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--