91Ó°ÊÓ

Let \(f: \overline{\mathbb{C}} \rightarrow \overline{\mathbb{C}}\) be a rational function. Show: $$ \sum_{p \in \bar{C}} \operatorname{Res}(f ; p)=0 \quad \text { (The exactity relation) } $$

Fix \(R>0\), consider the closed disk \(\bar{U}_{R}(0):=\\{z \in \mathbb{C} ;|z| \leq R\\}\) and the continuous functions \(f, g: \bar{U}_{R}(0) \rightarrow \mathbb{C}\) which are analytic on the open disk \(U_{R}(0)\), and have coinciding absolute values on its boundary: $$ |f(z)|=|g(z)| \quad \text { for all }|z|=R $$ Show: If \(f\) and \(g\) have no zeros in \(\bar{U}_{R}(0)\), then there exists a constant \(\lambda \in \mathbb{C}\) with \(|\lambda|=1\) and \(f=\lambda g .\)

Let \(f: \mathbb{C} \rightarrow \mathbb{C}\) be an entire function, which is injective. Show that \(f\) is of the type $$ f(z)=a z+b, \quad a \neq 0 $$ and deduce that each such map is a conformal map from \(\mathbb{C}\) onto itself. The, group Aut(C) of conformal maps \(\mathbb{C} \rightarrow \mathbb{C}\) consists exactly of the affine maps \(z \mapsto a z+b, a, b \in \mathbb{C}, a \neq 0\)