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  "body": "\n## Definition\n\nLet $p_1, \\ldots, p_n$ be meromorphic functions near $z_0 \\in \\C$, and consider the [[linear equation|0QQRP17Z7S]]\n$$y^{(n)} + p_1(z)\\, y^{(n-1)} + \\cdots + p_n(z)\\, y = 0.$$\nThe point $z_0$ is an *irregular singular point* (or irregular singularity) if it is a [[singular point|QK2J038VB4]] but not a [[regular singular point|H96HXG6CA0]].\nEquivalently, $(z - z_0)^k\\, p_k(z)$ is not analytic at $z_0$ for at least one $k \\in \\set{1, \\ldots, n}$.\nThe point $\\infty$ is irregular singular if $w = 0$ is irregular singular for the equation obtained by the substitution $z = 1/w$.\n\n## Remarks\n\nExample: $z^2 y'' + y' = 0$ has $p_1 = \\frac{1}{z^2}$, and $z\\, p_1 = \\frac{1}{z}$ is not analytic at $0$, so $0$ is an irregular singular point.\nIts solutions are the constants and $\\int e^{1/z}\\, dz$, and the derivative $e^{1/z}$ of the latter has an essential singularity at $0$.\n\nNon-example: $z^2 y'' + z y' - y = 0$ has $z p_1 = 1$ and $z^2 p_2 = -1$, both analytic, so $0$ is a regular singular point and not irregular; its solutions are $z$ and $z^{-1}$.\n\nFor $n = 2$, the point $\\infty$ is irregular for $y'' + p y' + q y = 0$ exactly when $z\\, p(z)$ or $z^2 q(z)$ is not analytic at $\\infty$.\nFor example, $\\infty$ is irregular for the Bessel equation, the Airy equation $y'' = z y$ and the equation $y'' = y$, whose solutions $e^{\\pm z}$ have essential singularities at $\\infty$.\n",
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