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

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

${\displaystyle a_{0))$

${\displaystyle a_{n))$

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

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

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

Remarks

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