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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.
Contents
Formal definition
Formally, a biholomorphic function is a function
If there exists a biholomorphism
Riemann mapping theorem and generalizations
If
Alternative definitions
In the case of maps f : U → C defined on an open subset U of the complex plane C, some authors (e.g., Freitag 2009, Definition IV.4.1) define a conformal map to be an injective map with nonzero derivative i.e., f’(z)≠ 0 for every z in U. According to this definition, a map f : U → C is conformal if and only if f: U → f(U) is biholomorphic. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, without requiring that the map be injective. According to this weaker definition of conformality, a conformal map need not be biholomorphic even though it is locally biholomorphic. For example, if f: U → U is defined by f(z) = z2 with U = C–{0}, then f is conformal on U, since its derivative f’(z) = 2z ≠ 0, but it is not biholomorphic, since it is 2-1.