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 $f_1=f$, andnextnote that $f$ is a map from the open unit disc onto $ℂ\backslash \backslash (-\mathrm{\infty },-1/4]$. 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 thesefunctions 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.