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  "body": "\n## Statement\n\nLet $\\mathbb{D} = \\set{z \\in \\C : \\abs{z} \u003c 1}$.\nLet $f$ be a [[holomorphic function|0NAHXR1FRH]] on $\\mathbb{D}$ that extends continuously to the closed disc $\\overline{\\mathbb{D}}$, and let $u = \\operatorname{Re} f$.\nThen for every $z \\in \\mathbb{D}$,\n$$f(z) = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\frac{e^{it} + z}{e^{it} - z}\\, u(e^{it})\\, dt + i \\operatorname{Im} f(0).$$\n\n## Remarks\n\nCall the integral $S(z)$.\nThe kernel $(e^{it} + z)/(e^{it} - z)$ is holomorphic in $z$, so $S$ is holomorphic on $\\mathbb{D}$, and its real part is the Poisson integral of $u$ with the [[Poisson kernel|W19567AVYG]].\nSince $u$ is [[harmonic|0DEB0JMN0W]] in $\\mathbb{D}$ and continuous on $\\overline{\\mathbb{D}}$, the Poisson integral of $u$ equals $u$.\nSo $f - S$ is holomorphic with real part $0$, and hence it is a purely imaginary constant.\nAt $z = 0$ the integral is the mean of $u$ over the circle, which is the real number $u(0)$, so the constant is $i \\operatorname{Im} f(0)$.\nExample: for $f(z) = z$ one has $u(e^{it}) = \\cos t$, and the formula gives $\\frac{1}{2\\pi}\\int_0^{2\\pi} \\frac{e^{it} + z}{e^{it} - z}\\cos t \\, dt = z$.\n\n## Explanations\n\n### L1 human\n\nA holomorphic function on the disc is determined, up to an imaginary constant, by its real part on the boundary circle.\nThe Schwarz formula writes the function explicitly as an integral of that real part.\nSo the real part on the boundary carries all the information about the function inside.\n\n### L2 human\n\nThe formula is the holomorphic counterpart of the Poisson integral: taking real parts of both sides gives the Poisson representation of $u$.\nIt solves the problem of finding a holomorphic function with prescribed real part on the boundary, which is a simple Riemann–Hilbert type problem.\nIt is used to prove the [[Herglotz representation theorem|9ZBAX4ZPX0]] and the Borel–Carathéodory theorem, and it has versions for the upper half-plane.\n",
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