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 from to is called biholomorphic if there exists a holomorphic function which is a two-sided inverse function: that is,

and
.

Examples of biholomorphic functions

Linear function

A linear function is a function such that there exist complex numbers and such that .

When , such a function is biholomorpic in the whole complex plane: in the definition we may take .

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

Quadratic function

The quadratic function from to such that .

Examples of non-biholomorphic functions

Quadratic function

The quadratic function from to such that .

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