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Mathieu's differential equation, y" + (a - 2q cos 2x)y 0, admits of solutions of period xr or 27r for four countable sets of characteristic values, a(q), which can be ordered as aT(q), r = O, 1, *. The power series expansions for the ar(q) converge up to the first double point for that order in the complex plane. (At a double point, ar(q) = ar+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 ar(q) at the singular point. The method is quite general and can be used in other problems as well. U 1. Introduction. Mathieu's differential equation (1.0) y" + (a-2q cos 2x)y = 0 admits of four countable sets of characteristic values, ar(q), corresponding to which the solutions y(x) are periodic, and of period wr or 27r. These four sets are associated with solutions defined below. 00

The double points of Mathieu’s differential equation