# Biholomorphism

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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 ${\displaystyle f}$ from ${\displaystyle A\subseteq \mathbb {C} }$ to ${\displaystyle B\subseteq \mathbb {C} }$ is called biholomorphic if there exists a holomorphic function ${\displaystyle g=f^{-1}}$ which is a two-sided inverse function: that is,

${\displaystyle f{\big (}g(z){\big )}\!=\!z~\forall z\in B~}$ and
${\displaystyle g{\big (}f(z){\big )}\!=\!z~\forall z\in A~}$.

## Examples of biholomorphic functions

### Linear function

A linear function is a function ${\displaystyle f}$ such that there exist complex numbers ${\displaystyle a\in \mathbb {C} }$ and ${\displaystyle b\in \mathbb {C} }$ such that ${\displaystyle f(z)\!=\!a\!+\!b\cdot z~\forall z\in \mathbb {C} ~}$.

When ${\displaystyle b\neq 0}$, such a function ${\displaystyle f}$ is biholomorpic in the whole complex plane: in the definition we may take ${\displaystyle A=B=\mathbb {C} }$.

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

The quadratic function ${\displaystyle f}$ from ${\displaystyle A=\{z\in \mathbb {C} :\Re (z)\!>\!0\}}$ to ${\displaystyle B=\{z\in \mathbb {C} :|\arg(z)|\!<\!\pi \}}$ such that ${\displaystyle f(z)=z^{2}=z\cdot z~\forall z\in A}$.

## Examples of non-biholomorphic functions

The quadratic function ${\displaystyle f}$ from ${\displaystyle A=\{z\in \mathbb {C} \}}$ to ${\displaystyle B=\{z\in \mathbb {C} \}}$ such that ${\displaystyle f(z)=z^{2}=z\cdot z~\forall z\in A}$.

Note that the quadratic function is biholomorphic or non-biholomorphic dependending on the domain ${\displaystyle A}$ under consideration.