Calculating a complex integral

[Robin04]

Homework Statement

$f(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dt \ e^{-\frac{t^2}{2\sigma^2}+t(i\omega-\alpha)}cos(\Omega t)$

Relevant Equations

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As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?

 

[vela]

Looks like you can express it in terms of the Fourier transform of a Gaussian.