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I am able to compute the Fourier transform of the general form Exp[-Sqrt[t]] as shown here:

FourierTransform[E^(G (-Sqrt[ B^2 F Abs[t]]))/G, t, ω, 
  Assumptions -> 
    {A >= 0, B >= 0, F >= 0, G >= 0, 
     A ∈ Reals, B ∈ Reals, F ∈ Reals, G ∈ Reals, t ∈ Reals}]

But when I add a constant into the square root term, Exp[-Sqrt[const + t]], I do not get a result.

FourierTransform[E^(G (-Sqrt[A B + B^2 F Abs[t]]))/G, t, ω, 
   Assumptions -> 
     {A >= 0, B >= 0, F >= 0, G >= 0, 
      A ∈ Reals, B ∈ Reals, F ∈ Reals, G ∈ Reals, t ∈ Reals}]

I've also attached a screenshot of the outcome below. Does anybody have any suggestions has to how to make the second expression workable? Thanks.

enter image description here

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  • $\begingroup$ it might take longer time as with me $\endgroup$ – Alrubaie Mar 29 at 17:19
  • $\begingroup$ the second takes so loong i think there's problem $\endgroup$ – Alrubaie Mar 29 at 17:21
  • 1
    $\begingroup$ The explicit assumptions that your constants are real is unnecessary because they are implied by the preceding assumptions involving inequalities. $\endgroup$ – m_goldberg Mar 29 at 17:46
  • $\begingroup$ even trying this then plug it k in Fourier still dosn't work! k = FullSimplify[ ComplexExpand[ Re[Refine[ E^(G (-Sqrt[A*B + B^2 F Abs[t]]))/G, {A >= 0, B >= 0, F >= 0, G >= 0, Element[A, Reals], Element[B, Reals], Element[F, Reals], Element[G, Reals], Element[t, Reals] }]]]] $\endgroup$ – Alrubaie Mar 29 at 17:49
  • $\begingroup$ it takes so loong $\endgroup$ – Alrubaie Mar 29 at 17:49

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