Fourier Cosine Series

If $f(x)$ is an Even Function, then $b_n = 0$ and the Fourier Series collapses to

$$f(x) = {\textstyle{1\over 2}}a_0 + \sum_{n=1}^\infty a_n\cos(nx),$$ (1)

where

$$a_0 = {1\over\pi} \int_{-\pi}^\pi f(x)\,dx = {2\over\pi} \int^\pi_0 f(x)\,dx$$ (2)

$$a_n = {1\over\pi} \int_{-\pi}^\pi f(x)\cos(nx)\,dx$$

$$= {2\over\pi} \int^\pi_0 f(x)\cos(nx)\,dx,$$ (3)

where the last equality is true because

$$f(x)\cos(nx) = f(-x)\cos(-nx).$$ (4)

Letting the range go to $L$,

$$a_0 = {2\over L} \int^L_0 f(x)\,dx$$ (5)

$$a_n = {2\over L} \int^L_0 f(x)\cos\left({n\pi x\over L}\right)\,dx.$$ (6)

See also Even Function, Fourier Cosine Transform, Fourier Series, Fourier Sine Series