Biholomorphism: Difference between revisions
imported>Dmitrii Kouznetsov |
imported>Dmitrii Kouznetsov m (→Definiton: spacebar) |
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Using the [[mathematical notations]], biholomorphic function can be defined as follows: | Using the [[mathematical notations]], biholomorphic function can be defined as follows: | ||
Function <math>f</math> from <math>A\subseteq \mathbb{C}</math> to <math> B \subseteq \mathbb{C} </math>is called biholomorphic if there exist [[holomorphic function]] <math> g=f^{-1}</math> | Function <math>f</math> from <math>A\subseteq \mathbb{C}</math> to <math> B \subseteq \mathbb{C} </math> is called biholomorphic if there exist [[holomorphic function]] <math> g=f^{-1}</math> | ||
such that | such that | ||
: <math> f\big(g(z)\big)\!=\!z ~ \forall z \in B ~ </math> and | : <math> f\big(g(z)\big)\!=\!z ~ \forall z \in B ~ </math> and |
Revision as of 18:03, 24 November 2008
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.