% LaTeX file -- PM 352, Winter 2010, Assignment 1

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 352 \hfill Assignment 1 \hfill Due Monday January 18.}} 

\vskip1em 

\begin{Ex}

%\item Let $U$ be a non-empty open subset of $\bC$, and let $f$ and $g$ be analytic functions 
%on $U$.  Prove that $h(z) = f(z)g(z)$ is analytic and derive the product rule.


\item Let $U$ be a non-empty open subset of $\bC$, and let $f$ be an analytic function 
on $U$.  Define $g$ on $\tilde U = \{ z : \ol{z} \in U \}$ by $g(z) = \ol{f(\ol{z})}$.
Prove that $g$ is analytic.\\
\hint check the Cauchy-Riemann equations.


\item Let $U$ be the region below.
Suppose that $g$ is an analytic function on $U$ 
such that $g(z)^6 = z$ and $g(-1) = -i$.  
Find $g(i)$ and $g(3i/4)$ in Cartesian form.
\begin{figure}[h]
\centering
\includegraphics[height = 2in]{fig1}
\end{figure}


\item 
\begin{parts}
\item Compute $\frac{\partial}{\partial z} z$,  $\frac{\partial}{\partial \bar z} z$,
$\frac{\partial}{\partial z}  \bar z$ and  $\frac{\partial}{\partial \bar z} \bar z$.
Hence find a formula for $\frac{\partial}{\partial z} (z^j \bar z^k)$ and 
$\frac{\partial}{\partial \bar z} (z^j \bar z^k)$.
\item Show by induction that for $m,n \ge 0$, 
\[
  \frac{\partial^m}{\partial z^m} \frac{\partial^n}{\partial \bar z^n} (z^j \bar z^k) =
  \begin{cases} \frac{j!k!}{(j-m)!(k-n)!} z^{j-m} \bar z^{k-n} &\IF m \le j \AND n \le k\\
                       0 &\IF m > j \OR n > k. \end{cases}
\]
\item Let $p(z,\bar z) = \sum_{j,k = 0}^N a_{jk} z^j \bar z^k$ be a polynomial in $z$ and $\bar z$.
Show that $\frac{\partial p}{\partial \bar z} \equiv 0$ (i.e. $p$ is analytic) 
if and only if $p = \sum_{j=0}^N a_{j0} z^j$ (i.e.\ no $\bar z$ terms at all).
\end{parts}

\item  Define $\phi(z) =\dfrac{1-z}{1+z}$ for $z \ne -1$.  
\begin{parts}
\item Show that $\phi$ is analytic on $\bC \setminus\{-1\}$ and that $\phi'(z)$ is never $0$.
\item Solve $\phi(z) = w$.  Deduce that $\phi(\phi(z)) = z$, and that $\phi$ maps 
$\bC \setminus\{-1\}$  one-to-one and onto itself.
\item Show that $\phi$ maps the unit disk $\bD = \{ z \in \bC : |z|<1\}$ one-to-one and onto
the right half plane $\bH = \{ z : \re z > 0 \}$.
\hint take $z$ in polar form and compute $\re \phi(z)$.
\end{parts}


\item Let $\bA = \{ z \in \bC : \frac12 < |z| < 2 \}$, and let $f(z) = z + \frac1z$.
\begin{parts}
\item Show that $f$ is 2-to-1 except at the zeroes of $f'$.
\item Show that the image of the circle of radius $r$ is an ellipse except for $r=1$.
What is the image in this case?
\hint An ellipse is the locus of $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ or, 
setting $c^2 = a^2-b^2$,  the set $\{ z=x+iy \in \bC : |z-c| + |z+c| = a \}$.
\item Show that $f(A)$ is the interior of an ellipse.
\end{parts}

\end{Ex}
\end{document}