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  "body": "\n## Definition\n\nLet $p_1, \\ldots, p_n$ be [[rational functions|0XDK9CGYKJ]] of $z$ with complex coefficients.\nThe linear equation\n$$y^{(n)} + p_1(z)\\, y^{(n-1)} + \\cdots + p_n(z)\\, y = 0$$\nis *Fuchsian* (or of Fuchsian class) if it satisfies the following condition.\nEvery point of the [[Riemann sphere|0J78KBPFZB]] $\\C \\cup \\set{\\infty}$ is an [[ordinary point|QK2J038VB4]] or a [[regular singular point|H96HXG6CA0]] of the equation.\n\n## Remarks\n\nAt a finite point $a$, the condition says that each $p_k$ has at most a pole of order $k$ at $a$.\nAt $\\infty$, it says that $z^k p_k(z)$ stays bounded as $z \\to \\infty$, for each $k$.\nSince the $p_k$ are rational, there are only finitely many singular points.\n\nExample: the hypergeometric equation $z (1 - z) y'' + \\left( c - (a + b + 1) z \\right) y' - a b\\, y = 0$ is Fuchsian.\nIts normalised coefficients are $p = \\frac{c - (a + b + 1) z}{z (1 - z)}$ and $q = \\frac{-a b}{z (1 - z)}$.\nThey have simple poles at $0$ and $1$, and $z p(z) \\to a + b + 1$ and $z^2 q(z) \\to a b$ as $z \\to \\infty$.\nSo the singular points $0$, $1$ and $\\infty$ are all regular.\n\nNon-example: the Bessel equation $z^2 y'' + z y' + (z^2 - \\nu^2) y = 0$ has $q(z) = 1 - \\frac{\\nu^2}{z^2}$.\nThen $z^2 q(z) = z^2 - \\nu^2$ is unbounded as $z \\to \\infty$, so $\\infty$ is an irregular singular point and the equation is not Fuchsian.\n",
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