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  "body": "\n## Statement\n\nLet $D \\subset \\C$ be a bounded open set whose boundary $\\partial D$ consists of finitely many piecewise $C^1$ closed curves.\nOrient $\\partial D$ so that $D$ lies on the left.\nLet $f$ be a [[continuously differentiable|06SKCHDQT5]] complex-valued function on an open set that contains the closure $\\bar D$.\nThen for each $z \\in D$,\n$$f(z) = \\frac{1}{2\\pi i} \\oint_{\\partial D} \\frac{f(\\zeta)}{\\zeta - z} \\, d\\zeta - \\frac{1}{\\pi} \\iint_D \\frac{\\partial f / \\partial \\bar \\zeta \\, (\\zeta)}{\\zeta - z} \\, dA(\\zeta),$$\nwhere $\\partial f / \\partial \\bar \\zeta$ is the [[Wirtinger derivative|698R12MY4G]] and $dA$ is the area measure.\n\n## Remarks\n\nProof idea: apply [[Green's theorem|0HR3N02317]], in the form $\\oint_{\\partial \\Omega} g \\, d\\zeta = 2i \\iint_\\Omega \\partial g / \\partial \\bar \\zeta \\, dA$, to $g(\\zeta) = f(\\zeta)/(\\zeta - z)$ on $D$ minus a small disc around $z$.\nSince $1/(\\zeta - z)$ is holomorphic there, $\\partial g / \\partial \\bar \\zeta = (\\partial f / \\partial \\bar \\zeta)/(\\zeta - z)$; the small circle contributes $2\\pi i f(z)$ in the limit.\nIf $f$ is holomorphic, then $\\partial f / \\partial \\bar \\zeta = 0$ and the formula is [[Cauchy's integral formula|0DG10TVYPN]].\nIf $f$ has compact support in $\\C$, the boundary term vanishes and $f(z) = \\frac{1}{\\pi} \\iint \\frac{\\partial f / \\partial \\bar \\zeta}{z - \\zeta} \\, dA(\\zeta)$.\nOne deduces from it that, for $g$ of class $C^1$ with compact support, $u(z) = \\frac{1}{\\pi} \\iint \\frac{g(\\zeta)}{z - \\zeta} \\, dA(\\zeta)$ solves $\\partial u / \\partial \\bar z = g$.\n\n## Explanations\n\n### L1 human\n\nCauchy's formula recovers a holomorphic function inside a region from its values on the boundary.\nFor a smooth function that is not holomorphic, the boundary values are not enough.\nThe Cauchy–Pompeiu formula adds an area integral that measures how far the function is from being holomorphic.\n\n### L2 human\n\nThe formula is the basic tool for the inhomogeneous Cauchy–Riemann equation $\\partial u / \\partial \\bar z = g$.\nThe kernel $\\frac{1}{\\pi z}$ is a fundamental solution of the operator $\\partial / \\partial \\bar z$.\nThis is the starting point of the $\\bar\\partial$ method, of the theory of the Beltrami equation through the Cauchy and Beurling transforms, and of generalizations to several complex variables.\n",
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