# Fourier sine and cosine series

In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.

## Notation

In this article, f denotes a real valued function on $[0,L]$.

## Sine series

The Fourier sine series of f is defined to be

$\sum_{n=1}^\infty c_n \sin \frac{n\pi x}{L}$

where

$c_n=\frac{2}{L}\int_0^L f(x) \sin \frac{n\pi x}{L} \, dx, n\in \mathbb{N}$.

If f is continuous and $f(0)=f(L)=0$, then the Fourier sine series of f is equal to f on $[0,L]$, odd, and periodic with period $2L$.

## Cosine series

The Fourier cosine series is defined to be

$\frac{c_0}{2} + \sum_{n=1}^\infty c_n \cos \frac{n\pi x}{L}$

where

$c_n=\frac{2}{L}\int_0^L f(x) \cos \frac{n\pi x}{L} \, dx, n\in \mathbb{N}_0$.

If f is continuous, then the Fourier cosine series of f is equal to f on $[0,L]$, even, and periodic with period $2L$.

## Remarks

This notion can be generalized to functions which are not continuous.