Fourier sine and cosine series
# 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 $\mathbb {R}$ which is periodic with period 2L.

## Sine series

If f is an odd function with period $2L$, then the Fourier Half Range sine series of f is defined to be

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

which is just a form of complete Fourier series with the only difference that $a_{0}$ and $a_{n}$ are zero, and the series is defined for half of the interval.

In the formula we have

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

## Cosine series

If f is an even function with a period $2L$, then the Fourier cosine series is defined to be

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

where

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

## Remarks

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

