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.

Answer 1

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.