If f(x) is an even function, then b_n = 0 and the Fourier series collapses to f(x) = 1/2 a_0 + sum_(n = 1)^∞ a_n cos(n x), where a_0 | = | 1/π integral_(-π)^π f(x) d x | = | 2/π integral_0^π f(x) d x a_n | = | 1/π integral_(-π)^π f(x) cos(n x) d x | = | 2/π integral_0^π f(x) cos(n x) d x, where the last equality is true because f(x) cos(n x) = f(-x) cos(-n x).

even function | Fourier cosine transform | Fourier series | Fourier sine series