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  "body": "\n## Summary\n\nThe Kelvin transform turns a function $u$ on a region of $\\R^n \\setminus \\set{0}$ into the function $x \\mapsto \\abs{x}^{2-n} u(x / \\abs{x}^2)$ on the inverted region.\nIt sends [[harmonic functions|0DEB0JMN0W]] to harmonic functions, and it exchanges the neighbourhoods of $0$ and of infinity.\nIt is the main tool to study harmonic functions near infinity, exterior Dirichlet problems and the Green's function of a ball.\nIn the plane it is just composition with the inversion $z \\mapsto 1/\\bar z$.\n\n## Explanations\n\n### L1 human\n\nInversion in the unit sphere maps a point at distance $r$ from the origin to the point in the same direction at distance $1/r$.\nComposing a harmonic function with this map does not keep it harmonic in dimension three or more; one has to multiply by the weight $\\abs{x}^{2-n}$.\nWith that weight, harmonicity survives.\nFor instance, in $\\R^3$ the Kelvin transform of the constant $1$ is $1/\\abs{x}$, the potential of a point charge.\n",
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