If f(x) is an odd function, then a_n = 0 and the Fourier series collapses to f(x) = sum_(n = 1)^∞ b_n sin(n x), where b_n | = | 1/π integral_(-π)^π f(x) sin(n x) d x | = | 2/π integral_0^π f(x) sin(n x) d x for n = 1, 2, 3, .... The last equality is true because f(x) sin(n x) | = | [-f(-x)][-sin(-n x)] | = | f(-x) sin(-n x).

Fourier cosine series | Fourier series | Fourier sine transform