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Re: FourierTransform and removable singularities

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  • Subject: [mg75345] Re: FourierTransform and removable singularities
  • From: "David W.Cantrell" <DWCantrell at>
  • Date: Thu, 26 Apr 2007 03:27:58 -0400 (EDT)
  • References: <f0kbmk$qvt$>

Roman <rschmied at> wrote:
> It seems to me that Mathematica 5.2 is not careful enough when doing
> Fourier transforms of functions with delta functions at removable
> singularities: if you call
>     FourierTransform[DiracDelta[t], t, w]
> you get the right answer,
>     1/Sqrt[2*Pi]
> But if you call something of the sort of
>     FourierTransform[DiracDelta[t]*(Sin[t]/t), t, w]
> which has a removable singularity at the point where the Dirac delta
> function acts, the answer is zero, which is wrong.
> Does anyone know how to resolve this by reformulating the problem? (a
> workaround)

I've got two suggestions, neither of which I like.

1)  This is similar to what Jens-Peer suggested.

Limit[FourierTransform[DiracDelta[t] Sin[t - c]/(t - c), t, w], c -> 0]



FourierTransform[DiracDelta[t](Sin[t] - t)/t + DiracDelta[t], t, w]


Note that, _formally_, the argument of the transform is indeed
your DiracDelta[t]*(Sin[t]/t), as the following (using diracDelta,
rather than DiracDelta) shows.

Simplify[diracDelta[t] (Sin[t] - t)/t + diracDelta[t]]


Perhaps the best way to handle you problem would be to have the sine
cardinal function

|                {  1          if  x = 0,
|    sinc(x)  =  {
|                {  sin(x)/x   otherwise

implemented in Mathematica. But defining that function yourself, it does
not work as desired with FourierTransform.

David W. Cantrell

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