Compute the fourier transform of the function $\ f(x)=x^2e^{-x^2} $

Question

So I have taken advantage of the Fourier Transform table $\ e^{-ax^2/2}$ $->$ $\ \left(\frac{\sqrt 2π}{a}\right)e^{-ω^2/2a} $

For $x^2$, I used twice from fourier table: $xf(x)$ $->$ $i\widehat{f}$'(ω)

Finally I got $\ (\sqrt πe^{{-ω^2/4}}(ω^2-2))$ divided by 4.

This doesnt quite fit the wolframalpha calculation of Fourier Transform, and I have to be missing out totally. Do I have to consider another constraint on the fourier table? Is this a convolution thing? Proper and stepwise help would be appreciated! And also, is f in the space $L^1(R)$?

Thanks!

DoppeDee · Answer 1 · 2016-10-12T16:06:36.650

0

There are different definitions of the Fourier Transform yielding different results.

Type the integral directly into Wolfram Alpha and you will get your result, but divided by -4.

edited Oct 12 '16 at 16:06

answered Oct 12 '16 at 16:01

DoppeDee

65

I see, can you explain what you mean by "different definitions of the f.t yielding different results" please? – kamikazi Oct 12 '16 at 21:32
Since I believe we are in the same class, you are probably using Folland's book. Look at the bottom of page 223. I believe that Wolfram Alpha uses one of those definitions (or maybe some other entirely!). – DoppeDee Oct 12 '16 at 22:26

Compute the fourier transform of the function $\ f(x)=x^2e^{-x^2} $

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