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  "body": "\n## Summary\n\nA regular singular point of a linear differential equation is a [[singular point|QK2J038VB4]] where the coefficients blow up only mildly.\nIn the normalised equation, the coefficient of $y^{(n-k)}$ has a pole of order at most $k$ there.\nAt such a point the solutions grow at most like a power of the distance to the point.\nThe [[Frobenius method|W8ZWXC2HQS]] then finds solutions of the form $(z - z_0)^r$ times a power series.\nBessel, Legendre and hypergeometric equations all have regular singular points.\nSee the [[definition|BMD7KQQA9C]].\n\n## Explanations\n\n### L1 human\n\nSome singular points are tame and some are wild.\nAt a tame one, the solutions behave like powers $z^r$, perhaps with a logarithm, times nice series.\nThe model is the Euler equation $z^2 y'' + a z y' + b y = 0$, solved by $y = z^r$ whenever $r(r - 1) + a r + b = 0$.\nAt a wild, irregular point, solutions can behave like $e^{1/z}$ instead.\n",
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