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

Asked Oct 03 '16 at 17:05

Active Oct 03 '16 at 17:05

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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?

asked Oct 03 '16 at 17:05

z.v.

2

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 03 '16 at 17:29
http://math.stackexchange.com/questions/1266856/gaussian-integral-using-contour-integration-with-a-parallelogram-contour/1267512#1267512 – Ron Gordon Oct 03 '16 at 17:31
I think a closed form solution might be available if one replaces $\exp(ibx)$ by $\cos(bx)$ – tired Oct 04 '16 at 07:39
but without residue theorem – tired Oct 04 '16 at 07:43

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

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