Finding the Fourier transform with the Cauchy residue theorem or integration in the complex plane

Question

I need to find the Fourier transform of the function $f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp(-\frac{(x-\mu)^2}{2\sigma^2}), x, \mu \in \mathbb{R}, \sigma>0$ by using the Cauchy residue theorem. The Fourier transform will have the expression $$\frac{1}{\sqrt{2\pi\sigma^2}}\int_{\mathbb{R}} e^{-(x-\mu)^2/2\sigma^2 + itx}\, dx.$$ I do not see an isolated singularity except for $x \rightarrow \infty.$ Can somebody provide some hint or propose a solution ? Thanks.