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Re: Fourier Transform

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93491] Re: Fourier Transform
  • From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
  • Date: Tue, 11 Nov 2008 07:46:25 -0500 (EST)
  • Organization: Uni Leipzig
  • References: <gf8rgp$poq$1@smc.vnet.net>
  • Reply-to: kuska at informatik.uni-leipzig.de

Hi,

ClearSystemCache[]
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]}, GenerateConditions -> False] // Timing

is not so different.

Regards
   Jens

Nikolaus Rath wrote:
> 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
> 
> --
>  =C2=BBIt is not worth an intelligent man's time to be in the majority.
>   By definition, there are already enough people to do that.=C2=AB
>                                                          -J.H. Hardy
> 
>   PGP fingerprint: 5B93 61F8 4EA2 E279 ABF6  02CF A9AD B7F8 AE4E 425C
> 


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