Applied Mathematics/Fourier Integral Transforms

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

suppose

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

Then we have the functions below.

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

This function $f(t)$ is referred to as Fourier integral.

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

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