The Fourier Transform is a mathematical tool that transforms a time-domain function into a frequency-domain representation. It expresses how much of each frequency exists in a signal.
If \( f(x) \) is a function of time or space, then its Fourier Transform \( F(s) \) is given by:
\[ F(s) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i x s} dx \]
\[ f(x) \iff F(s) \]
\[ F(s) = \int_{-\infty}^{\infty} f(x) e^{-ixs} dx \quad ; \quad f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(s) e^{ixs} ds \]
\[ F(s) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ixs} dx \quad ; \quad f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(s) e^{ixs} ds \]