Biholomorphism

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

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

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

Examples of biholomorphic functions

Linear function

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

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

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

Quadratic function

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

Examples of non-biholomorphic functions

Quadratic function

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

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