Let ${\displaystyle f(x)=(-\infty ,\infty )}$ and

suppose

${\displaystyle \int _{-\infty }^{\infty }|f(t)|dt\leq M}$.

Then we have the functions below.

${\displaystyle f(t)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}(\omega )e^{i\omega t}d\omega }$

This function ${\displaystyle f(t)}$ is referred to as Fourier integral.

${\displaystyle {\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt}$

This function ${\displaystyle {\hat {f}}(\omega )}$ is referred to as fourier transform as we previously learned.