Fourier series

In mathematics, the Fourier series, named after Joseph Fourier (1768—1830), of a complex-valued periodic function f of a real variable, is an infinite series

$\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi inx/T}$

defined by

$c_{n}={\frac {1}{T}}\int _{0}^{T}f(x)\exp \left({\frac {-2\pi inx}{T}}\right)\,dx,$

where T is the period of f.

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

$f(x)=\sum _{n=-\infty }^{\infty }c_{n}e^{2\pi inx/T},$

and usually do not worry too much about the conditions to be imposed on the arbitrary function f(x) of period T for this expansion to converge to it.