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Re: Problem with FourierParameters

  • To: mathgroup at smc.vnet.net
  • Subject: [mg87108] Re: [mg86998] Problem with FourierParameters
  • From: Devendra Kapadia <dkapadia at wolfram.com>
  • Date: Tue, 1 Apr 2008 03:20:44 -0500 (EST)
  • References: <200803280817.DAA04820@smc.vnet.net>

On Fri, 28 Mar 2008, Peter Breitfeld wrote:

> I get a wrong result using FourierParameters:
>
> FourierTransform[f, x, w]
> InverseFourierTransform[%, w, x]
>
> gives
>
> 1/2 Sqrt[\[Pi]/2] (Sign[1 - w] + Sign[1 + w])
> Sin[x]/x
>
> which is what I expected. But with the setting of FourierParameters:
>
> FourierTransform[f, x, w, FourierParameters -> {1, -1}]
> InverseFourierTransform[%, w, x, FourierParameters -> {1, -1}]
>
> gives
>
> 1/2 \[Pi] Sign[1 - w] + 1/2 \[Pi] Sign[1 + w]     <--- This is OK
> 0                                                 <--- Bug?
>
> Other functions e.g. f=1/(1+x^2) transform correctly in both cases.
>
> Gruss Peter
>
Hello Peter,

The answer given by InverseFourierTransform for your example with
FourierParameters -> {1, -1} is indeed incorrect.

We use symbolic integration for computing the inverse transform
in both the cases given above. The incorrect answer appears to
be caused by the fact that the output from FourierTransform in
the second case is not in factored form. Hence, a partial
workaround for the problem is to use Factor in the call
to InverseFourierTransform, as shown below:

===========================

In[2]:= FourierTransform[Sin[x]/x, x, w, FourierParameters -> {1, -1}]

         Pi Sign[1 - w]   Pi Sign[1 + w]
Out[2]= -------------- + --------------
               2                2

In[3]:= InverseFourierTransform[Factor[%], w, x,
            FourierParameters -> {1, -1}]

         Sin[x]
Out[3]= ------
           x

=====================

Thank you for reporting the problem. We are sorry for the inconvenience
caused by it.

Sincerely,

Devendra Kapadia,
Wolfram Research, Inc.


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