How do you integrate Gaussian integral with contour integration method?

Question

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method?

I do not even know how to setup the problem.

As a start : Consider the contour integral $$ \oint_C e^{i\pi z^2}\csc\pi z\ dz $$ where $C$ is the two parallel lines passing through $x=\pm\dfrac12$, making an angle of $\dfrac\pi4$ with respect to the real axis, and closed with vertical lines at $x=\pm R$, where we will take the limit $R\to\infty$ — Tunk-Fey, Aug 01 '14 at 18:27
@Tunk-Fey That is one of the ways to evaluate it using a parallelogram. But a rectangle can be used as well. — Random Variable, Aug 01 '14 at 20:19
@RandomVariable I didn't know that. Anyway, I'm still learning contour and residue integral so could you refer me a book(s) for a beginner to learn those methods. I'm having trouble to understand how to choose the appropriate contour for a specific integral. Thanks. — Tunk-Fey, Aug 02 '14 at 07:25

score 5 · Answer 1 · edited Jun 01 '16 at 21:47

Sine the integrand is an even function then we have

$$ I = \int_{-\infty}^{\infty} e^{-x^2} dx = {2}\int_{0}^{\infty} e^{-x^2} dx. $$

Making the change of variables $u=x^2$ the integral under consideration becomes

$$ I = \int_{0}^{\infty} \frac{e^{-u}}{\sqrt{u}} du.$$

So we can consider the complex integral

$$ \int_{C}\frac{e^{-z}}{\sqrt{z}} dz $$

Now you need to choose the right contour, noting that we have $z=0$ as a branch point.

1 Answers1