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  "body": "\n## Summary\n\nAnalytic continuation in several variables extends a [[holomorphic function|507C34W0ZT]] on an open set of $\\C^n$ step by step through overlapping domains.\nIt generalises [[analytic continuation|0M3YQJ81NB]] in one variable, and by the identity theorem each step is unique.\nAs in one variable, continuation along different paths can lead to different values.\nSo the result lives on a domain spread over $\\C^n$ rather than on a subset of $\\C^n$.\nUnlike in one variable, for $n \\ge 2$ every holomorphic function on certain domains continues to a strictly larger domain.\n\n## Explanations\n\n### L1 human\n\nStart with a holomorphic function on a small region and try to push it further, like extending a road piece by piece.\nEach new piece is forced: there is only one holomorphic way to extend on an overlap.\nIn several variables something new happens: for some regions, such as a ball with a hole, every function can be extended across the hole.\nFor example, a holomorphic function on $\\set{z \\in \\C^2 : 1 \u003c \\abs{z} \u003c 2}$ always continues to the whole ball $\\abs{z} \u003c 2$.\n",
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