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MathGroup Archive 1998

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Re: FourierTransform on UnitStep

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14482] Re: [mg14449] FourierTransform on UnitStep
  • From: David Withoff <withoff>
  • Date: Fri, 23 Oct 1998 20:59:01 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

> Hi,
> 
>   I am a new EE Student and I'm just begining with Fourier transforms. I
> was using Mathematica at school when I saw something I just couldn't
> figure out.
> 
>   FourierTransform[UnitStep[t],t,w] gives me I/w + pi DiracDelta[w]
> 
>   I learned that it should give me pi DiracDelta[w]-I/w or 1/Iw + pi
> DiracDelta[w].
> 
> Am I missing something or is there a bug in the FourierTransform
> package?  They are using Mathematica 3.0
> 
> Normand Leclerc
> lecn1306 at ele.etsmtl.ca

There are several equivalent definitions of a Fourier integral
transform.  The one that you are probably using is available in
Mathematica by setting the FourierFrequencyConstant option.

In[1]:= << Calculus`FourierTransform`

In[2]:= ?FourierTransform
FourierTransform[expr, t, w] gives a function of w, which is the Fourier
   transform of expr, a function of t.  It is defined by
   FourierTransform[expr, t, w] = FourierOverallConstant *
   Integrate[Exp[FourierFrequencyConstant I w t] expr, {t, -Infinity,
   Infinity}].

In[3]:= FourierTransform[UnitStep[t],t,w]

        I
Out[3]= - + Pi DiracDelta[w]
        w

In[4]:= FourierTransform[UnitStep[t],t,w,FourierFrequencyConstant -> -1]

        -I
Out[4]= -- + Pi DiracDelta[w]
        w

Dave Withoff
Wolfram Research


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