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

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 $$ b\ne 0$$, 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.

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

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.