Fourier transform $G(f)$ of a Gaussian signal

Question

Using the relationship between the width of a signal and that of its Fourier transform, find the Fourier transform of $g (t) = e^{- \pi a t^2}$, where $a$ is a real positive number.

Answer 1

This Fourier transform is available in every table of them, or through a simple integral:

$$G(\omega) = \frac{e^{-\frac{\omega ^2}{4 \pi a}}}{\sqrt{2 \pi } \sqrt{a}}$$

For $a=1$: