Odd periodic extension
= Odd periodic extension
{title2=$F(x+2L)=F(x),\quad F(-x)=-F(x)$}
Given a <function> on $(0,L)$, reflect it oddly onto $(-L,0)$ and repeat with <period> $2L$. Nonzero endpoint limits produce jumps at the joins. The resulting <Fourier series> is the <Fourier sine series> of the original <function>; its value at a jump converges to the mean of the one-sided limits under the <Dirichlet convergence theorem for Fourier series>.