Biholomorphism: Difference between revisions
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'''Biholomorphism''' is a property of a [[holomorphic function|holomorphic]] [[function of a complex variable]]. | |||
==Definition== | |||
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=== | ||
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>. | ||
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]], | In particular, the [[identity function]], which always returns a value equal to its argument, is biholomorphic. | ||
===Quadratic function=== | ===Quadratic function=== | ||
The [[ | The [[quadratic function]] <math>f</math> from | ||
<math>A= \{ z \in \mathbb{C} : \Re(z) \! > \!0 \}</math> to | <math>A= \{ z \in \mathbb{C} : \Re(z) \! > \!0 \}</math> to | ||
<math>B= \{ z \in \mathbb{C} : |\arg(z)| \! < \! \pi \}</math> | <math>B= \{ z \in \mathbb{C} : |\arg(z)| \! < \! \pi \}</math> | ||
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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 | Note that the quadratic function is biholomorphic or non-biholomorphic dependending on the [[domain]] <math>A</math> under consideration. | ||
Latest 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.
