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MathGroup Archive 2008

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Fourier Transform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93476] Fourier Transform
  • From: Nikolaus Rath <Nikolaus at rath.org>
  • Date: Mon, 10 Nov 2008 03:31:28 -0500 (EST)

Hello,

Consider the following expression:

expr = (c^2 Sqrt[2 \[Pi]]
    DiracDelta[ky + sy] DiracDelta[sz])/(-c^2 sx^2 - c^2 sy^2 -
   c^2 sz^2 + \[Omega]^2);
$Assumptions = {{x, y, z} \[Element] Reals};
InverseFourierTransform[expr, {sx, sy, sz}, {x, y, z}] // Timing
Integrate[
  1/Sqrt[2 \[Pi]] Exp[-I sx x] Exp[-I sy y] Exp[-I sz z]
   expr, {sx, -\[Infinity], \[Infinity]}, {sy, -\[Infinity], \
\[Infinity]}, {sz, -\[Infinity], \[Infinity]}] // Timing

On my system with Mathematica 6, the explicit integration takes 3
times as long as the InverseFourierTransform and also gives several
additional required assumptions for the same result (e.g. Im[-ky^2 +
\[Omega]^2/c^2] != 0 || Re[-ky^2 + \[Omega]^2/c^2] <= 0).

How is this possible? Is Mathematica using some special tricks when
evaluating the InverseFourierTransform?


Best,

   -Nikolaus

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