Overview
The Fourier transform is given by
{% \hat{f}(s) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} f(t) e^{ist} dt = \frac{1}{\sqrt{2 \pi}} \int f(t) (cos(st)+ i sin(st)) dt %}
The inverse Fourier transform is used to recover the original function {% f(t) %} from the Fourier transform {% \hat{f}(s) %}.
{% f(t) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} \hat{f}(t) e^{-ist} ds %}