Re: Fourier Transform

*To*: mathgroup at smc.vnet.net*Subject*: [mg13774] Re: Fourier Transform*From*: David Withoff <withoff>*Date*: Mon, 24 Aug 1998 05:07:23 -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