Unexpected result from FourierTransform?

Questions
,
1

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,

10139FT2Theta.png
10139FT2Theta.png619×314 6.76 KB

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.

2

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.

3

Thank you very much for your reply.

4

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))*)
5

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!