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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $D \\subseteq \\C^n$ be a bounded domain with $C^1$ boundary $\\partial D$, oriented as the boundary of $D$.\nFor $\\zeta \\ne z$ in $\\C^n$, let $\\omega(\\zeta, z)$ be the Bochner–Martinelli kernel, the $(n, n-1)$-form in $\\zeta$\n$$\\omega(\\zeta, z) = \\frac{(n-1)!}{(2\\pi i)^n} \\frac{1}{\\abs{\\zeta - z}^{2n}} \\sum_{j=1}^n (\\bar\\zeta_j - \\bar z_j) \\, d\\zeta_j \\wedge \\bigwedge_{k \\ne j} \\left( d\\bar\\zeta_k \\wedge d\\zeta_k \\right).$$\nLet $f$ be a function of class $C^1$ on a neighbourhood of the closure $\\overline{D}$.\nThen for every $z \\in D$,\n$$f(z) = \\int_{\\partial D} f(\\zeta) \\, \\omega(\\zeta, z) - \\int_{D} \\bar\\partial f(\\zeta) \\wedge \\omega(\\zeta, z),$$\nwhere $\\bar\\partial$ is the [[$\\bar\\partial$ operator|M1VKCW3RFW]] in $\\zeta$.\n\n## Remarks\n\nThe forms $d\\bar\\zeta_k \\wedge d\\zeta_k$ have degree $2$, so the order of the factors in the product does not matter.\nFor fixed $z$, the kernel satisfies $d_\\zeta \\, \\omega(\\zeta, z) = 0$ on $\\zeta \\ne z$.\nApply Stokes' theorem on $D$ minus a small ball around $z$ and let the radius tend to $0$; the small sphere contributes $f(z)$ in the limit.\nIf $f$ is [[holomorphic|507C34W0ZT]] near $\\overline{D}$, then $\\bar\\partial f = 0$ and $f(z) = \\int_{\\partial D} f(\\zeta) \\, \\omega(\\zeta, z)$.\nFor $n = 1$, $\\omega(\\zeta, z) = \\frac{1}{2\\pi i} \\frac{d\\zeta}{\\zeta - z}$ and the formula is the Cauchy–Pompeiu formula, which contains the Cauchy integral formula.\nFor $n \\ge 2$ the kernel is not holomorphic in $z$, unlike the Cauchy kernel of the polydisc, but it works on every bounded domain with $C^1$ boundary.\n\n## Explanations\n\n### L1 human\n\nThe formula recovers the values of a function inside a domain from its values on the boundary.\nA correction term measures how far the function is from being holomorphic.\nFor holomorphic functions the correction is zero.\nIt is a version of the Cauchy integral formula that works on any reasonable bounded domain of $\\C^n$, not only on products of discs.\n\n### L2 human\n\nThe kernel is built from the fundamental solution of the Laplacian on $\\R^{2n}$, which is why it is not holomorphic in $z$ when $n \\ge 2$.\nThe formula is a main tool for solving the equation $\\bar\\partial u = g$ and for proving extension results such as the Hartogs–Bochner theorem.\nIt is the special case $\\eta_j = (\\bar\\zeta_j - \\bar z_j)/\\abs{\\zeta - z}^2$ of the [[Cauchy–Fantappiè formula|R9JJ44TMQB]] for holomorphic functions.\n",
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