Understanding the computation $\int_{-\infty}^{\infty}\exp(- \pi t^2)\exp(2 \pi i x t/\sqrt{2 \pi}) dt = \exp(-x^2/2)$

Question

I've been trying to figure out how the following is true, but I get stuck every time:

$$ \int_{-\infty}^{\infty}e^{- \pi t^2}e^\dfrac{2 \pi i x t}{\sqrt{2 \pi}} dt = e^{-x^2/2} $$

Answer 1

You are just stating that the gaussian function is a fixed point of the Fourier transform.
The usual ways for proving it are completing the square and apply a bit of complex analysis (contour integration along a rectangle), or define $f(x)$ as the LHS and check it fulfills a simple differential equation, by differentiation under the integral sign and integration by parts.