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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 $D$.\nFix $z \\in D$.\nLet $\\eta = (\\eta_1, \\dots, \\eta_n)$ be a map of class $C^1$ from a neighbourhood of $\\partial D$ to $\\C^n$ such that\n$$\\sum_{j=1}^n \\eta_j(\\zeta) (\\zeta_j - z_j) = 1 \\quad \\text{for all } \\zeta \\in \\partial D.$$\nPut\n$$\\omega_\\eta(\\zeta) = \\frac{(n-1)!}{(2\\pi i)^n} \\sum_{j=1}^n \\eta_j(\\zeta) \\, d\\zeta_j \\wedge \\bigwedge_{k \\ne j} \\left( \\bar\\partial \\eta_k \\wedge d\\zeta_k \\right),$$\nwhere $\\bar\\partial$ is the [[$\\bar\\partial$ operator|M1VKCW3RFW]] in $\\zeta$.\nLet $f$ be a [[holomorphic function|507C34W0ZT]] on a neighbourhood of $\\overline{D}$.\nThen\n$$f(z) = \\int_{\\partial D} f(\\zeta) \\, \\omega_\\eta(\\zeta).$$\n\n## Remarks\n\nThe forms $\\bar\\partial \\eta_k \\wedge d\\zeta_k$ have degree $2$, so the order of the factors in the product does not matter.\nWith $\\eta_j(\\zeta) = (\\bar\\zeta_j - \\bar z_j)/\\abs{\\zeta - z}^2$, the form $\\omega_\\eta$ is the kernel of the [[Bochner–Martinelli formula|YMHYAEJQ80]], and the statement reduces to that formula for holomorphic $f$.\nFor a general $\\eta$, a homotopy argument joining $\\eta$ to the Bochner–Martinelli choice shows that the boundary integral does not depend on $\\eta$.\nFor $n = 1$ the condition forces $\\eta = 1/(\\zeta - z)$, and the formula is the Cauchy integral formula.\nOn a convex domain with $C^2$ boundary, one can choose $\\eta$ depending holomorphically on $z$, which gives integral formulas with kernels holomorphic in $z$.\n\n## Explanations\n\n### L1 human\n\nThere are many integral formulas that recover a holomorphic function inside a domain from its boundary values.\nThe Cauchy–Fantappiè formula produces one formula for each choice of an auxiliary map $\\eta$ satisfying a single linear condition.\nDifferent choices of $\\eta$ adapt the formula to the shape of the domain.\n\n### L2 human\n\nThe Bochner–Martinelli kernel is universal but not holomorphic in $z$, which limits its use.\nThe freedom in $\\eta$ allows kernels that are holomorphic in $z$ on convex or strongly pseudoconvex domains.\nSuch kernels, built by Henkin, Ramírez and others, give solutions of the $\\bar\\partial$ equation with estimates and results on approximation by holomorphic functions.\n",
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