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Unexpected result from FourierTransform?

Posted 4 months ago

I have try to do a Fourier tranform with Mathematica for θ(t1^2 + t2^2 + t3^2 - a), where a>0 and θ(x) is the Heaviside theta function. It seems that the Mathematica gave an incorrect answer. The code was,

FourierTransform[
 HeavisideTheta[t1^2 + t2^2 + t3^2 - a], {t1, t2, t3}, {ω1, ω2, ω3}, 
 Assumptions -> a > 0]

and the result gaven by Mathematica was (2π)^(3/2) δ(ω1) δ(ω2) δ(ω3). However, as definition, the Fourier transform for θ(t1^2 + t2^2 + t3^2-a) is,

enter image description here

where ω={ω1,ω2,ω3}.
The last item of I2(ω) is not zero. I'd like to know from which the problem arisen.

Thank you very much for your help.

POSTED BY: Jianhua Yang
4 Replies

Maybe:

 InverseLaplaceTransform[
  FourierTransform[
   LaplaceTransform[HeavisideTheta[t1^2 + t2^2 + t3^2 - a], a, s], {t1,
     t2, t3}, {\[Omega]1, \[Omega]2, \[Omega]3}, 
   Assumptions -> s > 0], s, a]

  (*2 Sqrt[2] \[Pi]^(3/2)
     DiracDelta[\[Omega]1] DiracDelta[\[Omega]2] DiracDelta[\[Omega]3] \
  - (a^(3/4) (2 Cosh[
        Sqrt[a] Sqrt[-\[Omega]1^2 - \[Omega]2^2 - \[Omega]3^2]] - (
      2 Sinh[Sqrt[a] Sqrt[-\[Omega]1^2 - \[Omega]2^2 - \[Omega]3^2]])/(
      Sqrt[a] Sqrt[-\[Omega]1^2 - \[Omega]2^2 - \[Omega]3^2])))/(
   Sqrt[2 \[Pi]] Sqrt[
    Sqrt[a] Sqrt[-\[Omega]1^2 - \[Omega]2^2 - \[Omega]3^2]] \
  (-\[Omega]1^2 - \[Omega]2^2 - \[Omega]3^2)^(3/4))*)
POSTED BY: Mariusz Iwaniuk
Posted 4 months ago

It's a resourceful way to solve the problem. It can be employed to identify the result of integrating the residual part of I(ω) directly. Yes, they gave the same answer. Thanks!

POSTED BY: Jianhua Yang
Posted 4 months ago

My Mathematica 13.2 result is mathematical nonsense

FourierTransform[  HeavisideTheta[t1^2 + t2^2 + t3^2 - a],
       {t1, t2,  t3},
       {\[Omega]1, \[Omega]2, \[Omega]3} ]

2 Sqrt[2] \[Pi]^(3/2) DiracDelta[\[Omega]1] DiracDelta[\[Omega]2] DiracDelta[\[Omega]3]

The problem is the transformation from cartesian to spherical coordinates.

By spherical symmetry, the problem can be simplified to the simple choice of the result vector

\[Omega]  =  {0 ,0 , \Omega3} 

Then you have to calculate the integral over the z-axis with

z=cos theta,  dz = sin \theta d\theta

2 Pi \Integrate[  t^2 * Exp[ I \[Omega  ] t  z ]  ,{z,-1,1}, {t,a, Infinity}

Keep in mind, this is a spherical distribution in spherical coordinates, where \CapitalOmega is the absolute value of the cartesian vector \omega, tas a map onto the complex numbers to be applied onto smooth functions with compact support, that projects to the mass outside of a sphere.

In order to get the limits, a cut-off factor of the t-integral has to be introduced

t -> exp[-t^2/s^2]/\sqrt[2 PI s^2] 

The

\int_a^Infinity  t^2 Exp[ i \omega t ]

does not make any sense.

Perhaps, anywhere the idea is introduced errorneously, that

Fourier[  t^n   f[ t ] , t ,  omega  ]  =  D[  Fourier[  f[ t  ] ,  t,  \omega  ] ,  { \omega, n }  ] 

The formula is not applicable of course, if a non-constant measure of integration is part of the game.

POSTED BY: Roland Franzius
Posted 4 months ago

Thank you very much for your reply.

POSTED BY: Jianhua Yang
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