Re:Fourier Transform PDF Characteristic Function

• To: mathgroup at smc.vnet.net
• Subject: [mg14994] Re:[mg14955] Fourier Transform PDF Characteristic Function
• From: "Tomas Garza" <tgarza at mail.internet.com.mx>
• Date: Wed, 2 Dec 1998 03:59:14 -0500
• Sender: owner-wri-mathgroup at wolfram.com

```Yves: I am pasting a copy of a session in my machine (PC, Windows 95,
Mathematica 3.0), where the InverseFourierTransform gives the correct
result. Perhaps you should contact Wolfram support.

In[8]:=
<<Calculus`FourierTransform`
In[14]:=
ft=FourierTransform[\!\(E\^\(-\(x\^2\/2\)\)\/\ at \(2\ \[Pi]\)\),x,t]
Out[14]=
\!\(E\^\(-\(t\^2\/2\)\)\)
In[15]:=
InverseFourierTransform[ft,t,x]
Out[15]=
\!\(E\^\(-\(x\^2\/2\)\)\/\ at \(2\ \[Pi]\)\)

Good luck,

Tomas Garza
Mexico City

Yves Gavreau wrote:

>I'm trying to find the PDF from a Characteristic Function using
>Mathematica. I'm not a mathematician just a curious guy. Here is the
>problem (this is a paste from the notebook).

----------------------------------------------------------------------------
--------------------------------------------------------
<<Statistics`NormalDistribution`
<<Calculus`FourierTransform`
>This is the CharacteristicFunction of a NormalDistribution In[3]:=
>CF=CharacteristicFunction[NormalDistribution[\[Mu],\[Sigma]],t] Out[3]=
\!\(E\^\(I\ t\ \[Mu] - \(t\^2\ \[Sigma]\^2\)\/2\)\) Also this is suppose
>to be the FourierTransform of a NormalDistribution PDF In[5]:=
ft=FourierTransform[PDF[NormalDistribution[\[Mu],\[Sigma]],x],x,t]
Out[5]=
\!\(E\^\(I\ t\ \[Mu] - \(t\^2\ \[Sigma]\^2\)\/2\)\) Now with the
InverseFourierTransform we are suppose to get back the
NormalDistribution PDF
In[6]:=
InverseFourierTransform[ft,t,x]
Out[6]=
\!\(InverseFourierTransform[E\^\(I\ t\ \[Mu] - \(t\^2\
\[Sigma]\^2\)\/2\), t,
x]\)
>As you can see it doesn't give back the NormalDistribution PDF

```

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