= Solution
Use the <weighted inner product>
$$
\boxed{\langle R_m,R_n\rangle_r=\int_0^ar\overline{R_m(r)}R_n(r)\,dr.}
$$
Multiply 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,
$$
(\lambda_n-\lambda_m)\langle R_m,R_n\rangle_r=\left[r\left(\overline{R_m'}R_n-\overline R_mR_n'\right)\right]_0^a=0.
$$
The boundary term vanishes as before. Thus \b[eigenfunctions with distinct eigenvalues are orthogonal]. The full disk <inner product> differs only by the angular factor $2\pi$.
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