Fourier series

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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.

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