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  "body": "\n## Statement\n\nLet $n \\ge 2$ and write $z = (z', z_n)$ with $z' \\in \\C^{n-1}$ and $\\abs{z'} = \\max_{j \u003c n} \\abs{z_j}$.\nLet $\\Delta^n$ be the unit [[polydisc|DJBWP7D3M8]] in $\\C^n$, let $0 \u003c r \u003c 1$ and $0 \u003c s \u003c 1$, and let\n$$H = \\set{z \\in \\Delta^n : \\abs{z'} \u003c r \\text{ or } \\abs{z_n} \u003e s}.$$\nThen every [[meromorphic function|W6FTSF57T1]] on $H$ is the restriction of a unique meromorphic function on $\\Delta^n$.\n\n## Remarks\n\nThe set $H$ is the [[Hartogs figure|GHJC53RFBT]].\nFor holomorphic functions the extension is given by a Cauchy integral in $z_n$ over a circle in the shell $s \u003c \\abs{z_n} \u003c 1$.\nFor meromorphic functions one must also control the poles of the one-variable functions $z_n \\mapsto f(z', z_n)$ as $z'$ varies, which makes the proof harder.\nUniqueness follows from the identity theorem, since $H$ is a nonempty open subset of the connected set $\\Delta^n$.\n\nExample: $f(z) = 1/(z_n - z_1)$ is meromorphic on $H$, and it extends to the meromorphic function $1/(z_n - z_1)$ on $\\Delta^n$.\nWhen $r \\le s$, its polar set $\\set{z_n = z_1}$ passes through the region between the column and the shell.\n\nThe hypothesis $n \\ge 2$ is needed: in one variable, $e^{1/z}$ is holomorphic on the annulus $s \u003c \\abs{z} \u003c 1$ but has no meromorphic extension to the unit disc.\n\n## Explanations\n\n### L1 human\n\nHartogs's extension theorem says that holomorphic functions on a Hartogs figure fill in the missing region.\nLevi's theorem says the same for meromorphic functions, which are allowed to have poles.\nThe poles can run through the missing region, but the function still extends there as a quotient of holomorphic functions.\n\n### L2 human\n\nThe theorem shows that extension phenomena in several variables are not limited to holomorphic functions.\nMany continuation arguments in several variables move a Hartogs figure step by step through a domain.\nLevi's theorem lets these arguments run for meromorphic functions as well as for holomorphic ones.\nIt is used, for example, in the study of meromorphic functions on complex manifolds and of the singularities of meromorphic maps.\n",
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