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  "body": "\n## Example\n\nLet $\\lambda \\in \\C$. *Legendre's equation* is\n$$(1 - z^2)\\, y'' - 2 z\\, y' + \\lambda\\, y = 0,$$\nusually written with $\\lambda = n (n + 1)$.\nIn normalised form, $p(z) = \\frac{-2z}{1 - z^2}$ and $q(z) = \\frac{\\lambda}{1 - z^2}$, which are rational.\nThey have simple poles at $z = \\pm 1$, so $\\pm 1$ are regular singular points, and every other finite point is ordinary.\nAs $z \\to \\infty$, $z p(z) \\to 2$ and $z^2 q(z) \\to -\\lambda$, so $\\infty$ is ordinary or regular singular.\nHence Legendre's equation is a [[Fuchsian equation|ZBVH2YE50B]], with singular points among $-1$, $1$ and $\\infty$.\n\n## Remarks\n\nAt the [[ordinary point|QK2J038VB4]] $0$, a [[power series solution|08RT9GDDPQ]] $\\sum a_k z^k$ satisfies $(k + 2)(k + 1)\\, a_{k+2} = \\left( k (k + 1) - \\lambda \\right) a_k$.\nThe series converge for $\\abs{z} \u003c 1$, the distance from $0$ to the nearest singular point.\nIf $\\lambda = n (n + 1)$ with $n \\ge 0$ an integer, the recurrence stops, and one solution is the Legendre polynomial $P_n$ of degree $n$.\nFor $n = 2$, $P_2(z) = \\frac{3 z^2 - 1}{2}$: then $y'' = 3$ and $y' = 3z$, and $(1 - z^2) \\cdot 3 - 2z \\cdot 3z + 6 \\cdot \\frac{3 z^2 - 1}{2} = 0$.\nThe equation arises from the Laplace equation in spherical coordinates.\n\n## Explanations\n\n### L1 human\n\nLegendre's equation appears when one studies gravity or electric potential around a sphere.\nIts only bad points are the poles $z = \\pm 1$ of the sphere and the point at infinity, and all of them are tame.\nFor special values of the parameter it has polynomial solutions, the Legendre polynomials.\n",
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