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  "body": "\n## Definition\n\nLet $z_0$ be a [[regular singular point|H96HXG6CA0]] of $y'' + p(z) y' + q(z) y = 0$.\nWrite $(z - z_0)\\, p(z) = \\sum_{k \\ge 0} p_k (z - z_0)^k$ and $(z - z_0)^2 q(z) = \\sum_{k \\ge 0} q_k (z - z_0)^k$ as convergent [[power series|0MZB06ST2B]].\nThe *Frobenius method* seeks solutions of the form\n$$y = (z - z_0)^r \\sum_{k \\ge 0} a_k (z - z_0)^k, \\qquad a_0 \\ne 0,$$\nwith an exponent $r \\in \\C$ and coefficients $a_k \\in \\C$, as follows.\n\n1. Put $F(s) = s (s - 1) + p_0 s + q_0$. The exponent $r$ must be a root of the *indicial equation* $F(r) = 0$.\n2. The coefficients satisfy, for $k \\ge 1$, the recurrence\n$$F(r + k)\\, a_k = - \\sum_{j=0}^{k-1} \\left( (r + j)\\, p_{k-j} + q_{k-j} \\right) a_j.$$\n3. When $F(r + k) \\ne 0$ for all $k \\ge 1$, the recurrence determines all $a_k$ from $a_0$.\n\n## Remarks\n\nThe rules come from inserting the series into $(z - z_0)^2 y'' + (z - z_0) \\cdot (z - z_0) p\\, y' + (z - z_0)^2 q\\, y = 0$ and comparing the coefficients of $(z - z_0)^{r + k}$.\nThe roots of the indicial equation are the *exponents* of the equation at $z_0$.\nFuchs' theorem says when the series converges and how to get a second solution.\n\nExample: for $z^2 y'' + z y' + (z^2 - \\frac{1}{4}) y = 0$ at $z_0 = 0$, one has $p_0 = 1$, $q_0 = -\\frac{1}{4}$, $q_2 = 1$, and all other $p_k, q_k$ vanish.\nSo $F(s) = s^2 - \\frac{1}{4}$, with roots $r = \\pm \\frac{1}{2}$.\nFor $r = \\frac{1}{2}$ the recurrence is $k (k + 1) a_k = - a_{k-2}$, with $a_1 = 0$, which gives $y = z^{-1/2} \\sin z$ when $a_0 = 1$.\n\nNon-example: at an irregular singular point the method can fail.\nFor $z^2 y'' + y' = 0$ at $0$, the only solutions of the form above are constants, while $y = \\int e^{1/z}\\, dz$ is a second solution whose derivative has an essential singularity at $0$.\n",
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