For a non-periodic function , the corresponding Fourier Integral can be written as:


Or, by plugging and into the first equation, one can write the Fourier Integral of in a single expression:

Fourier Integral Mean Value Theorem:

If a function is piecewise continuous and converges, then the Fourier Integral of converges to the mean value at point where both left-hand and right-hand first derivatives of exist.