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  "body": "\n## Definition\n\nLet $d \\ge 2$ and $\\ell \\ge 0$ be integers, and let $S^{d-1} = \\set{x \\in \\R^d : \\norm{x} = 1}$ be the unit [[sphere|T62369RFH6]].\nA *spherical harmonic of degree $\\ell$* is a function $Y \\colon S^{d-1} \\to \\C$ such that $Y = p|_{S^{d-1}}$ for some polynomial $p$ on $\\R^d$ that is:\n\n1. a [[homogeneous polynomial|2C3PAQ7H50]] of degree $\\ell$, and\n2. [[harmonic|0DEB0JMN0W]], that is, $\\Delta p = \\sum_{j=1}^{d} \\frac{\\partial^2 p}{\\partial x_j^2} = 0$.\n\nThe spherical harmonics of degree $\\ell$ form a complex vector space $\\mathcal{H}_\\ell(S^{d-1})$.\n\n## Remarks\n\nExample: on $S^2$, $Y(x, y, z) = x^2 - y^2$ is a spherical harmonic of degree $2$, since $\\Delta(x^2 - y^2) = 2 - 2 = 0$.\nNon-example: $x^2$ is homogeneous of degree $2$ but $\\Delta x^2 = 2 \\ne 0$.\nOn $S^2$ it equals $\\left( x^2 - \\frac13 (x^2 + y^2 + z^2) \\right) + \\frac13$, a sum of harmonics of degrees $2$ and $0$, so it is not a spherical harmonic of degree $2$.\nFor $d \\ge 3$, the dimension of $\\mathcal{H}_\\ell(S^{d-1})$ is $\\binom{\\ell + d - 1}{d - 1} - \\binom{\\ell + d - 3}{d - 1}$, which is $2\\ell + 1$ for $d = 3$; for $d = 2$ it is $2$ when $\\ell \\ge 1$.\nEach $Y \\in \\mathcal{H}_\\ell$ satisfies $\\Delta_{S^{d-1}} Y = -\\ell(\\ell + d - 2) Y$ for the Laplace–Beltrami operator of the sphere.\nThe spaces $\\mathcal{H}_\\ell$ are pairwise orthogonal in $L^2(S^{d-1})$, and their sum is dense.\nFor $d = 3$, in spherical coordinates, a standard orthonormal basis is $Y_\\ell^m(\\theta, \\varphi) = \\sqrt{\\frac{2\\ell + 1}{4\\pi} \\frac{(\\ell - m)!}{(\\ell + m)!}} \\, P_\\ell^m(\\cos \\theta) \\, e^{im\\varphi}$ for $-\\ell \\le m \\le \\ell$, where $P_\\ell^m$ are the associated Legendre polynomials.\n",
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