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MathGroup Archive 1998

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Re: FourierTransform on UnitStep

  • To: mathgroup at smc.vnet.net
  • Subject: [mg14503] Re: FourierTransform on UnitStep
  • From: Edward Neuman <edneuman at siu.edu>
  • Date: Fri, 23 Oct 1998 20:59:21 -0400
  • Organization: SIUC
  • References: <70k58j$eon@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Normand Leclerc wrote:

> 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

Normand,

There is no bug in the `FourierTransform` package. Definition of the
integral Fourier transform leaves a room for specification of the
Fourier frequency constant. In some books it is defined either as 1 or
-1 or 2 or -2.
Try this:

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

In[3]:=
Options[FourierTransform]
Out[3]=
{FourierFrequencyConstant -> 1,
  FourierOverallConstant -> 1, DefiniteIntegral -> False,
  Assumptions -> {}}

Change the Fourier frequency constant to -1.

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

Out[4]=
-(I/w) + Pi*DiracDelta[w]

Edward Neuman



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