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  "body": "\n## Statement\n\nLet $n \\ge 2$, let $r \u003e 0$, and let $u$ be a positive [[harmonic function|0DEB0JMN0W]] on the punctured ball $B(0, r) \\setminus \\set{0}$ in $\\R^n$.\nThen there are a constant $b \\ge 0$ and a harmonic function $v$ on the whole ball $B(0, r)$ such that, for $0 \u003c \\abs{x} \u003c r$,\n$$u(x) = v(x) + b \\log \\frac{1}{\\abs{x}} \\quad (n = 2), \\qquad u(x) = v(x) + \\frac{b}{\\abs{x}^{n-2}} \\quad (n \\ge 3).$$\n\n## Remarks\n\nSo an isolated singularity of a positive harmonic function is a nonnegative multiple of the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], up to a harmonic term.\nIf $b = 0$, the singularity is removable, in agreement with the [[removable singularity theorem|3H6SQA6JWM]].\nProof idea: averaging $u$ over the spheres about $0$ gives a radial harmonic function on the punctured ball.\nIt has the form $a + b\\abs{x}^{2-n}$, or $a + b \\log(1/\\abs{x})$ in the plane.\nPositivity of $u$ is then used to show that $b \\ge 0$ and that the difference of $u$ and the singular term has a removable singularity at $0$.\nPositivity is needed: $u(x) = x_1 / \\abs{x}^n$ is harmonic on $\\R^n \\setminus \\set{0}$, takes both signs, and is not of the stated form, since it grows like $\\abs{x}^{1-n}$ near $0$.\nExample: $u(x) = 1 + \\abs{x}^{2-n}$ is positive on the punctured unit ball for $n \\ge 3$, with $v = 1$ and $b = 1$.\n\n## Explanations\n\n### L1 human\n\nA positive harmonic function can only blow up at an isolated point in one way: like the potential of a point charge.\nEvery other part of it is a perfectly regular harmonic function.\nSo positivity forbids wild singularities such as dipoles, which take both large positive and large negative values.\n\n### L2 human\n\nThe theorem classifies isolated singularities of positive harmonic functions, and it is the local form of the Riesz decomposition of positive superharmonic functions.\nA global corollary: a positive harmonic function on $\\R^n \\setminus \\set{0}$, $n \\ge 3$, has the form $a + b\\abs{x}^{2-n}$ with $a, b \\ge 0$, by combining Bôcher's theorem with the Kelvin transform and Liouville's theorem.\nIt is also the model for the description of the Martin boundary at an isolated boundary point.\n",
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