Biholomorphism

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Biholomorphism is property of a holomorphic function of complex variable.

Definiton

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

Function from to is called biholomorphic if there exist holomorphic function such that

and
.

Examples of biholomorphic functions

Linear function

The linear function is such function that there exist complex numners and such that ~.

At , such function is biholomorpic in the whole complex plane. Then, in the definition, the case is reallized.

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