biholomorphism

right|thumb|The complex exponential function mapping biholomorphically a rectangle to a quarter-annulus. 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.

==Formal definition== Formally, a biholomorphic function is a function \phi defined on an open subset U of the n-dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set V in Cn and the inverse \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.