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Bug introduced in 8.0.4 or earlier and fixed in 9.0.1 or earlier

I am trying to compute the inverse Fourier transform of this expression

fhat[w_] = (Exp[-2*I*w] + Exp[2*I*w])

using 

f[x_] = InverseFourierTransform[fhat[w], w, x, FourierParameters -> {1, -1}]

Here, Mathematica correctly returns 

DiracDelta[-2 + x] + DiracDelta[2 + x]

However, when (when using Mathematica 8) I try to compute this

fhat[w_] = 5*(Exp[-2*I*w] + Exp[2*I*w])

I get 0 as the inverse Fourier Transform, which is clearly incorrect. The result should be 5*(DiracDelta[-2 + x] + DiracDelta[2 + x]).

Can anyone explain to me this behavior? Do you know a way to circumvent this problem?

Improve this question
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    $\begingroup$ I guess you are in v8? I can reproduce this in this version. v9 and v10 give the correct result. $\endgroup$
    – xzczd
    Sep 17, 2014 at 12:25
  • $\begingroup$ You are right, I am in v8. So I will have to update... Is this a known bug of v8? $\endgroup$
    – Goulifet
    Sep 17, 2014 at 12:26
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    $\begingroup$ I think it's not, at least I can't find an exact duplicate. Here's just a possible related one: mathematica.stackexchange.com/q/34027/1871 $\endgroup$
    – xzczd
    Sep 17, 2014 at 12:33
  • $\begingroup$ This is clearly incorrect behaviour. $\endgroup$
    – djp
    Feb 26, 2015 at 1:19

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