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  "body": "\n## Summary\n\nThe Green's function of a domain for the [[Laplacian|0E0B6RA8K0]] is the potential of a unit point charge in the domain.\nThe boundary of the domain is held at potential zero.\nIn a plane domain $\\Omega$ with pole $w$, it is a function $g_\\Omega(z, w)$ that behaves like $\\log(1/\\abs{z - w})$ near $w$.\nIt is [[harmonic|0DEB0JMN0W]] elsewhere and vanishes on the boundary.\nIt solves the Poisson equation with zero boundary values: $\\Delta_z g_\\Omega(z, w) = -2\\pi \\delta_w$.\nIn free space $\\R^3$ the corresponding function is $1/(4\\pi\\abs{x - y})$, the fundamental solution of the three-variable Laplace equation.\nGreen's functions give integral formulas for the [[Dirichlet problem|8D56GQ4AG8]] and are invariant under [[conformal maps|0YYT36R4YD]] in the plane.\n\n## Explanations\n\n### L1 human\n\nPlace a single point charge inside a grounded metal box.\nThe electric potential it creates inside the box is the Green's function, with the charge's position as the pole.\nNear the charge the potential is huge, and on the walls of the box it is zero.\nFor the unit disc with the pole at the centre, the Green's function is $\\log(1/\\abs{z})$, which is $0$ on the unit circle.\n",
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