Applied Mathematics/Fourier Integral Transforms

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


\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 refferred to as fourier transform as we previously learned.