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  "body": "\n## Summary\n\nLegendre functions are the solutions of the general Legendre differential equation, which has two parameters, a degree $\\nu$ and an order $\\mu$.\nThe solutions of the first kind, $P_\\nu^\\mu$, and of the second kind, $Q_\\nu^\\mu$, are the standard basis of solutions.\nWhen $\\mu = 0$ and $\\nu$ is a nonnegative integer, the first kind gives the [[Legendre polynomials|QFYGJVA996]].\nThey arise when the Laplace equation or the Helmholtz equation is separated in spherical coordinates.\n\n## Explanations\n\n### L1 human\n\nLegendre polynomials solve [[Legendre's equation|W6VE30V10Y]] only for special integer values of a parameter.\nIf the parameter is any number, the equation still has solutions, but they are no longer polynomials.\nThese solutions are the Legendre functions.\nFor example, $Q_0(x) = \\frac12 \\ln \\frac{1 + x}{1 - x}$ on $(-1, 1)$ solves the equation with parameter $0$, and it blows up at both ends of the interval.\n",
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