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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $D \\subseteq \\C^n$ be a [[bounded|09QSZVVM26]] open set with $C^1$ boundary $\\partial D$ such that $\\C^n \\setminus \\bar D$ is [[connected|0SPE8WNWFV]].\nLet $f \\colon \\partial D \\to \\C$ be a $C^1$ function that satisfies the [[tangential Cauchy–Riemann equations|X7GP9BECD8]] on $\\partial D$.\nThen there is a continuous function $F$ on $\\bar D$ that is [[holomorphic|507C34W0ZT]] on $D$ and satisfies $F = f$ on $\\partial D$.\n\n## Remarks\n\nThe proof uses the [[Bochner–Martinelli formula|YMHYAEJQ80]].\nThe Bochner–Martinelli integral of $f$ over $\\partial D$ defines a function $F_-$ on $D$ and a function $F_+$ on $\\C^n \\setminus \\bar D$.\nThe tangential Cauchy–Riemann equations make both functions holomorphic, and the jump formula gives $F_- - F_+ = f$ on $\\partial D$.\nThe function $F_+$ tends to $0$ at infinity.\nOn each complex line that misses $\\bar D$ it is an entire function of one variable tending to $0$, hence zero by [[Liouville's theorem|0EHKVRF4J6]].\nThese lines fill a nonempty open set.\nSince $\\C^n \\setminus \\bar D$ is connected, $F_+ = 0$ by the identity theorem, so $F = F_-$ has boundary values $f$.\n\nExample: if $f$ is the restriction to $\\partial D$ of a function holomorphic near $\\bar D$, then $f$ satisfies the tangential Cauchy–Riemann equations and $F$ is that function.\n\nThe hypothesis $n \\ge 2$ is necessary.\nFor $n = 1$ the tangential condition is empty.\nThe function $f(z) = \\bar z$ on the unit circle has no holomorphic extension to the disc that is continuous up to the boundary.\nIndeed $\\int_{\\abs{z} = 1} \\bar z \\, dz = 2\\pi i \\ne 0$, whereas Cauchy's theorem would make the integral of such an extension zero.\n\n## Explanations\n\n### L1 human\n\nOn the boundary of a domain in $\\C^n$ with $n \\ge 2$, a function may already satisfy the part of the Cauchy–Riemann equations that only involves directions along the boundary.\nThe theorem says that such a function is always the boundary value of a holomorphic function inside.\nSo, in several variables, holomorphic functions on a domain are determined by, and can be built from, suitable functions on its boundary.\n\n### L2 human\n\nThe theorem is a boundary version of [[Hartogs's extension theorem|FSGES670AZ]].\nApplied to the restriction of a holomorphic function defined near $\\partial D$, it gives the extension of that function to $D$.\nIt is the starting point of the theory of CR functions on real hypersurfaces and of their holomorphic extension.\n",
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