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MathGroup Archive 2007

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Re: FourierTransform and removable singularities

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75344] Re: FourierTransform and removable singularities
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Wed, 25 Apr 2007 05:46:47 -0400 (EDT)
  • References: <f0kbmk$qvt$1@smc.vnet.net>

Hi.

The only thing that comes in my mind is to replace the Dirac delta
generalized
function with a delta secuence.

So for example, let

In[64]:=
delta[n_][x_] := n/(Sqrt[Pi]/E^((-n^2)*x^2))

Then

In[66]:=
FourierTransform[delta[n][t], t, w]
Limit[%, n -> Infinity]

Out[66]=
n/(E^(w^2/(4*n^2))*(Sqrt[n^2]*Sqrt[2*Pi]))

Out[67]=
1/Sqrt[2*Pi]

which is correct. Also as you stated

In[82]:=
({(1/Sqrt[2*Pi])*Integrate[Exp[I*w*t]*#1, {t, -Infinity, Infinity}],
FourierTransform[#1, t, w]} & )[DiracDelta[t]]

Out[82]=
{1/Sqrt[2*Pi], 1/Sqrt[2*Pi]}

Now, the example with the removable singularity

In[83]:=
FourierTransform[delta[n][t]*(Sin[t]/t), t, w]
Limit[%, n -> Infinity]

Out[83]=
(n*((-(1/2))*Sqrt[Pi/2]*Erf[(-1 + w)/(2*Sqrt[n^2])] + (1/2)*Sqrt[Pi/
2]*Erf[(1 + w)/(2*Sqrt[n^2])]))/Sqrt[Pi]

Out[84]=
1/Sqrt[2*Pi]

which is the desired (correct) result. On the contrary

In[85]:=
({(1/Sqrt[2*Pi])*Integrate[Exp[I*w*t]*#1, {t, -Infinity, Infinity}],
FourierTransform[#1, t, w]} & )[(DiracDelta[t]*Sin[t])/t]

Out[85]=
{0, 0}

which is incorect as you have already said.

I hope I helped you!

Regards
Dimitris




=CF/=C7 Roman =DD=E3=F1=E1=F8=E5:
> 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)
>
> Cheers!
> Roman.



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