\documentclass{amsart}

\DeclareMathOperator{\cis}{cis}

%\DeclareMathOperator{\Im}{Im}

\title{Complex Analysis HW I}



\begin{document}

\maketitle

   This homework is due by class time on Friday 24.  Your solutions must be typeset (preferably
   in \LaTeX) and submitted by email as a PDF file: your name and the homework number should
   appear at the top of the file and in the subject line of your message, and the filename should be
   ``[YourAndrewID]-[number of the HW].pdf''. 


\begin{enumerate} 

\item  Prove that there is no continuous function $f: {\mathbb C} \rightarrow {\mathbb C}$ 
   such that $f(z)^2 = z$ for all $z$. Hint: start with the observation that $f(\cis(\theta))$
   is a continuous function of $\theta$. 

\item  The {\em upper half plane} is the set 
   ${\mathbb H} = \{ z \in {\mathbb C}: \Im(z) > 0 \}$.
   $SL_2({\mathbb R})$ is the set of $2 \times 2$ real matrices with determinant $1$.

   If 
\[
    A = \left ( \begin{array}{rr} a & b \\ c & d \end{array} \right )
\]
   and $z \in {\mathbb H}$, then 
\[
    A \cdot z = \frac{a z + b}{c z + d}.
\]

\begin{enumerate}

\item  Prove that $A \cdot z \in {\mathbb H}$.

\item  Prove that $A \cdot (B \cdot z) = (A B) \cdot z$.

\item  Identify the set of $A$ such that $A \cdot z = z$ for all $z$.

\end{enumerate} 


\item  Let $f: U \rightarrow {\mathbb C}$ be holomorphic, where $U$ is closed under
  complex conjugation, and define 
  a function $g: U \rightarrow {\mathbb C}$ by
  $g(z) = \overline{f(\bar z)}$. Prove that $g$ is holomorphic and find the
  derivative of $g$.

\item  Let $a \in {\mathbb C}$ with $\vert a \vert < 1$, and let
   $U$ be the open unit disk $\{ z \in {\mathbb C} : \vert z \vert < 1 \}$   

\begin{enumerate}

\item  Prove that the map $\phi$ 
  given by $\phi: z \mapsto \frac{z -a}{1 - {\bar a} z}$
 is holomorphic on $U$.

\item  Prove that $\phi$ is a bijection from $U$ to $U$, and that $\phi^{-1}$ is also
  holomorphic on $U$.

\end{enumerate} 

\item Let $U$ be the open unit disk, and let $f :U \rightarrow {\mathbb C}$ 
   be a holomorphic function such that $f$ takes only real values.
   Prove that $f$ is constant. Hint: Cauchy-Riemann. 



\end{enumerate}

\end{document} 


