Half-range Fourier cosine series ($f(x)=a_0/2+{\sum }_{n\ge 1}{a}_n\cos (n\pi x/L)$)

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.

For a positive integer $m$, take $f(x)=\sin (mx)$ on $0<x<\pi$ and form its half-range Fourier cosine series. The product-to-sum identity givesThus $a_n=4m/[\pi (m^2-n^2)]$ for opposite parity and zero for the same parity. The exceptional integral $a_m=(2/\pi ){\int }_0^{\pi }\sin (mx)\cos (mx)dx$ is zero. The constant term is $a_0/2$, not $a_0$. These coefficients represent the even periodic extension, not the original sine wave on negative arguments. Their quadratic decay gives absolute convergence.