Applied Mathematics/Fourier Integral Transforms

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

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

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.