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  "body": "\n## Definition\n\nLet $\\ell$ and $m$ be integers with $0 \\le m \\le \\ell$, and let $P_\\ell$ be the [[Legendre polynomial|QFYGJVA996]] of degree $\\ell$.\nThe *associated Legendre polynomial* $P_\\ell^m$ is the function on $[-1, 1]$ given by\n$$P_\\ell^m(x) = (-1)^m (1 - x^2)^{m/2} \\frac{d^m}{dx^m} P_\\ell(x).$$\nFor $0 \u003c m \\le \\ell$ one also sets\n$$P_\\ell^{-m}(x) = (-1)^m \\frac{(\\ell - m)!}{(\\ell + m)!} P_\\ell^m(x).$$\n\n## Remarks\n\nThe factor $(-1)^m$ is the Condon–Shortley phase; some authors leave it out.\nExamples: $P_\\ell^0 = P_\\ell$, $P_1^1(x) = -(1 - x^2)^{1/2}$, $P_2^1(x) = -3x (1 - x^2)^{1/2}$ and $P_2^2(x) = 3(1 - x^2)$.\nCheck for $P_2^2$: $P_2(x) = (3x^2 - 1)/2$ has second derivative $3$, so $P_2^2(x) = (1 - x^2) \\cdot 3$.\nEach $P_\\ell^m$ solves the associated Legendre equation $(1 - x^2) y'' - 2x y' + \\left( \\ell(\\ell + 1) - \\frac{m^2}{1 - x^2} \\right) y = 0$ on $(-1, 1)$.\nFor fixed $m$ they are orthogonal: $\\int_{-1}^{1} P_k^m(x) P_\\ell^m(x) \\, dx = \\frac{2 (\\ell + m)!}{(2\\ell + 1)(\\ell - m)!} \\delta_{k\\ell}$.\nNon-example: the case $m \u003e \\ell$ is excluded, because then the $m$-th derivative of $P_\\ell$ is zero and the formula gives only the zero function.\n",
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