Biholomorphically equivalent domains: Difference between revisions

Revision as of 18:26, 16 April 2008

Definition

Let $U, V \subset C^{n}$ be domains (open connected subset). We say that $U, V$ are biholomorphically equivalent if there exists a holomorphic function $φ : U \to V$ with a holomorphic inverse $φ^{- 1} : V \to U$ .

(By a holomorphic function from $U$ to $V$ , we mean a holomorphic function from $U$ to $C^{n}$ , whose image lies completely inside $V$ . In this case, we get a bijection from $U$ to $V$ that is holomorphic both ways.

Revision as of 18:25, 16 April 2008 (view source) Vipul (talk \| contribs) (New page: ==Definition== Let <math>U,V \subset \mathbb{C}^n</math> be domains (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[h...)		Revision as of 18:26, 16 April 2008 (view source) Vipul (talk \| contribs) No edit summary Newer edit →
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	Let <math>U,V \subset \mathbb{C}^n</math> be [[domain]]s (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[holomorphic function]] <math>\varphi:U \to V</math> with a holomorphic inverse <math>\varphi^{-1}:V \to U</math>.		Let <math>U,V \subset \mathbb{C}^n</math> be [[domain]]s (open connected subset). We say that <math>U,V</math> are '''biholomorphically equivalent''' if there exists a [[holomorphic function]] <math>\varphi:U \to V</math> with a holomorphic inverse <math>\varphi^{-1}:V \to U</math>.

	(By a holomorphic function from <math>U</math> to <math>V</math>, we mean a holomorphic function from <math>U</math> to <math>\C^n</math>, whose image lies completely inside <math>V</math>. In this case, we get a bijection from <math>U</math> to <math>V</math> that is holomorphic both ways.		(By a holomorphic function from <math>U</math> to <math>V</math>, we mean a holomorphic function from <math>U</math> to <math>\mathbb{C}^n</math>, whose image lies completely inside <math>V</math>. In this case, we get a bijection from <math>U</math> to <math>V</math> that is holomorphic both ways.