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# Unexpected result from FourierTransform?

Posted 1 year 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, 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.
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Posted 1 year ago
 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 1 year 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 1 year 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 1 year ago