Mathieu function

Summary

Mathieu functions are solutions of Mathieu's equation $y'' + (a - 2q \cos 2x) y = 0$. For each $q$, only special values of $a$, the characteristic values, admit solutions of period $\pi$ or $2\pi$. These periodic solutions $\operatorname{ce}_r(x, q)$ and $\operatorname{se}_r(x, q)$ are the Mathieu functions in the narrow sense. They arise when the Helmholtz equation is separated in elliptic coordinates, for instance for vibrations of an elliptic drum. Mathieu's equation is the basic example of a Hill differential equation, studied by Floquet theory.