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  "body": "\n## Definition\n\nLet $\\nu, \\mu \\in \\C$.\nThe *general Legendre equation* is the [[ordinary differential equation|0NT64JV96A]]\n$$(1 - z^2) w'' - 2z w' + \\left( \\nu(\\nu + 1) - \\frac{\\mu^2}{1 - z^2} \\right) w = 0.$$\nA *Legendre function* of degree $\\nu$ and order $\\mu$ is a solution $w$ of this equation on a domain of $\\C \\setminus \\set{-1, 1}$.\nThe *Legendre function of the first kind* of degree $\\nu$ and order $0$ is\n$$P_\\nu(z) = {}_2F_1\\left( -\\nu, \\nu + 1; 1; \\frac{1 - z}{2} \\right),$$\nwhere ${}_2F_1$ is the [[hypergeometric function|463X5V31D9]].\nThe series converges for $\\abs{1 - z} \u003c 2$, and $P_\\nu$ extends analytically to $\\C \\setminus (-\\infty, -1]$.\n\n## Remarks\n\nFor $\\nu = n$ a nonnegative integer, the factor $(-n)_k$ vanishes for $k \u003e n$, the series stops, and $P_n$ is the [[Legendre polynomial|QFYGJVA996]] of degree $n$.\nCheck for $n = 1$: $P_1(z) = 1 + \\frac{(-1)(2)}{1} \\cdot \\frac{1 - z}{2} = z$.\nFor $\\mu = 0$ the equation is [[Legendre's equation|W6VE30V10Y]] with $\\lambda = \\nu(\\nu + 1)$.\nA second, independent solution is the Legendre function of the second kind $Q_\\nu$; for example $Q_0(x) = \\frac12 \\ln \\frac{1 + x}{1 - x}$ for $-1 \u003c x \u003c 1$.\nNon-example: $w(z) = z^2$ is not a Legendre function of order $0$ for any $\\nu$, since the equation would require $2 - 6z^2 + \\nu(\\nu + 1) z^2 = 0$ for all $z$, which fails at $z = 0$.\nFor general order $\\mu$ the functions $P_\\nu^\\mu$ and $Q_\\nu^\\mu$ are also given by hypergeometric functions, with a factor $\\left( \\frac{z + 1}{z - 1} \\right)^{\\mu/2}$ in front.\n",
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