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  "body": "\n## Definition\n\nLet $a$ and $q$ be real numbers.\n*Mathieu's equation* is the [[linear differential equation|0QQRP17Z7S]]\n$$\\frac{d^2 y}{dx^2} + (a - 2q \\cos 2x)\\, y = 0.$$\nFix $q$.\nThe values of $a$ for which the equation has a nonzero even solution of [[period|0ATF6Q0633]] $\\pi$ or $2\\pi$ form an increasing sequence $a_0(q) \u003c a_1(q) \u003c \\cdots$.\nFor each $a_r(q)$, such a solution is unique up to a constant factor.\nThe *even Mathieu function* $\\operatorname{ce}_r(x, q)$ is that solution for $a = a_r(q)$, normalised by $\\int_0^{2\\pi} \\operatorname{ce}_r(x, q)^2\\, dx = \\pi$.\nIts 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 $r \\ge 1$.\nIn the same way, odd solutions of period $\\pi$ or $2\\pi$ exist for an increasing sequence $b_1(q) \u003c b_2(q) \u003c \\cdots$ of values of $a$.\nThe *odd Mathieu function* $\\operatorname{se}_r(x, q)$ is the odd solution for $a = b_r(q)$, with the same normalisation and with $\\operatorname{se}_r(x, 0) = \\sin rx$.\n\n## Remarks\n\nThe numbers $a_r(q)$ and $b_r(q)$ are the *characteristic values*; they are eigenvalues of [[Sturm–Liouville|7WV6B2502K]] problems on $[0, \\pi]$ or $[0, \\frac{\\pi}{2}]$ with suitable boundary conditions.\nThe function $\\operatorname{ce}_r$ or $\\operatorname{se}_r$ has period $\\pi$ when $r$ is even and period $2\\pi$ when $r$ is odd.\nExample: for $q = 0$ one has $a_r(0) = r^2$ and $\\operatorname{ce}_r(x, 0) = \\cos rx$ for $r \\ge 1$; indeed $y = \\cos rx$ satisfies $y'' + r^2 y = 0$ and $\\int_0^{2\\pi} \\cos^2 rx\\, dx = \\pi$.\nFor $r = 0$ the constant must be $\\frac{1}{\\sqrt 2}$, since $\\int_0^{2\\pi} \\frac12\\, dx = \\pi$.\nNon-example: for $q = 0$ and $a = \\frac14$, the solution $\\cos \\frac{x}{2}$ has period $4\\pi$, not $\\pi$ or $2\\pi$, so it is not a Mathieu function in this sense.\nFor most other values of $a$, [[Floquet theory|BJVZXPB297]] gives a basis of solutions of the form $e^{i\\mu x} p(x)$ and $e^{-i\\mu x} p(-x)$, with $p$ of period $\\pi$ and $\\mu$ the characteristic exponent.\n",
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