# 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 is an odd function with period ${\displaystyle 2L}$, then the Fourier Half Range sine series of f is defined to be

${\displaystyle f(x)=\sum _{n=1}^{\infty }b_{n}\sin {\frac {n\pi x}{L}}}$
which is just a form of complete Fourier series with the only difference that ${\displaystyle a_{0}}$ and ${\displaystyle a_{n}}$ are zero, and the series is defined for half of the interval.

In the formula we have

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

## Cosine series

If f is an even function with a period ${\displaystyle 2L}$, 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
${\displaystyle c_{n}={\frac {2}{L}}\int _{0}^{L}f(x)\cos {\frac {n\pi x}{L}}\,dx,\quad n\in \mathbb {N} _{0}.}$

## Remarks

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