The double points of Mathieu’s differential equation

Abstract:

Mathieu’s differential equation, $y'' + (a - 2q\cos 2x)y = 0$, admits of solutions of period $\pi$ or $2\pi$ for four countable sets of characteristic values, $a(q)$, which can be ordered as ${a_r}(q)$, $r = 0$, $1$, $\dots$. The power series expansions for the ${a_r}(q)$ converge up to the first double point for that order in the complex plane. [At a double point, ${a_r}(q) = {a_r} + 2(q)$.] The present work furnishes the double points for orders $r$ up to and including 15. These double points are singular points, and the usual methods of determining the characteristic values break down at a singular point. However, it was possible to determine two smooth functions in which one could interpolate for both $q$ and ${a_r}(q)$ at the singular point. The method is quite general and can be used in other problems as well.