Koebe function

Example

The Koebe function is $$k(z) = \frac{z}{(1 - z)^2} = \sum_{n=1}^{\infty} n z^n, \qquad \abs{z} < 1.$$ It is a univalent function on the unit disc $\mathbb{D}$ in the class $S$, and $k(\mathbb{D}) = \C \setminus (-\infty, -1/4]$.

Check: one has $$k(z) = \frac{1}{4}\left( \left(\frac{1 + z}{1 - z}\right)^2 - 1 \right).$$ The Möbius map $z \mapsto (1+z)/(1-z)$ maps $\mathbb{D}$ bijectively onto the right half-plane $\set{\operatorname{Re} w > 0}$. Squaring maps the right half-plane bijectively onto $\C \setminus (-\infty, 0]$. Finally $w \mapsto (w - 1)/4$ maps $\C \setminus (-\infty, 0]$ onto $\C \setminus (-\infty, -1/4]$. A composition of injective holomorphic maps is injective and holomorphic, and $k(0) = 0$, $k'(0) = 1$.