Biholomorphism

Biholomorphism is property of a holomorphic function of complex variable.

Definiton

Using the mathematical notations, biholomorphic function can be defined as follows:

Function $f$ from $A\subseteq \mathbb {C}$ to $B\subseteq \mathbb {C}$ is called biholomorphic if there exist holomorphic function $g=f^{-1}$ such that

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

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

Examples of biholomorphic functions

Linear function

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

At $b\neq 0$, such function $f$ is biholomorpic in the whole complex plane. Then, in the definition, the case $A=B=E=\mathbb {C}$ is reallized.

In particular, the identity function, whith always return values 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 dependently on the range $A$ in the definition.