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  "body": "\n## Summary\n\nFix a point $y$ of the sphere and a degree $\\ell$.\nThe zonal spherical harmonic $Z_y^{(\\ell)}$ is the spherical harmonic of degree $\\ell$ that reproduces the value at $y$ of every harmonic of degree $\\ell$.\nIt is invariant under the rotations that fix $y$, so it depends only on the angle between $x$ and $y$.\nIt is a multiple of a [[Gegenbauer polynomial|5TRFA6F79B]] in $x \\cdot y$, and of the [[Legendre polynomial|QFYGJVA996]] $P_\\ell(x \\cdot y)$ on the sphere in $\\R^3$.\nIt is the building block of expansions in [[spherical harmonics|3F32Y21HP4]] and of the Poisson kernel of the ball.\n\n## Explanations\n\n### L1 human\n\nPick a point $y$ on a globe, as if it were the north pole.\nA zonal function is one that is constant along each circle of latitude around $y$, so it only depends on the distance from $y$.\nAmong the spherical harmonics of a given degree there is, up to a factor, exactly one zonal one.\nWith the right factor, integrating any harmonic of that degree against it returns the value of that harmonic at $y$.\nOn the ordinary sphere it is $\\frac{2\\ell + 1}{4\\pi} P_\\ell(\\cos \\theta)$, where $\\theta$ is the angle from $y$.\n",
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