Fourier series

In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), refers to an infinite series representation of a periodic function ƒ of a real variable ξ, of period P:

$f(\xi +P)=f(\xi )\ .$

In the case of a complex-valued function ƒ(ξ), Fourier's theorem states that an infinite series, known as a Fourier series, is equivalent (in some sense) to such a function:

$f(\xi )=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi in\xi /P}$

where the coefficients {c_n} are defined by

$c_{n}={\frac {1}{P}}\int _{0}^{P}f(\xi )\exp \left({\frac {-2\pi in\xi }{P}}\right)\,d\xi \ .$

In what sense it may be said that this series converges to ƒ(ξ) is a complicated question. However, physicists being less delicate than mathematicians in these matters, simply write

$f(\xi )=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi in\xi /P}\ ,$

and usually do not worry too much about the conditions to be imposed on the arbitrary function ƒ(ξ) of period P in order that this expansion converge to the function.