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  "body": "\n## Summary\n\nThe Poisson kernel is the integral kernel that produces a [[harmonic function|0DEB0JMN0W]] in a disc from its values on the boundary circle.\nOn the unit disc it is $P(z, \\zeta) = (1 - \\abs{z}^2)/\\abs{\\zeta - z}^2$ for $\\abs{z} \u003c 1$ and $\\abs{\\zeta} = 1$.\nIntegrating a continuous function on the circle against it solves the [[Dirichlet problem|8D56GQ4AG8]] for the disc.\nSimilar kernels exist for the upper half-plane and for balls in $\\R^n$.\nThey are central in harmonic analysis and in the theory of [[Hardy spaces|PNNNFJVF6W]].\n\n## Explanations\n\n### L1 human\n\nSuppose you know the temperature at every point of the rim of a round plate and the plate is in equilibrium.\nThe temperature at an interior point is a weighted average of the rim temperatures.\nThe Poisson kernel gives the weights: rim points close to the interior point count more.\nAt the centre all rim points count equally, so the centre temperature is the plain average of the rim.\n",
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