Is there a non-trivial Möbius transformation from the upper half-plane to itself that fixes both the points \(1+i\) and \(-1+i\)? Either find an example of such a map or show that none exists.
Find an automorphism of the unit disc that takes \(\frac{1}{2}\) to \(0\) and (when considered as a map \(\hat{\mathbb C}\to \hat{\mathbb C}\)) also takes \(-1\) to \(-i\).
Find an automorphism of the unit disc that takes \(-\frac i 2\) to \(0\) and (when considered as a map \(\hat{\mathbb C}\to \hat{\mathbb C}\)) also takes \(1\) to \(i\).
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 \(f(z)\).
Find a Möbius transformation \(f\) from the unit disc \(\mathbb D\) onto the upper half-plane \(\mathbb H\) that takes \(0\) to \(i\) and (when considered as a map \(\hat{\mathbb C}\to \hat{\mathbb C}\)) also takes \(i\) to \(2\). Give an explicit formula for \(f(z)\).
Let \(\Omega\subset\mathbb{C}\) be the set of all
complex numbers \(z\) for which \(\mathop{\text{Re}}(z)>-1\) and \(\mathop{\text{Im}}(z)>-1\).
(a) Find a biholomorphic map \(f\) from
\(\Omega\) onto the upper half-plane.
Give an explicit formula for \(f(z)\).
(b) Hence, find a biholomorphic map \(\tilde
f\) from \(\Omega\) onto the
open unit disc. Give an explicit formula for \(\tilde f(z)\).
Does there exist a biholomorphic map taking the closed upper half of the unit disc onto the closed unit disc; i.e. from \(\left\{z\in\mathbb{C}:|z|\leq1,\text{ Im}\,z\geq0\right\}\) onto \(\left\{w\in\mathbb{C}:|w|\leq1\right\}\)? Either find an example of such a map or show that none exists. Hint: The boundary would be mapped to the boundary.
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 \(\mathbb D\).
Consider the map \(z\to z^2\).
(i) Find and sketch the images of the lines \(\mathop{\text{Im}}z=b\) (for \(0<b<1\)).
Hint: find a parametrisation for the lines.
(ii) Find the image of \(\{z:
0<\mathop{\text{Im}}z<1\}\) under this map.
Describe the image of
(i) \(\{z: |z-1|>1\}\) under \(z\to w=\frac{z}{z-2}\)
(ii) \(\{z: |z-i|<1, \hbox{Re }
z<0\}\) under \(z\to
w=\frac{z-2i}{z}\)
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 \(f(z)\).
Find the unique Möbius transformation \(f(z)\) taking the ordered set of points \(\{ 0,-1, -i \}\) to the ordered set of points \(\{ 1,\infty,i \}\) in \(\hat{\mathbb C}\).
Let \(C_1\) be the circle through \(0\), \(-1\) and \(i\), and let \(C_2\) be the circle through \(0\), \(-1\) and \(-i\). Let \(\mathcal R\) be the intersection of the interiors of the two circles. Find the image of \(\mathcal R\) under your map \(f\), and hence construct a biholomorphic map from \(\mathcal R\) to the set \(\Omega: = \{w \in \mathbb{C}: -\pi/4 < \mathop{\text{Arg}}(w) < \pi/4 \}\).
Find a biholomorphic map from \(\mathcal R\) to the upper half-plane \(\mathbb H\).
Consider the region \(P \subset \mathbb
C\) defined by \(P = \Bigl\{
z\in{\mathbb D} \>:\>
-3\pi/4<\mathop{\text{Arg}}z<3\pi/4\Bigr\} \>.\)
(a) Draw \(P\) in the complex
plane.
(b) Find a biholomorphic map from \(P\)
onto the upper half-plane \(\mathbb{H}\).
(c) Find a biholomorphic map from \(P\)
onto the lower half-plane \(\mathbb{H}_L:=\{w
\in \mathbb C: \mathop{\text{Im}}(z)<0 \}\).