# Koebe function

###### Definition.

The analytic function^{}

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

on the unit disc in the complex plane^{} is called the Koebe function. For some $|\alpha |=1$, 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 $\u2102\backslash (-\mathrm{\infty},-1/4]$. The maps ${f}_{\alpha}(z)$ can be also given as
${f}_{\alpha}(z)=\overline{\alpha}{f}_{1}(\alpha z)$.
Further note that the power series^{} representation of these
functions is given by

$${f}_{\alpha}(z)=\frac{z}{{(1-\alpha z)}^{2}}=\sum _{n=1}^{\mathrm{\infty}}n{\alpha}^{n-1}{z}^{n}.$$ |

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 |
---|---|

Canonical name | KoebeFunction |

Date of creation | 2013-03-22 14:23:30 |

Last modified on | 2013-03-22 14:23:30 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Definition |

Classification | msc 30C45 |

Synonym | Köbe function |

Defines | rotation of the Koebe function |

Defines | rotation of the Köbe function |