PlanetMath: Koebe function

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Koebe function

(Definition)

Definition 1 The analytic function

$\displaystyle f(z) := \frac{z}{(1-z)^2}$

on the unit disc in the complex plane is called the Koebe function. For some $\lvert \alpha \rvert = 1$ , the functions

$\displaystyle f_\alpha(z) := \frac{z}{(1- \alpha z)^2}$

are called rotations of the Koebe function.

Firstly note that , and next note that 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

$\displaystyle f_\alpha(z) = \frac{z}{(1-\alpha z)^2} = \sum_{n=1}^\infty n \alpha^{n-1} z^n .$

Also note that these functions belong to the class of Schlicht functions.

Bibliography

1: John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.

"Koebe function" is owned by jirka.

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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, 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 5066 times total.

Classification:

AMS MSC:

30C45 (Functions of a complex variable :: Geometric function theory :: Special classes of univalent and multivalent functions )

Pending Errata and Addenda

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