# Mathematica does not calculate inverse fourier transformation or the convolution of two functions

I am trying to take convolution of a Gaussian function with an exponential function. Mathematica couldn't calculate it.

g2[τ_] :=  1 + μ*Exp[-τ/τpop] - (μ + 1)
Cos[Ω*τ] Exp[-τ/τcoh];

detectorRes[τ_] := Exp[-τ^2/(2*σ^2)]

Convolve[g2[τ],detectorRes[τ],τ,t,
Assumptions->{μ ∈ Reals && τpop ∈
Reals && τcoh ∈
Reals && w ∈ Reals && τ ∈ Reals && Ω > 0}]


As it was not working, I tried taking Fourier of the functions, multiply them and then take the inversefourier of it to obtained the desired result. Unfortunately, now Mathematica is unable to take the inversefourier of the final function.

gw = Assuming[{μ ∈ Reals && τpop ∈ Reals && τcoh ∈ Reals
&& w ∈ Reals && τ ∈ Reals && Ω > 0},
FourierTransform[g2[τ], τ, w]];

responsew = Assuming[{σ > 0 && τ ∈ Reals && w ∈ Reals},
FourierTransform[detectorRes[τ], τ, w]];

Assuming[{ μ ∈ Reals && τpop ∈
Reals && τcoh ∈
Reals && w ∈ Reals && τ ∈ Reals && Ω >
0 && σ > 0, w ∈ Reals},
InverseFourierTransform[gw responsew, w, t]]


Any suggestion would be highly appreciated.

NOTE: Ok, I figured out the issue. For some strange reason, mathematica doesn't like to take integration with Sin, Cosine functions. Instead, if you convert them into complex exponents and do the integration, mathematica immediately spits out the results.

• Thanks ybeltukov for the edit. I am new to stackexchange, so I am not familiar with the formatting. – nepaliketo Sep 29 '14 at 18:27
• You are welcome! There was a typo in the first formula (Assumptions instead of Assumption). Now it works for me. Could you check it? – ybeltukov Sep 29 '14 at 18:30
• Do you get the convolution of the functions? – nepaliketo Sep 29 '14 at 18:39
• – ybeltukov Sep 29 '14 at 19:35
• Oh yes I also get those two terms, but the third term doesn't get calculated. I get this, -(Sqrt[2 [Pi]]/Sqrt[-(1/[Sigma]^2)]) - ( E^(-(([Sigma]^2 + 2 [Tau] [Tau]pop)/(2 [Tau]pop^2))) Sqrt[ 2 [Pi]] [Mu])/Sqrt[-( 1/[Sigma]^2)] + (1 + [Mu]) Convolve[ E^(-(t/[Tau]coh)) Cos[t [CapitalOmega]], E^(t^2/(2 [Sigma]^2)), t, [Tau], Assumptions -> {[Mu] [Element] Reals && [Tau]pop [Element] Reals && [Tau]coh [Element] Reals && w [Element] Reals && [Tau] [Element] Reals && [CapitalOmega] > 0 && [CapitalOmega] [Element] Reals}] – nepaliketo Sep 29 '14 at 19:51