I am trying to compute the following Fourier transform
$$\int_{-\infty}^\infty\text{d}x\,e^{i k x}e^{-x^2/a^2}\frac{P(x)}{Q(x)}$$
where $\text{deg}P(x)+1\leq\text{deg}Q(x)$, and the roots of $Q(x)$ are all complex. I have tried doing integration by residues, but I have found that, when doing the natural extension $x\rightarrow z$ there is no way of defining a contour passing through infinity in which the integrand vanishes. I have also tried doing $x\rightarrow z,\,x^2\rightarrow |z|^2$, but then the integrand is not holomorphic.
Does anyone have any idea on a better extension to the complex plane or a better contour for doing the integration?