(Assignment sheet 7 problem 4) Find a Möbius transformation \(f\) from the upper half-plane \(\mathbb H\) onto the unit disc \(\mathbb D\) that takes \(1+i\) to \(0\) and (when considered as a map \(\hat{\mathbb C}\to \hat{\mathbb C}\)) also takes \(1\) to \(-i\). Give an explicit formula for \(f(z)\).
(Assignment sheet 7 problem 8) Use standard examples to find a biholomorphic map from the upper half \(\Omega:=\{z \in \mathbb D: \mathop{\text{Im}}(z) > 0 \}\) of the unit disc onto the unit disc \(\mathbb D\).
(Assignment sheet 7 problem 11) Construct a biholomorphic map \(f\) from \(\mathcal{R}\) onto \(\mathcal{R}'\), where \(\mathcal{R}=\{z: \mathop{\text{Im}}z<\frac{1}{2}\}\) and \(\mathcal{R}'=\{z\ :\ |z-1|<2\}\). Give an explicit formula for \(f(z)\).
(Assignment sheet 8 problem 3)
Show that for any \(\rho>0\) the sequence \(\bigl\{\frac{1}{nz}\bigr\}_{n\in\mathbb N}\) converges uniformly on \(A=\{z \in \mathbb C: \ {\left|z\right|}\geq \rho\}\).
Does \(\bigl\{\frac{1}{nz}\bigr\}_{n\in\mathbb N}\) converge uniformly on \(\mathbb C^\ast:=\mathbb C\setminus \{ 0\}\)?
(Assignment sheet 8 problem 6) For every \(n \in \mathbb N\), let \(f_n(z)=\sin(z/n)\) for \(z\in \mathbb C\). Show that \(\{f_n\}_{n\in\mathbb N}\) converges pointwise on \(\mathbb C\). Let \(\rho\) be a positive real number. Show that \(\{f_n\}_{n\in\mathbb N}\) converges uniformly on \(A=\{z:{\left|z\right|}\le \rho\}\). Show that \(\{f_n\}_{n\in\mathbb N}\) does not converge uniformly on \(\mathbb C\).
(Assignment sheet 8 problem 9) Prove that \(\sum_{n=0}^{\infty} e^{nz}\) converges uniformly on \(A=\{z \in \mathbb C: \: \mathop{\text{Re}}(z)\le -1\}\), but not on \(B=\{z \in \mathbb C: \: \mathop{\text{Re}}(z)\le 0\}\).
(Assignment sheet 8 problem 10) Let \(R\) satisfy \(0<R<1\). Show that the series \(\displaystyle\sum_{n=1}^{\infty}\frac{z^{n}}{1+z^n}\) converges uniformly on \(A= \{z \in \mathbb C: \: |z|<R \}\). Conclude that the infinite series defines a continuous function on the unit disc \(\mathbb D\).