Osgood's theorem (function theory)

The set of Osgood (after William Osgood ) is a statement of function theory and states that any injective holomorphic function a biholomorphic map is on her image.

sentence

Be open and an injective holomorphic function. Then it is open and the inverse mapping is holomorphic, i.e. the mapping is biholomorphic. ${\displaystyle \Omega \subseteq \mathbb {C} ^{n}}$ ${\displaystyle f\colon \Omega \to \mathbb {C} ^{n}}$ ${\displaystyle \Omega ':=f(\Omega )\subseteq \mathbb {C} ^{n}}$ ${\displaystyle f^{-1}\colon \Omega '\to \Omega }$ ${\displaystyle f}$

Since holomorphism is a local property, the theorem also applies to mappings between complex manifolds .

Difference from the real case

The statement of the theorem does not hold for real - analytic functions . For example, with is bijective and analytical, but the inverse function is no longer analytical at the zero point. ${\displaystyle f\colon (-1,1)\to (-1,1)}$ ${\displaystyle x\mapsto x^{3}}$

literature

• Raghavan Narasimhan: Several Complex Variables. , University of Chicago Press, Chicago 1971, ISBN 0-226-56817-2