For a periodic function with fundamental period , the corresponding Fourier Series representation is given by:

The Fourier Coefficients and can be determined from the following integrals:

where can be interpreted as the average value of over the interval .

Also Note that the above integration interval from - to can actually be any interval of length , such as from 0 to , which may be more convenient in some cases.

Mean Value Convergence Theorem:

If a periodic function with period is piecewise continuous over the interval , the Fourier Series of converges to the mean value at point where both the left-hand and right-hand first derivatives of exist.

Important: For non-periodic functions, one can argue that they are periodic with an infinite period, that is, . The Fourier Series then becomes the Fourier Integral.

Complex Form of Fourier Series

If one uses the complex forms of and , the Fourier Series of function becomes:


Fourier Series of Selected Functions