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General Identities

General Mathematical Identities for Analytic Functions

General Identities

Integral representations

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: