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# 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. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.

Defines:

rotation of the Koebe function, rotation of the K\"obe function

Synonym:

K\"obe function

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

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30C45*no label found*

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