategorized.*

Tags: Fourier transform, Tempered distributions

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Tags: Fourier transform, Tempered distributions

add a comment

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 on the real line:

Differentiating in the sense of distributions, we have where 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 “)

(In particular, we’ve actually computed the integral of to be corresponding to )

—- 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.