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

Posted by Phi. Isett in Uncategorized.
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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.

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