Services & ResourcesWolfram ForumsMathGroup Archive

[Date Index] [Thread Index] [Author Index]

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"



(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 4.0,
MathReader 4.0, or any compatible application. The data for the notebook 
starts with the line containing stars above.

To get the notebook into a Mathematica-compatible application, do one of 
the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[      2846,         90]*)
(*NotebookOutlinePosition[      3484,        113]*)
(*  CellTagsIndexPosition[      3440,        109]*)
(*WindowFrame->Normal*)



Notebook[{

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"]
}, Open  ]]
},
FrontEndVersion->"4.0 for X",
ScreenRectangle->{{0, 800}, {0, 600}},
WindowSize->{620, 455},
WindowMargins->{{Automatic, 58}, {Automatic, 15}}
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{

Cell[CellGroupData[{
Cell[1739, 51, 69, 0, 154, "Title"],
Cell[1811, 53, 36, 0, 32, "Text"],
Cell[1850, 55, 129, 2, 27, "Input"],
Cell[1982, 59, 179, 4, 59, "Input"],
Cell[2164, 65, 93, 3, 32, "Text"],
Cell[2260, 70, 217, 5, 75, "Input"],
Cell[2480, 77, 131, 3, 27, "Input"],
Cell[2614, 82, 60, 0, 32, "Text"],
Cell[2677, 84, 153, 3, 27, "Input"]
}, Open  ]]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)


--------------B5014080F33D51D9BAE03B21--