Orthogonality of Sturm-Liouville eigenfunctions (${\int }_a^{b}w\overline{y_m}{y}_{n}dx=0$)

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 $\int w\overline{u}v$. Integration by parts proves this by equating $\int \overline{u}Lv$ and $\int \overline{Lu}v$.