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Homework 7: Complex Analysis

Problem 1

The map $latex \Phi(z) = i\dfrac{1+z}{1-z}$ is a biholomorphic mapping from the unit disk $latex \mathbb D$ onto the upper half-plane $latex \mathbb H$. $latex \Phi$ is called the Cayley transform.

Problem 2

Let $latex A=\begin{pmatrix} a & b\\c & d \end{pmatrix}\in SL(2,\mathbb R)$ and $latex \phi_A$ the Möbius transformation $latex \phi_A(z) = \dfrac{az+b}{cz+d}$. Then $latex \phi_A\in\text{Aut}(\mathbb H)$.

Problem 3

    For $latex a\in\mathbb H$, give an explicit $latex A\in SL(2,\mathbb R)$ so that $latex \phi_A$ takes $latex a$ to $latex i$.

    Problem 4

    If $latex \psi\in\text{Aut}(\mathbb H)$, then there exists $latex A\in SL(2,\mathbb R)$ so that $latex \psi = \phi_A$.
    (Hint: Consider the matrix $latex A = \begin{pmatrix} \cos\theta & -\sin\theta\\\sin\theta & \cos\theta \end{pmatrix}$ and verify that $latex \Phi^{-1}\circ\phi_A\circ\Phi$ is a rotation in $latex \mathbb D$. Use the previous problem and the discussion in class, with mappings fixing $latex 0$ in $latex \mathbb D$ replaced by mappings fixing $latex i$ in $latex \mathbb H$.)

    Problem 5

    Discuss the "uniqueness" of the matrix $latex A$ in theprevious problem, and thus find a subgroup $latex N\subset SL(2,\mathbb R)$ such that
    $latex \text{Aut}(\mathbb H)\cong SL(2,\mathbb R)/N$.