Biholomorphism

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

1 Definition
2 Examples of biholomorphic functions
- 2.1 Linear function
- 2.2 Quadratic function
3 Examples of non-biholomorphic functions
- 3.1 Quadratic function

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 $b\ne 0$ , 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 $f(z)=z^2=z\cdot z ~\forall z\in A$ .