In the

mathematical theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

of functions of

one 1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sp ...

or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

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 de ...

whose inverse is also holomorphic.

Formal definition

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

\phi

defined on an

open subset In mathematics, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point in it, contains all points of the met ...

''U'' of the

n

-dimensional complex space C^''n'' with values in C^''n'' which is holomorphic and one-to-one, such that its

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

is an open set

V

in C^''n'' and the inverse

\phi^:V\to U

is also holomorphic. More generally, ''U'' and ''V'' can be

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

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 or Corollary E.10 pg. 57). 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 (topology), path between two points can be continuously transformed into any other such path while preserving ...

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 \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...

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

unit ball Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...

s and open unit polydiscs 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 (mathematics), 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-prese ...

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. 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. Other authors (e.g., Conway 1978) define a conformal map as one with nonzero derivative, but without requiring that the map be injective. According to this weaker definition, a conformal map need not be biholomorphic, even though it is locally biholomorphic, for example, by the inverse function theorem. 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