Fourier series

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

This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

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.