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  "body": "\n## Example\n\nLet $n \\ge 2$ and write a point of $\\C^n$ as $z = (z', z_n)$ with $z' \\in \\C^{n-1}$, and $\\abs{z'} = \\max_{j \u003c n} \\abs{z_j}$.\nLet $\\Delta^n = \\set{z : \\abs{z'} \u003c 1, \\abs{z_n} \u003c 1}$ be the unit [[polydisc|DJBWP7D3M8]], and fix $0 \u003c r \u003c 1$ and $0 \u003c s \u003c 1$.\nThe *Hartogs figure* is the connected open set\n$$H = \\set{z \\in \\Delta^n : \\abs{z'} \u003c r \\text{ or } \\abs{z_n} \u003e s}.$$\nIt is the union of a thin column $\\set{\\abs{z'} \u003c r}$ and a shell $\\set{s \u003c \\abs{z_n} \u003c 1}$ inside $\\Delta^n$.\n\nEvery [[holomorphic function|507C34W0ZT]] $f$ on $H$ extends to a holomorphic function on all of $\\Delta^n$.\nFix $t$ with $s \u003c t \u003c 1$ and put\n$$F(z', z_n) = \\frac{1}{2\\pi i} \\int_{\\abs{\\zeta} = t} \\frac{f(z', \\zeta)}{\\zeta - z_n} \\, d\\zeta \\quad \\text{for } \\abs{z'} \u003c 1, \\ \\abs{z_n} \u003c t.$$\nThe circle $\\abs{\\zeta} = t$ lies in the shell, so the integrand is defined, and $F$ is holomorphic by differentiation under the integral sign.\nFor $\\abs{z'} \u003c r$, the one-variable function $\\zeta \\mapsto f(z', \\zeta)$ is holomorphic on the whole unit disc, so the Cauchy integral formula gives $F = f$ there.\nThe sets where $F$ and $f$ are both defined form a connected open set, so the identity theorem gives $F = f$ on all of it.\nGluing $F$ and $f$ gives the extension to $\\Delta^n$.\n\n## Remarks\n\nSo $H$ is not a [[domain of holomorphy|4487CSC0MB]]: every holomorphic function on $H$ continues to the larger set $\\Delta^n$.\nThis is the basic instance of [[Hartogs's extension theorem|FSGES670AZ]].\nThe extra variables are essential: in one variable the shell alone is the annulus $s \u003c \\abs{z} \u003c 1$, and $1/z$ is holomorphic there but does not extend to the unit disc.\n\n## Explanations\n\n### L1 human\n\nPicture a solid cylinder made of a thin vertical pole together with a thick outer ring, with the space between them missing.\nIn several complex variables, every holomorphic function defined on this shape automatically extends to fill the missing part.\nThe reason is that values on the outer ring determine, through Cauchy's integral, a function on the whole cylinder.\nThis picture is the starting point for all extension phenomena in several variables.\n",
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    "name": "Hartogs figure",
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