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  "body": "\n## Summary\n\nA biholomorphism is a bijective [[holomorphic map|3MQTWKBBBT]] between open subsets of $\\C^n$ whose inverse is also holomorphic.\nTwo domains related by a biholomorphism are biholomorphically equivalent: they are the same for complex analysis.\nIn one variable every simply connected proper domain of $\\C$ is biholomorphic to the disc, by the [[Riemann mapping theorem|02WHMEG5ZR]].\nIn several variables this fails: for $n \\ge 2$ the [[unit ball|PDQ9KW9FAP]] and the unit [[polydisc|DJBWP7D3M8]] are [[not biholomorphic|1NF8T07H9X]].\n\n## Explanations\n\n### L1 human\n\nA biholomorphism is a change of complex coordinates that can be undone.\nIt matches the points of two domains one to one, and it carries holomorphic functions on one domain to holomorphic functions on the other.\nFor example, the map $(z_1, z_2) \\mapsto (2 z_1, \\ z_2 + z_1^2)$ is a biholomorphism of $\\C^2$ onto itself, with inverse $(w_1, w_2) \\mapsto (w_1/2, \\ w_2 - w_1^2/4)$.\nBiholomorphic domains cannot be told apart by complex analysis.\n",
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