Biholomorphism: Difference between revisions

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'''Biholomorphism''' is property of a [[holomorphic]] [[function of complex variable]].
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==Definiton==
'''Biholomorphism''' is a property of a [[holomorphic function|holomorphic]] [[function of a complex variable]].
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>
==Definition==
such that
Using [[mathematical notation]], a biholomorphic function can be defined as follows:
 
A [[holomorphic function]] <math>f</math> from <math>A\subseteq \mathbb{C}</math> to <math> B \subseteq \mathbb{C} </math> is called ''biholomorphic'' if there exists a [[holomorphic function]] <math> g=f^{-1}</math> which is a two-sided [[inverse function]]: that is,
: <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
: <math> g\big(f(z)\big)\!=\!z ~ \forall z \in A ~ </math>.
: <math> g\big(f(z)\big)\!=\!z ~ \forall z \in A ~ </math>.
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===Linear function===
===Linear function===
The [[linear function]] is such function <math>f</math> that there exist [[complex numner]]s
A [[linear function]] is a function <math>f</math> such that there exist [[complex number]]s
<math>a \in \mathbb{C}</math> and
<math>a \in \mathbb{C}</math> and
<math>b \in \mathbb{C}</math> such that  <math>f(z)\!=\!a\!+\!b\cdot z~ \forall z \in \mathbb{C}</math>~.
<math>b \in \mathbb{C}</math> such that  <math>f(z)\!=\!a\!+\!b\cdot z~ \forall z \in \mathbb{C}~</math>.


At <math> b\ne 0</math>, such function <math>f</math> is biholomorpic in the whole [[complex plane]]. Then, in the definition, the case <math>A=B=E=\mathbb{C}</math> is reallized.
When <math> b\ne 0</math>, such a function <math>f</math> is biholomorpic in the whole [[complex plane]]: in the definition we may take <math>A=B=\mathbb{C}</math>.


In particular, the [[identity function]], whith always return values equal to its argument, is biholomorphic.
In particular, the [[identity function]], which always returns a value equal to its argument, is biholomorphic.
===Quadratic function===
===Quadratic function===
The [[quadratic function]] <math>f</math> from  
The [[quadratic function]] <math>f</math> from  
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===Quadratic function===
===Quadratic function===
The [[quadratic function]] <math>f</math> from  
The [[quadratic function]] <math>f</math> from  
<math>A= \{ z \in \mathbb{C} </math> to   
<math>A= \{ z \in \mathbb{C} \}</math> to   
<math>B= \{ z \in \mathbb{C} </math>
<math>B= \{ z \in \mathbb{C} \}</math>
such that <math>f(z)=z^2=z\cdot z ~\forall z\in A </math>.
such that <math>f(z)=z^2=z\cdot z ~\forall z\in A </math>.


Note that the quadratic function is biholomorphic or non-biholomorphic dependently on the range <math>A</math> in the definition.
Note that the quadratic function is biholomorphic or non-biholomorphic dependending on the [[domain]] <math>A</math> under consideration.[[Category:Suggestion Bot Tag]]

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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.