As far as I know this is the formula for FT : 
On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please enlighten me?
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Anarkie
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The cosine comes into play because the Gaussian is even and so integrating it against sine gives 0. – Cameron Williams Jan 25 '14 at 21:10
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For finding the Fourier transform, just use Gauss integrals. See here. – Mhenni Benghorbal Jan 25 '14 at 21:24
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@CameronWilliams there is no sine ... – Anarkie Jan 25 '14 at 21:39
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@MhenniBenghorbal there is no cos(wt) on the link you gave. – Anarkie Jan 25 '14 at 21:40
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@Anarkie: Use the identity $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$. – Mhenni Benghorbal Jan 25 '14 at 21:43
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2You are aware that $e^{ix} = \cos x + i\sin x$, right? – Cameron Williams Jan 25 '14 at 21:54
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@CameronWilliams because it is even the isinx vanishes? – Anarkie Jan 25 '14 at 21:57
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Yep. Integrating an even function against an odd function over a symmetric interval gives zero. – Cameron Williams Jan 25 '14 at 21:58
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@CameronWilliams There are zillions of formulas I need to keep in mind, everything is equal to everything in one way ... facepalm... But thanks a lot... – Anarkie Jan 25 '14 at 21:59