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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be open, $z_0 \\in U$, and let $p_1, \\ldots, p_n$ be [[meromorphic functions|0ZZ7ZYDR8W]] on $U$.\nConsider the homogeneous [[linear equation|0QQRP17Z7S]]\n$$y^{(n)} + p_1(z)\\, y^{(n-1)} + \\cdots + p_n(z)\\, y = 0.$$\nThe point $z_0$ is an *ordinary point* of the equation if every $p_k$ is [[analytic|081WKEDK6C]] at $z_0$ (after removing removable singularities).\nOtherwise $z_0$ is a *singular point*: some $p_k$ has a pole at $z_0$.\n\n## Remarks\n\nThe point $\\infty$ of the [[Riemann sphere|0J78KBPFZB]] is ordinary or singular according as $w = 0$ is ordinary or singular for the equation obtained by the substitution $z = 1/w$.\nFor real equations with real-analytic coefficients the same words are used, with $z$ replaced by a real variable $x$.\nThe definition applies to the normalised equation; for $a_0(z) y'' + a_1(z) y' + a_2(z) y = 0$ one first divides by $a_0$.\n\nExample: for $y'' - 2 z y' + 4 y = 0$, the coefficients $-2z$ and $4$ are polynomials, so every $z_0 \\in \\C$ is an ordinary point.\n\nNon-example: for $z^2 y'' + z y' + (z^2 - 1) y = 0$, the normalised coefficient $p_1(z) = \\frac{1}{z}$ has a pole at $0$, so $0$ is not an ordinary point but a singular point.\n\nAt an ordinary point $z_0$, every solution is analytic, and its Taylor series at $z_0$ converges at least on the largest disc around $z_0$ on which all $p_k$ are analytic.\n",
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