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  "body": "\n## Summary\n\nLet $\\Omega \\subset \\C^n$ be a bounded domain with smooth boundary.\nIts Szegő kernel is the reproducing kernel of the [[Hardy space|PNNNFJVF6W]] of holomorphic functions with square-integrable boundary values on $\\partial\\Omega$.\nIt plays for boundary integrals the role that the [[Bergman kernel|8RWDHW0CXC]] plays for volume integrals.\nIt gives a canonical integral formula that recovers a holomorphic function inside $\\Omega$ from its values on the boundary.\nOn the unit disc it is the Cauchy kernel, and on the unit ball it has an explicit closed form.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function on a nice domain is fixed by its values on the boundary.\nThe Szegő kernel is the weight that tells how much each boundary point contributes to the value at an inside point $z$.\nTo get $f(z)$, average the boundary values of $f$ against this weight.\nOn the unit disc the weight is $\\frac{1}{2\\pi(1 - z\\bar\\zeta)}$, and this average is exactly Cauchy's integral formula.\n",
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