Re: Fourier Transform
- To: mathgroup at smc.vnet.net
- Subject: [mg13733] Re: [mg13689] Fourier Transform
- From: David Withoff <withoff>
- Date: Wed, 19 Aug 1998 01:38:24 -0400
- Sender: owner-wri-mathgroup at wolfram.com
> Let f(x) = 1/x. If f is regarded as the generalized function, then its
> Fourier transform is:
> -Pi*I*Sign[t] (see, e.g., G.B.
> Folland, "Fourier Analysis and Its Applications," p. 337).
> Using Mathematica 3.0 we get:
> In[1]:=
> << "Calculus`FourierTransform`"
>
> In[2]:=
> FourierTransform[1/x, x, t]
>
> Out[2]=
> 2*I*Pi*(-(1/2) + UnitStep[t, ZeroValue -> 1/2]).
>
> This agrees with the above result only if t = 0. Bug?
>
> Edward Neuman
You can use the FourierFrequencyConstant option to get the definition of
Fourier transform from the reference that you quoted. The result is
then equivalent to -Pi*I*Sign[t].
In[20]:= FourierTransform[1/x, x, t, FourierFrequencyConstant -> -1]
1 1 Out[20]= 2 I Pi
(-(-) + UnitStep[-t, ZeroValue -> -])
2 2
Different authors use difference choices for these constants.
Dave