Fourier series

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

Fourier series express a piecewise continuous, bounded, periodic function as a linear combination of orthogonal sine and cosine functions.

The Fourier series of a function f(t) is of the form:

$f(t)={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }[a_{n}\cos(\omega _{n}t)+b_{n}\sin(\omega _{n}t)]$

where, n is an integer and

$\omega _{n}=n{\frac {2\pi }{T}}$

$a_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\cos(\omega _{n}t)\,dt$

$b_{n}={\frac {2}{T}}\int _{t_{1}}^{t_{2}}f(t)\sin(\omega _{n}t)\,dt$

Fourier series can be generalized to Fourier transformations for other classes of functions, for instance square integrable functions.