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  "body": "\n## Statement\n\nLet $R \u003e 0$ and $z_0 \\in \\C$, and let $D = \\set{z : 0 \u003c \\abs{z - z_0} \u003c R}$.\nLet $p, q$ be functions such that $P(z) = (z - z_0)\\, p(z)$ and $Q(z) = (z - z_0)^2 q(z)$ are analytic on $\\abs{z - z_0} \u003c R$.\nLet $r_1, r_2$ be the roots of the indicial equation $r (r - 1) + P(z_0)\\, r + Q(z_0) = 0$, labelled so that $\\operatorname{Re} r_1 \\ge \\operatorname{Re} r_2$.\nFix a branch of $\\log (z - z_0)$ on $D$ minus a ray from $z_0$, and put $(z - z_0)^r = e^{r \\log (z - z_0)}$.\nThen the equation $y'' + p(z) y' + q(z) y = 0$ has two linearly independent solutions $y_1, y_2$ there, of the following form, where all the [[power series|0MZB06ST2B]] converge on $\\abs{z - z_0} \u003c R$.\n\n1. $y_1 = (z - z_0)^{r_1} \\sum_{k \\ge 0} a_k (z - z_0)^k$ with $a_0 = 1$.\n2. If $r_1 - r_2 \\notin \\Z$, then $y_2 = (z - z_0)^{r_2} \\sum_{k \\ge 0} b_k (z - z_0)^k$ with $b_0 = 1$.\n3. If $r_1 = r_2$, then $y_2 = y_1 \\log (z - z_0) + (z - z_0)^{r_1} \\sum_{k \\ge 1} b_k (z - z_0)^k$.\n4. If $r_1 - r_2$ is a positive integer, then $y_2 = c\\, y_1 \\log (z - z_0) + (z - z_0)^{r_2} \\sum_{k \\ge 0} b_k (z - z_0)^k$ with $b_0 = 1$ and a constant $c$, which may be $0$.\n\n## Remarks\n\nThe coefficients come from the recurrence of the [[Frobenius method|W8ZWXC2HQS]].\nFor $r = r_1$, the factor $F(r_1 + k)$ in front of $a_k$ never vanishes for $k \\ge 1$, because $r_1$ has the largest real part.\nConvergence on the whole disc follows by comparing the recurrence with a majorant series built from Cauchy estimates for the coefficients of $P$ and $Q$.\nThe second solution comes from reduction of order, which is where the [[logarithm|02QGGD5G87]] enters.\n\nExample: for the Bessel equation of order $0$, $z^2 y'' + z y' + z^2 y = 0$, the indicial equation is $r^2 = 0$, so case 3 applies.\nThe solutions are $J_0(z) = \\sum_{m \\ge 0} \\frac{(-1)^m}{(m!)^2} \\left( \\frac{z}{2} \\right)^{2m}$ and a second solution with a term $J_0(z) \\log z$.\n\nIf $z_0$ is an ordinary point, then $P(z_0) = Q(z_0) = 0$, the exponents are $0$ and $1$, and the theorem gives two analytic solutions, with [[radius of convergence|0N288P2TB5]] at least $R$.\nThe hypothesis that $z_0$ is at worst a regular singular point is needed: at the irregular point $0$ of $z^2 y'' + y' = 0$, the solution $\\int e^{1/z} dz$ has no such form.\nFor real equations, the same holds on $0 \u003c \\abs{x - x_0} \u003c R$ with $(x - x_0)^r$ replaced by $\\abs{x - x_0}^r$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the Frobenius method always works at a regular singular point.\nThere is always one solution of the form $z^r$ times a convergent power series.\nThe second solution has the same form, unless the two exponents differ by a whole number; then a logarithm may appear.\nThe series converge as far as the coefficients of the equation stay nice.\n\n### L2 human\n\nThe theorem is the basic local result for linear equations in the complex domain.\nIt explains the behaviour of Bessel, Legendre, Laguerre and hypergeometric functions near their singular points.\nGoing around $z_0$ once multiplies $(z - z_0)^r$ by $e^{2 \\pi i r}$ and adds $2 \\pi i$ to the logarithm.\nSo the theorem computes the local [[monodromy|C8HD62AB6Q]] of the equation from the exponents.\nAt an irregular singular point, solutions are described instead by asymptotic expansions in sectors, which leads to the [[Stokes phenomenon|TTGF8KCA8C]].\n",
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