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The FourierTransform doesn't seem to work on the following function:

    fun[x_, y_, t_] = 
  1/((x - 4 y) (x + 4 y) (x^2 + 2 y^2))
    E^(-(1/4)
      t (3 x + Sqrt[
      x^2 - 16 y^2])) (-E^(1/4 t (3 x + Sqrt[x^2 - 16 y^2]))
        x^2 (x - 4 y) (x + 4 y) + x^3 Sqrt[x^2 - 16 y^2] - 
     E^(1/2 t Sqrt[x^2 - 16 y^2]) x^3 Sqrt[x^2 - 16 y^2] + 
     5 x y^2 Sqrt[x^2 - 16 y^2] - 
     5 E^(1/2 t Sqrt[x^2 - 16 y^2]) x y^2 Sqrt[
      x^2 - 16 y^2] + (1 + E^(1/2 t Sqrt[x^2 - 16 y^2])) (x - 
        4 y) (x + 4 y) (x^2 + y^2));

Doing FourierTransform[fun[x, y, t], t, \[Omega]] simply returns the same thing (i.e., FourierTransform[expression of the function]. How can I calculate the FourierTransform of this function and plot it?

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    $\begingroup$ Using Expand beforehand might work. $\endgroup$
    – Julien Kluge
    May 25, 2021 at 19:34
  • $\begingroup$ Thanks @JulienKluge, it did give an expression. But I am not able to plot it with respect to $\omega$! $\endgroup$
    – Mike
    May 25, 2021 at 19:55
  • $\begingroup$ To plot it, you will need to assign numerical values to x and y. If its imaginary part is non-zero, you will need to convert it to real numbers (e.g. using Abs, Re, Im or ReIm) $\endgroup$
    – mikado
    May 25, 2021 at 19:59
  • $\begingroup$ I tried that in the first place. The problem seems to come from the DiracDelta in the expression of FourierTransform. $\endgroup$
    – Mike
    May 25, 2021 at 20:03
  • $\begingroup$ I think for most of values of x and y that FourierTransform simply does not exist. For example, putting {x -> 1, y -> 1}], we otain a meaningless result -8 I Sqrt[(2 \[Pi])/15] DiracDelta[-3 I + Sqrt[15] - 4 s] + 8/3 Sqrt[2 \[Pi]] DiracDelta[-3 I + Sqrt[15] - 4 s] - 1/3 Sqrt[2 \[Pi]] DiracDelta[s] + 8 I Sqrt[(2 \[Pi])/15] DiracDelta[3 I + Sqrt[15] + 4 s] + 8/3 Sqrt[2 \[Pi]] DiracDelta[3 I + Sqrt[15] + 4 s]. Here s instead of \[Omega] is used. Putting {x -> 5, y -> 1}, one obtains -(25/27) + 76/27 E^(-9 t/2) - (8 E^(-3 t))/9. Its FT does not exist. $\endgroup$
    – user64494
    May 26, 2021 at 5:05

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