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  "body": "\n## Summary\n\nThe associated Legendre polynomials $P_\\ell^m$ are the [[Legendre functions|69XMERM6KJ]] of integer degree $\\ell$ and integer order $m$ on the interval $[-1, 1]$.\nThey are obtained from the [[Legendre polynomials|QFYGJVA996]] by differentiating $m$ times and multiplying by $(1 - x^2)^{m/2}$.\nThey give the dependence on the polar angle of the [[spherical harmonics|3F32Y21HP4]] on the sphere in $\\R^3$.\n\n## Explanations\n\n### L1 human\n\nSpherical harmonics describe patterns on a sphere, such as the shapes of atomic orbitals.\nEach pattern splits into a wave around the equator and a profile from the north pole to the south pole.\nThe associated Legendre polynomials are these north-to-south profiles.\nFor example, $P_1^1(\\cos \\theta) = -\\sin \\theta$ for $0 \\le \\theta \\le \\pi$, a profile that is zero at both poles and largest in size at the equator.\nDespite the name, they are not polynomials when $m$ is odd, because of the square root $(1 - x^2)^{1/2}$.\n",
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