Biholomorphism: Difference between revisions
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'''Biholomorphism''' is a property of a [[holomorphic function|holomorphic]] [[function of a complex variable]]. | '''Biholomorphism''' is a property of a [[holomorphic function|holomorphic]] [[function of a complex variable]]. | ||
Revision as of 17:36, 25 January 2009
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.