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I found that 

FourierTransform[Cos[Abs[x]], x, ξ, FourierParameters -> {1, -1}]

gives 0, while

FourierTransform[Cos[x], x, ξ, FourierParameters -> {1, -1}]

yields π DiracDelta[-1 + ξ] + π DiracDelta[1 + ξ]

How does this happen?

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    $\begingroup$ What version of Mathematica are you using? I get the second answer for both statements using Windows 10, Mathematica 10.4. $\endgroup$
    – JimB
    Apr 8, 2016 at 15:07
  • $\begingroup$ @JimBaldwin, i'm using version 9. i'll check the new version, thanks~ $\endgroup$
    – davyjones
    Apr 8, 2016 at 15:28
  • $\begingroup$ @JimBaldwin, but on wolframalpha FourierTransform[Cos[Abs[t]],t,w] still results 0 $\endgroup$
    – davyjones
    Apr 8, 2016 at 15:33
  • $\begingroup$ @andre, the second result is surely ok, but why 0 in the first one? $\endgroup$
    – davyjones
    Apr 8, 2016 at 15:34
  • $\begingroup$ When I try this on v10.0, the first code returns unevaluated rather than evaluating to zero, but interestingly enough, Assuming[x \[Element] Reals, FourierTransform[Cos[Abs[x]], x, \[Xi], FourierParameters -> {1, -1}]] returns 0. $\endgroup$
    – march
    Apr 8, 2016 at 15:44

1 Answer 1

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This is not an answer.

FourierTransform[Cos[Abs[x]], x, ξ, FourierParameters -> {1, -1}]
(* π DiracDelta[-1 + ξ] + π DiracDelta[1 + ξ] *)

Assuming[x ∈ Reals, FourierTransform[Cos[Abs[x]], x, ξ, 
  FourierParameters -> {1, -1}]]
(* π DiracDelta[-1 + ξ] + π DiracDelta[1 + ξ] *)

FourierTransform[Cos[x], x, ξ, FourierParameters -> {1, -1}]
(* π DiracDelta[-1 + ξ] + π DiracDelta[1 + ξ] *)

Mathematica 10.4 on Windows 10 (x64).

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