Applied Mathematics/Fourier Integral Transforms

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

suppose

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

Then we have the functions below.

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

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

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

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