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  "body": "\n## Summary\n\nThe Frobenius method finds solutions of a second-order linear differential equation near a [[regular singular point|H96HXG6CA0]] in the form $(z - z_0)^r \\sum_{k \\ge 0} a_k (z - z_0)^k$.\nThe exponent $r$ comes from a quadratic equation, the indicial equation, and the coefficients $a_k$ from a recurrence.\nIt extends the [[power series method|08RT9GDDPQ]], which works only at ordinary points.\nIt produces Bessel functions, Legendre functions and hypergeometric functions.\nSee the [[definition|7D47PADY9N]].\n\n## Explanations\n\n### L1 human\n\nNear a regular singular point the solutions look like a power $z^r$ times an ordinary power series.\nThe power $r$ need not be a whole number, and it is found first by looking at the lowest terms of the equation.\nThen the series coefficients follow one after another.\nFor $z^2 y'' + z y' + (z^2 - \\frac{1}{4}) y = 0$, the exponents are $r = \\pm \\frac{1}{2}$, and the method gives $z^{-1/2} \\sin z$ and $z^{-1/2} \\cos z$.\n",
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