# Contours for the integral that contains $exp(-x^2)$

I'm interested in the integral

$$\int_0^\infty dx \frac{e^{i b x - c x^2}}{x^2 + a^2}$$

where $a$, $b$ and $c$ are all positive real numbers. What contours should I choose to evaluate this integral using residue theorem? Can it even be done in this case?

• It is not trivial that it is possible, since the value of the integral for $a=c=1$ and $b\to 0^+$ is non-elementary ($\frac{\pi e}{2}\text{Erfc}(1)$). – Jack D'Aurizio Oct 3 '16 at 17:29
• math.stackexchange.com/questions/1266856/… – Ron Gordon Oct 3 '16 at 17:31
• I think a closed form solution might be available if one replaces $\exp(ibx)$ by $\cos(bx)$ – tired Oct 4 '16 at 7:39
• but without residue theorem – tired Oct 4 '16 at 7:43