# Koebe function

###### Definition.
 $f(z):=\frac{z}{(1-z)^{2}}$

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

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

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

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

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

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1995.
Title Koebe function KoebeFunction 2013-03-22 14:23:30 2013-03-22 14:23:30 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 30C45 Köbe function rotation of the Koebe function rotation of the Köbe function