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  "body": "\n## Summary\n\nA function on an open subset of $\\R^n$ is harmonic at a point when it is harmonic on some neighbourhood of that point.\nThis local notion lets one say where a function is harmonic without fixing a domain in advance.\nA function is a [[harmonic function|0DEB0JMN0W]] on an open set exactly when it is harmonic at each point of the set.\n\n## Explanations\n\n### L1 human\n\nBeing harmonic at a point is not a condition on the value at that single point.\nIt asks that the function solves Laplace's equation in a small ball around the point.\nFor example, $\\log(x^2 + y^2)$ is harmonic at every point of the plane except the origin, where it is not even defined.\nThe function $x^3$ in the plane has Laplacian $6x$, which vanishes on the line $x = 0$ but on no ball, so it is harmonic at no point.\n",
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