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The infinite series:

1 2 a0 + ∞ ∑ n=1  ( an cos(nx) + bn sin(nx) )

Many periodic function can be represented using such series, by an appropriate choice of the coefficients. a_n, b_n.

The expansion of a periodic function f in terms of trigonometric functions: if f has period 2L and the Dirichlet conditions are satisfied (and quite possibly if not) then

f(x) = 1 2 a0+ ∞ ∑ n=1  an cos n πx L +bn sin n πx L

where the Fourier coefficients a_n and b_n are given by
a_n = [1/L] ∫₀^2Lf(x) cos[n πx/L]dx,
b_n = [1/L] ∫₀^2Lf(x) sin[n πx/L] dx.
By periodicity we may exchange ∫₀^2L with an integral across any interval of length 2L .
Alternatively,

f(x) = ∞ ∑ n=− ∞  cn e[in πx/L] dx

where

cn = 1 2L ⌠ ⌡ 2L 0  f(x)e− [in πx/L] dx.