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Fourier series

Fourier series can be used to approximate a general periodic function, even with discontinuities. It does this by using the sum of continuous sine and cosine waves.

Summary of mathematics used

The Fourier series of the periodic function $f (x)$ defined on the domain $[- π, π]$ such that $f (x) + f (x + 2 π)$ is given by:

$f (x) = \frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} \cos n x + b_{n} \sin n x)$

where

$a_{0} = \frac{1}{π} \int_{- π}^{π} f (x) d x$

and

$a_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \cos n x d x, and b_{n} = \frac{1}{π} \int_{- π}^{π} f (x) \sin n x d x for n = 1, 2, \dots .$

This formula can be extended to functions with an arbitrary period. In addition, any function defined on an interval $[a, b]$ can be extended to a periodic function, with period $b - a$ , for which a Fourier series can be calculated. For more details on the mathematics underlying Fourier series see https://mathworld.wolfram.com/FourierSeries.html and the references contained within it.