Half-range Fourier cosine series
= Half-range Fourier cosine series
{title2=$f(x)=a_0/2+\sum_{n\geq1}a_n\cos(n\pi x/L)$}
For a function prescribed on $0<x<L$, its half-range <Fourier cosine series> is the <Fourier series> of the even extension, made $2L$-periodic. Its coefficients are $a_n=(2/L)\int_0^L f(x)\cos(n\pi x/L)\,dx$, including $n=0$. At a jump the usual piecewise-smooth convergence theorem gives the average of the two one-sided limits of the periodic extension.