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Re: Incorrect Fourier Transforms?

• To: mathgroup at smc.vnet.net
• Subject: [mg24832] Re: Incorrect Fourier Transforms?
• From: Hendrik van Hees <h.vanhees at gsi.de>
• Date: Tue, 15 Aug 2000 03:04:04 -0400 (EDT)
• Organization: GSI Darmstadt
• References: <8n5j1q$n24@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

I have reproduced the bug. It seems to be InverseFourierTransform which
is buggy. But also FourierTransform shows the same bug. 

Again it seems to be a bug concerning elementary function theory. I do
not know how FourierTransform & Co. is implemented but since I'd use
Cauchy's theorem (residuum theorem) to calculate the back transformation
it looks as if the closing of the integration path is done the wrong way
in InverseFourierTransform (of course one has to close it on the upper
half plane for t<0 so that the contribution from the big half circle
cancels while the same is true for the lower half plane at t>0).

There is also a bug in Integrate when trying to do the fourier
transformation directly.

I attach the notebook where this defect is clearly shown. I hope the
developers of Mathematica answer this time how they want to fix that bug
and I hope also that there will be bug fixes to download from some
ftp-server (free of charge!).

-- 
Hendrik van Hees		Phone:  ++49 6159 71-2751
c/o GSI-Darmstadt SB3 3.183	Fax:    ++49 6159 71-2990
Planckstr. 1			mailto:h.vanhees at gsi.de
D-64291 Darmstadt		http://theory.gsi.de/~vanhees/index.html


 filename="fourier-bug.nb"



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Cell[CellGroupData[{
Cell["Bug in Fourier Transform and InverseFourierTransform", "Title"],

Cell["Here all looks fine:", "Text"],

Cell[BoxData[
    \(FourierTransform[UnitStep[t]*Exp[\(-b\)\ t], t, w, 
      FourierParameters \[Rule] {1, \(-1\)}]\)], "Input"],

Cell[BoxData[
    \(F[t_, b_] = 
      FullSimplify[
        InverseFourierTransform[1/\((b + I\ w)\), w, t, 
          FourierParameters \[Rule] {1, \(-1\)}], b > 0]\)], "Input"],

Cell["\<\
Surprisingly for a fixed value of b the same calculation goes wrong\
\
\>", "Text"],

Cell[BoxData[{
    \(\(a = 2;\)\), "\[IndentingNewLine]", 
    \(g[w_] = 
      FullSimplify[
        FourierTransform[UnitStep[t]*Exp[\(-a\)\ t], t, w, 
          FourierParameters \[Rule] {1, \(-1\)}]]\)}], "Input"],

Cell[BoxData[
    \(G[t_] = 
      InverseFourierTransform[g[w], w, t, 
        FourierParameters \[Rule] {1, \(-1\)}]\)], "Input"],

Cell["Using  Integrate produces also wrong output:", "Text"],

Cell[BoxData[
    \(Integrate[
      Exp[\(-I\)\ om\ t]/\((\(-I\)/\((\(-2\)\ I\  + 
                om)\))\), {om, \(-Infinity\), Infinity}]\)], "Input"]
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