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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $R \u003e 0$, and let $B = \\set{x \\in \\R^n : \\abs{x} \u003c R}$ be the [[open ball|0YE5868HZB]] of radius $R$ centred at $0$.\nLet $u$ be a [[harmonic function|0DEB0JMN0W]] on $B$.\nThen there is a unique sequence $(p_k)_{k \\ge 0}$ with the following property.\nEach $p_k$ is a [[harmonic|MVYFSYMECG]] [[homogeneous polynomial|2C3PAQ7H50]] of degree $k$ on $\\R^n$, and\n$$u(x) = \\sum_{k=0}^{\\infty} p_k(x), \\qquad x \\in B.$$\nThe series converges absolutely and [[uniformly|08VZ57PT26]] on each compact subset of $B$.\n\n## Remarks\n\nProof idea: on a smaller closed ball, $u$ is the Poisson integral of its boundary values.\nThe [[Poisson kernel|TCYSWCTYYM]] of the ball expands as a series of zonal harmonics, each a homogeneous harmonic polynomial in $x$; integrating term by term gives the $p_k$.\nUniqueness follows by comparing homogeneous parts of the Taylor series of $u$ at $0$, since $p_k$ is the degree $k$ part.\nIn polar form $x = r\\zeta$ with $\\abs{\\zeta} = 1$, the expansion reads $u(r\\zeta) = \\sum_k r^k p_k(\\zeta)$, and the restriction of $p_k$ to the unit sphere is a [[spherical harmonic|3F32Y21HP4]] of degree $k$.\nThe restrictions to the sphere of harmonic polynomials of different degrees are orthogonal in $L^2$ of the sphere.\nIn fact, $L^2$ of the sphere is the orthogonal direct sum of the spaces of spherical harmonics of each degree.\nExample: on the unit disc in $\\R^2 = \\C$, $u = \\operatorname{Re} \\frac{1}{1 - z}$ is harmonic, and $u(r e^{i\\theta}) = \\sum_{k \\ge 0} r^k \\cos k\\theta$, where $r^k \\cos k\\theta = \\operatorname{Re} z^k$ is a homogeneous harmonic polynomial of degree $k$.\n\n## Explanations\n\n### L1 human\n\nA harmonic function on a ball can be written as an infinite sum of harmonic polynomials, one for each degree.\nThis is like a Taylor series, but each piece is itself harmonic.\nOn the sphere, the pieces become the spherical harmonics, the analogues of $\\cos k\\theta$ and $\\sin k\\theta$ on a circle.\n\n### L2 human\n\nThis is the higher-dimensional version of the Fourier series of a harmonic function on the disc.\nIt solves the Dirichlet problem on a ball by separation of variables: expand the boundary data in spherical harmonics and multiply the degree $k$ part by $(\\abs{x}/R)^k$.\nIn $\\R^3$ the space of spherical harmonics of degree $k$ has dimension $2k + 1$, and the expansion is the one used for gravitational and electrostatic potentials.\nThe [[Fischer decomposition|JWNRB34ZGV]] describes how all polynomials, not only harmonic ones, split into harmonic pieces.\n",
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