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Re: FourierTransform and removable singularities


Roman <rschmied at gmail.com> 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.

In[62]:=
Limit[FourierTransform[DiracDelta[t] Sin[t - c]/(t - c), t, w], c -> 0]

Out[62]=
1/Sqrt[2*Pi]

2)

In[63]:=
FourierTransform[DiracDelta[t](Sin[t] - t)/t + DiracDelta[t], t, w]

Out[63]=
1/Sqrt[2*Pi]

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.

In[64]:=
Simplify[diracDelta[t] (Sin[t] - t)/t + diracDelta[t]]

Out[64]=
(diracDelta[t]*Sin[t])/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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