# Biholomorphism  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

Biholomorphism is a property of a holomorphic function of a complex variable.

## Definition

Using mathematical notation, a biholomorphic function can be defined as follows:

A holomorphic function from to is called biholomorphic if there exists a holomorphic function which is a two-sided inverse function: that is, and .

## Examples of biholomorphic functions

### Linear function

A linear function is a function such that there exist complex numbers and such that .

When , such a function is biholomorpic in the whole complex plane: in the definition we may take .

In particular, the identity function, which always returns a value equal to its argument, is biholomorphic.

The quadratic function from to such that .
The quadratic function from to such that .
Note that the quadratic function is biholomorphic or non-biholomorphic dependending on the domain under consideration.