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Koebe function
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(Definition)
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Definition 1 The analytic function \begin{equation*} f(z) := \frac{z}{(1-z)^2} \end{equation*}on the unit disc in the complex plane is called the Koebe function. For some $\lvert \alpha \rvert = 1$ the functions \begin{equation*} f_\alpha(z) := \frac{z}{(1- \alpha z)^2} \end{equation*}are called rotations of the Koebe function.
Firstly note that $f_1 = f$ and next note that $f$ is a map from the open unit disc onto ${\mathbb{C}} \backslash (-\infty,-1/4]$ The maps $f_\alpha (z)$ can be also given as $f_\alpha(z) = \bar{\alpha} f_1 (\alpha z )$ Further note that the power series representation of these functions is given by \begin{equation*} f_\alpha(z) = \frac{z}{(1-\alpha z)^2} =
\sum_{n=1}^\infty n \alpha^{n-1} z^n . \end{equation*} Also note that these functions belong to the class of Schlicht functions.
- 1
- John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.
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"Koebe function" is owned by jirka.
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(view preamble | get metadata)
| Other names: |
Köbe function |
| Also defines: |
rotation of the Koebe function, rotation of the Köbe function |
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Cross-references: schlicht functions, class, representation, power series, onto, open, map, functions, complex plane, unit disc, analytic function
There are 3 references to this entry.
This is version 3 of Koebe function, born on 2004-06-04, modified 2005-03-07.
Object id is 5888, canonical name is KoebeFunction.
Accessed 6075 times total.
Classification:
| AMS MSC: | 30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions ) |
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Pending Errata and Addenda
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