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.

In this article f denotes a real valued function on $\mathbb {R}$ 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

f(x)=\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(x) is an even function, 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,n\in \mathbb {N} _{0}

.

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