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- Biholomorphism abstract "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. Formally, a biholomorphic function is a function defined on an open subset U of the -dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set in Cn and the inverse 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 (e.g., see Gunning 1990, Theorem I.11).If there exists a biholomorphism , we say that U and V are biholomorphically equivalent or that they are biholomorphic. If every simply connected open set other than the whole complex plane is biholomorphic to the unit disc (this is the Riemann mapping theorem). The situation is very different in higher dimensions. For example, open unit balls and open unit polydiscs are not biholomorphically equivalent for 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. 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.".
- Biholomorphism thumbnail Biholomorphism_illustration.svg?width=300.
- Biholomorphism wikiPageID "6077963".
- Biholomorphism wikiPageRevisionID "605118562".
- Biholomorphism hasPhotoCollection Biholomorphism.
- Biholomorphism id "6032".
- Biholomorphism title "biholomorphically equivalent".
- Biholomorphism subject Category:Algebraic_geometry.
- Biholomorphism subject Category:Complex_manifolds.
- Biholomorphism subject Category:Functions_and_mappings.
- Biholomorphism subject Category:Several_complex_variables.
- Biholomorphism type Artifact100021939.
- Biholomorphism type ComplexManifolds.
- Biholomorphism type Conduit103089014.
- Biholomorphism type Manifold103717750.
- Biholomorphism type Object100002684.
- Biholomorphism type Passage103895293.
- Biholomorphism type PhysicalEntity100001930.
- Biholomorphism type Pipe103944672.
- Biholomorphism type Tube104493505.
- Biholomorphism type Way104564698.
- Biholomorphism type Whole100003553.
- Biholomorphism type YagoGeoEntity.
- Biholomorphism type YagoPermanentlyLocatedEntity.
- Biholomorphism comment "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. Formally, a biholomorphic function is a function defined on an open subset U of the -dimensional complex space Cn with values in Cn which is holomorphic and one-to-one, such that its image is an open set in Cn and the inverse is also holomorphic.".
- Biholomorphism label "Biholomorfisme".
- Biholomorphism label "Biholomorphe Abbildung".
- Biholomorphism label "Biholomorphism".
- Biholomorphism label "双正則写像".
- Biholomorphism sameAs Biholomorphe_Abbildung.
- Biholomorphism sameAs 双正則写像.
- Biholomorphism sameAs Biholomorfisme.
- Biholomorphism sameAs m.0fnzn6.
- Biholomorphism sameAs Q377166.
- Biholomorphism sameAs Q377166.
- Biholomorphism sameAs Biholomorphism.
- Biholomorphism wasDerivedFrom Biholomorphism?oldid=605118562.
- Biholomorphism depiction Biholomorphism_illustration.svg.
- Biholomorphism isPrimaryTopicOf Biholomorphism.
