TITLE: $\int_{-\infty}^{+\infty} e^{-x^2} dx$ with complex analysis
QUESTION [105 upvotes]: Inspired by this recently closed question, I'm curious whether there's a way to do the Gaussian integral using techniques in complex analysis such as contour integrals.
I am aware of the calculation using polar coordinates and have seen other derivations. But I don't think I've ever seen it done with methods from complex analysis. I am ignorant enough about complex analysis to believe it can somehow be done without knowing how it would be done.

REPLY [15 votes]: $\text{MR}1641980\, (99\text{h}:26002)$
Desbrow, Darrell.
On Evaluating $\displaystyle\int_{-\infty}^\infty e^{ax(x-2b)}\, dx$ by Contour Integration Round a Parallelogram. 
Amer. Math. Monthly $105\, (1998)$, no. $8,\, 726–731$.