Fourier integral representations
The Fourier integral is the continuous analogue of a Fourier series. This formulas can be derived from the Fourier series expansion of the function on interval as .
The substitution of and into the integral gives the following Fourier integral formulas:
The first definition can be rewritten in exponential form, leading to exponential direct and inverse Fourier transforms:
If the function is absolutely integrable on the real axis, you have the following equality: