Odd extension
= Odd extension
An odd extension of a function $f$ on the positive half-line is $\widetilde f(r)=f(r)$ for $r>0$ and $\widetilde f(r)=-f(-r)$ 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>, $r\phi(t,r)$ has a smooth <odd extension> whenever $\phi$ is smooth and radial.