Biholomorphic map

In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic.

Formally, a biholomorphic function is a function ${\displaystyle \phi }$ defined on an open subset U of the ${\displaystyle n}$-dimensional complex space Cⁿ with values in Cⁿ which is holomorphic and one-to-one, such that its image is an open set ${\displaystyle V}$ in Cⁿ and the inverse ${\displaystyle \phi ^{-1}:V\to U}$ is also holomorphic. More generally, U and V can be complex manifolds. As in the case of functions of a single complex variable, a sufficient condition for a holomorphic map to be biholomorphic onto its image is that the map is injective, in which case the inverse is also holomorphic (e.g., see Gunning 1990, Theorem I.11).

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