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  "body": "\n## Definition\n\nLet $m$ be a nonnegative integer and let $c \\ge 0$.\nThe *prolate spheroidal wave equation* is the [[linear differential equation|0QQRP17Z7S]]\n$$(1 - x^2) \\frac{d^2 y}{dx^2} - 2x \\frac{dy}{dx} + \\left( \\lambda - c^2 x^2 - \\frac{m^2}{1 - x^2} \\right) y = 0, \\qquad -1 \u003c x \u003c 1,$$\nwith a real parameter $\\lambda$.\nCall $\\lambda$ an eigenvalue if the equation has a nonzero solution that is bounded on $(-1, 1)$.\nThe eigenvalues form an increasing sequence $\\lambda_{m, m}(c) \u003c \\lambda_{m, m + 1}(c) \u003c \\lambda_{m, m + 2}(c) \u003c \\cdots$.\nFor each eigenvalue, the bounded solution is unique up to a constant factor.\nThe *angular prolate spheroidal wave function* $S_{mn}(c, x)$, for $n \\ge m$, is that bounded solution for $\\lambda = \\lambda_{mn}(c)$, with a chosen normalisation.\nThe *oblate* functions are defined in the same way with $c^2$ replaced by $-c^2$.\n\n## Remarks\n\nThe eigenvalue problem is a singular [[Sturm–Liouville|7WV6B2502K]] problem on $(-1, 1)$, which explains the discrete increasing sequence of eigenvalues.\nExample: for $c = 0$ the equation is the associated Legendre equation, $\\lambda_{mn}(0) = n(n + 1)$, and $S_{mn}(0, x)$ is a multiple of the [[associated Legendre polynomial|VPJCSVRY6S]] $P_n^m(x)$.\nFor $c = 0$, $m = 0$ and $n = 0$: $y = 1$ is bounded and satisfies $-2x \\cdot 0 + (0 - 0) \\cdot 1 = 0$ with $\\lambda = 0$.\nNon-example: for $c = 0$, $m = 0$ and $\\lambda = 0$, the function $y = \\frac12 \\log \\frac{1 + x}{1 - x}$ also solves the equation, but it is unbounded near $x = \\pm 1$, so it is not a spheroidal wave function.\nThe radial spheroidal wave functions solve the same equation on $x \u003e 1$ and describe the dependence on the other coordinate.\nNormalisations differ between authors; a common one makes $S_{mn}(c, x)$ tend to $P_n^m(x)$ as $c \\to 0$.\n",
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