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  "body": "\n## Statement\n\nLet $U \\subseteq \\R^n$ be an open set and let $u \\colon U \\to \\R$ be [[harmonic|0DEB0JMN0W]].\n\n1. The function $u$ is [[infinitely differentiable|0V33BNKZ9S]] on $U$, and every partial derivative of $u$ is harmonic on $U$.\n2. The function $u$ is real [[analytic|081WKEDK6C]] on $U$: each $a \\in U$ has a neighbourhood on which $u$ is a convergent power series in $x - a$.\n\n## Remarks\n\nProof idea for part 1: by the [[mean value property|0HYPR64JM2]], $u$ equals its convolution with a smooth radial mollifier away from the boundary of $U$.\nA convolution with a smooth function is smooth.\nSince $\\Delta$ commutes with partial derivatives, $\\Delta(\\partial^\\alpha u) = \\partial^\\alpha(\\Delta u) = 0$.\nProof idea for part 2: the [[gradient estimates|N76R7JWMNJ]] give $\\abs{\\partial^\\alpha u(a)} \\le C^{k+1} k^k \\sup_{B(a, r)} \\abs{u} / r^{k}$ for $\\abs{\\alpha} = k$, with $C$ depending only on $n$.\nThis growth makes the Taylor remainder tend to $0$ on a small ball.\nExample: $u(x, y) = e^x \\cos y$ is harmonic on $\\R^2$, and its Taylor series at $0$ converges to it everywhere.\nHarmonicity is needed and $C^2$ is not enough: $\\abs{x}^3$ on $\\R$ is twice continuously differentiable but not three times differentiable at $0$.\nIn contrast, a merely continuous function with the mean value property is already harmonic, hence analytic.\n\n## Explanations\n\n### L1 human\n\nLaplace's equation forces its solutions to be extremely smooth.\nEven if one only asks for two derivatives, a harmonic function automatically has derivatives of all orders, and it is given locally by a power series.\nSo a harmonic function cannot have a hidden corner or kink anywhere.\n\n### L2 human\n\nThis is the simplest case of elliptic regularity: solutions of elliptic equations with smooth or analytic coefficients are as regular as the coefficients allow.\nAnalyticity implies a unique continuation property: a harmonic function on a connected open set that vanishes on a nonempty open subset vanishes everywhere.\nThe heat equation shares smoothness but not analyticity in time, and the wave equation has neither, so this property marks the elliptic case.\n",
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