The questions:
(i) Given the functions
$$ \phi_{a} (z) = \frac{z - a}{1 - \overline{a} z} \quad \text{and} \quad \rho_{\alpha} (z) = e^{i \alpha} \cdot z$$
where $|a| <1$ and $\alpha \in \mathbb{R}$, compute $$( \phi_{a} \circ \rho_{\alpha} - \rho_{\alpha} \circ \phi_{a}) (z)$$
(ii) Suppose $$ f(z) = \phi_{a} \circ \rho_{\alpha} (z) \quad \text{and} \quad g(z) = \phi_{b} \circ \rho_{\beta} (z) $$
belong to $Aut(D)$, the group of holomorphic automorphisms of the unit disc. Find $c$ and $\gamma$, where $|c| < 1 $ and $ \gamma \in \mathbb{R}$, such that $$ f \circ g = \phi_{c} \circ \rho_{\gamma}$$
(iii) If $\mathbb{H} = \{ z \in \mathbb{C} \mid Im(z) > 0 \}$ is the upper half plane, use your knowledge of the elements of $Aut(D)$ to describe the elements of $Aut(\mathbb{H})$ as fractional linear transformations.
Where I am having trouble:
(i) The solution I have is very messy, and I'm unsure of what the point of computing it is. What are we trying to show? $$( \phi_{a} \circ \rho_{\alpha} - \rho_{\alpha} \circ \phi_{a}) (z) = \frac{(e^{i\alpha}a - a) + (a\overline{a} - e^{2i\alpha}a\overline{a})z + (e^{2i\alpha}\overline{a} - e^{i\alpha}\overline{a})z^{2}}{(1 - e^{i \alpha}\overline{a}z)(1 - \overline{a}z)}$$
(ii) I tried to compute this however it was a mess of variables and there wasn't any obvious way to separate them into the form: $$ \frac{e^{i\gamma}z - c}{1 - e^{i\gamma}\overline{c}z}$$
(iii) I imagine this part would be quite straightforward but I need a hint on how to make the jump from $Aut(D)$ to $Aut(H)$.
