Mathieu function

Definition

Let $a$ and $q$ be real numbers. Mathieu's equation is the linear differential equation $$\frac{d^2 y}{dx^2} + (a - 2q \cos 2x)\, y = 0.$$ Fix $q$. The values of $a$ for which the equation has a nonzero even solution of period $\pi$ or $2\pi$ form an increasing sequence $a_0(q) < a_1(q) < \cdots$. For each $a_r(q)$, such a solution is unique up to a constant factor. The even Mathieu function $\operatorname{ce}_r(x, q)$ is that solution for $a = a_r(q)$, normalised by $\int_0^{2\pi} \operatorname{ce}_r(x, q)^2\, dx = \pi$. Its sign is fixed by continuity in $q$ from $\operatorname{ce}_0(x, 0) = \frac{1}{\sqrt 2}$ and $\operatorname{ce}_r(x, 0) = \cos rx$ for $r \ge 1$. In the same way, odd solutions of period $\pi$ or $2\pi$ exist for an increasing sequence $b_1(q) < b_2(q) < \cdots$ of values of $a$. The odd Mathieu function $\operatorname{se}_r(x, q)$ is the odd solution for $a = b_r(q)$, with the same normalisation and with $\operatorname{se}_r(x, 0) = \sin rx$.