Overview
The Gaussian function is given by
{% f(x) = e^{-x^2/a^2} %}
The
Fourier Transform
of the Gaussian is then given by
{% \hat{f}(p) = \int_{-\infty}^{\infty} e^{-x^2/a^2} e^{2 \pi i p x} dx %}
The result is another Gaussian function
{% f(p) = e^{-\pi^2 p^2 a^2} \int_{-\infty}^{\infty} e^{-(x/a - \pi i p a)^2} dx = a \sqrt{\pi} e^{- \pi^2 a^2 p^2} %}