Biholomorphism

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.

-dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set

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 or Corollary E.10 pg.

every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (this is the Riemann mapping theorem).

In fact, there does not exist even a proper holomorphic function from one to the other.

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.

Notice that per definition of biholomorphisms, nothing is assumed about their derivatives, so, this equivalence contains the claim that a homeomorphism that is complex differentiable must actually have nonzero derivative everywhere.

This article incorporates material from biholomorphically equivalent on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.