In the

mathematical theory A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms. It can also simultaneously be a body of knowledge (e.g., based on known axioms and definitions), and so in this sense can refer to an area o ...

of functions of one or more complex variables, and also in

complex algebraic geometry In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and co ...

, a biholomorphism or biholomorphic function is a

bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex deriv ...

whose inverse is also

holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...

Formal definition

Formally, a ''biholomorphic function'' is a function

\phi

defined on an

open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...

''U'' of the

n

-dimensional complex space C^''n'' with values in C^''n'' which is

and one-to-one, such that its

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...

is an open set

V

in C^''n'' and the inverse

\phi^:V\to U

is also

. More generally, ''U'' and ''V'' can be

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...

s. 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

\phi \colon U \to V

, we say that ''U'' and ''V'' are biholomorphically equivalent or that they are biholomorphic.

Riemann mapping theorem and generalizations

n=1,

every

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

open set other than the whole complex plane is biholomorphic to the

unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...

(this is the

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphic ...

). The situation is very different in higher dimensions. For example, open

unit ball Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (al ...

s and open unit

polydisc In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs. More specifically, if we denote by D(z,r) the open disc of center ''z'' and radius ''r'' in the complex plane, then a ...

s are not biholomorphically equivalent for

n>1.

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

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 In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...

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'') = ''z''² with ''U'' = C–, then ''f'' is conformal on ''U'', since its derivative ''f''’(''z'') = 2''z'' ≠ 0, but it is not biholomorphic, since it is 2-1.

References

* * * * * {{PlanetMath attribution, urlname=BiholomorphicallyEquivalent, title=biholomorphically equivalent Several complex variables Algebraic geometry Complex manifolds Functions and mappings