A uniformly convergent series for Sturm-Liouville eigenvalues

For the regular Sturm-Liouville problem with equation $y” + (\lambda - q(x))y = 0$ on $0 \le x \le \pi$, there are well-known asymptotic expansions for the eigenvalues and eigenfunctions. We show that these asymptotic expansions can be replaced by convergent series for sufficiently large eigenvalues. Convergence is uniform on the interval $0 \le x \le \pi$ and uniform with respect to the eigenvalues, in the sense that a single majorant bounds all series. The basic idea is to replace the asymptotic results, which use an expansion of powers of ${n^{ - 1}}or{(n + 1/2)^{ - 1}}$ for integers $n$, by a series in powers of ${\mu ^{ - 1}}$, where ${\mu ^2}$ is an eigenvalue for the corresponding constant coefficient Sturm-Liouville problem with equation $y” + \lambda y = 0$.