Fourier series

In mathematics, the Fourier series, named after Joseph Fourier (1768&mdash;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.