In the mathematical theory

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 of functions of one

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

 or more complex variables

Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...

, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

 whose inverse

Inverse function

In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

 is also holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

.

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

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

  and one-to-one

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

, such that its image

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

 is an open set in Cⁿ and the inverse is also holomorphic

Holomorphic function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...

. More generally, U and V can be complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

s. One can prove that it is enough for to be holomorphic and one-to-one in order for it to be biholomorphic onto its image.

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

Proper map

In mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...

 holomorphic function from one to the other.

In the case of the complex plane C, some authors (e.g. Freitag 2009, Definition IV.4.1) define conformal map

Conformal map

In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

as a synonym for biholomorphism. The usual conditions for a function to be conformal on C - namely, that is holomorphic and that its derivative is nowhere zero - are equivalent to biholomorphism.