# 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 ${\displaystyle \mathbb {R} }$ which is periodic with period 2L.

## Sine series

If f(x) is an odd function, then the Fourier sine series of f is defined to be

${\displaystyle f(x)=\sum _{n=1}^{\infty }c_{n}\sin {\frac {n\pi x}{L}}}$

where

${\displaystyle c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\sin {\frac {n\pi x}{L}}\,dx,n\in \mathbb {N} }$.

## Cosine series

If f(x) is an even function, then the Fourier cosine series is defined to be

${\displaystyle f(x)={\frac {c_{0}}{2}}+\sum _{n=1}^{\infty }c_{n}\cos {\frac {n\pi x}{L}}}$

where

${\displaystyle c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos {\frac {n\pi x}{L}}\,dx,n\in \mathbb {N} _{0}}$.

## Remarks

This notion can be generalized to functions which are not even or odd, but then the above formulas will look different.