# Biholomorphism

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

## Contents

## 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.

### Quadratic function

The quadratic function from to such that .

## Examples of non-biholomorphic functions

### Quadratic function

The quadratic function from to such that .

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