Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 10 / iii / Solution

Use the weighted inner productMultiply the equation for $R_n$ by ${\overline{R}}_m$, subtract the complex conjugate equation for $R_m$ times $R_n$, and integrate. Since the eigenvalues are real,The boundary term vanishes as before. Thus eigenfunctions with distinct eigenvalues are orthogonal. The full disk inner product differs only by the angular factor $2\pi$.