Orthogonality of Sturm-Liouville eigenfunctions
= Orthogonality of Sturm-Liouville eigenfunctions
{title2=$\int_a^b w\overline{y_m}y_n\,dx=0$}
For a regular <self-adjoint> <Sturm-Liouville problem> with positive weight $w$ and self-adjoint boundary conditions, <eigenfunctions> with distinct <eigenvalues> are <orthogonal> in the weighted <inner product> $\int w\overline uv$. <Integration by parts> proves this by equating $\int\overline u Lv$ and $\int\overline{Lu}v$.