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  "body": "\n## Definition\n\nLet $n \\ge 2$ and write a point of $\\C^n$ as $(z', z_n)$ with $z' \\in \\C^{n-1}$ and $z_n \\in \\C$.\nA *Hartogs domain* is a [[connected|0SPE8WNWFV]] [[open set|02BH3MCBQB]] $D \\subseteq \\C^n$ such that $(z', e^{i\\theta} z_n) \\in D$ for all $(z', z_n) \\in D$ and all $\\theta \\in \\R$.\nIt is a *complete Hartogs domain* if moreover $(z', \\lambda z_n) \\in D$ for all $(z', z_n) \\in D$ and all $\\lambda \\in \\C$ with $\\abs{\\lambda} \\le 1$.\n\n## Remarks\n\nExample: $D = \\set{(z_1, z_2) \\in \\C^2 : \\abs{z_1} \u003c 1, \\ \\abs{z_2} \u003c 1 - \\abs{z_1}^2}$.\nIt is open and connected, and the condition on $z_2$ involves only $\\abs{z_2}$.\nThis number does not change under $z_2 \\mapsto e^{i\\theta} z_2$, and it does not increase under $z_2 \\mapsto \\lambda z_2$ with $\\abs{\\lambda} \\le 1$.\nSo $D$ is a complete Hartogs domain.\n\nNon-example: $D = \\set{(z_1, z_2) \\in \\C^2 : \\abs{z_1 - z_2} \u003c 1}$.\nThe point $(1, 1/2)$ lies in $D$, but its rotation $(1, -1/2)$ does not, since $\\abs{1 + 1/2} \u003e 1$.\n\nThe [[Hartogs figure|GHJC53RFBT]] is a Hartogs domain that is not complete.\nA [[Reinhardt domain|WA0SE56FVH]] is invariant under rotation of each coordinate separately, so it is a Hartogs domain in particular.\n",
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