Odd extension

An odd extension of a function $f$ on the positive half-line is $\tilde{f}(r)=f(r)$ for $r>0$ and $\tilde{f}(r)=-f(-r)$ for $r<0$. At zero it is set to zero when the boundary value is zero. Smoothness requires compatible even derivatives at zero. In the radial reduction of the three-dimensional wave equation, $r\varphi (t,r)$ has a smooth odd extension whenever $\varphi$ is smooth and radial.

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.