% LaTeX file -- PM 352, Winter 2010, Assignment 4

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 352 \hfill Assignment 4 \hfill Due Wednesday March 10.}} 

\vskip2em 

\begin{Ex}

\item Show that every  \textit{convex} region is simply connected.
\item Let $U$ be a simply connected open set in $\bC$,  and suppose that $f(z)$
is analytic on $U$ and never vanishes.  Show that there is an analytic function $g(z)$
on $U$ such that $g(z)^2 = f(z)$.


\item How many zeros does $p(z) = z^4 -6z + 3$ have in the annulus
$\bA = \{ z \in \bC : 1 < |z| < 2 \}$.


\item Let $p(z) = c \prod_{i=1}^k (z-a_i)^{n_i}$ be a non-constant polynomial.
\begin{parts}
\item Obtain a simplified formula for $\frac{p'(z)}{p(z)}$.
\item Suppose that $p'(b) = 0$. Use (a) to express $b$ as a \textit{convex combination} of the zeros $a_1,\dots,a_k$ of $p$. i.e.\ $Z(p') \subseteq \ol{\operatorname{conv}(Z(p))}$. \ 
\hint use the formula for $0 = \frac{p'(b)}{p(b)}$ from (a), rationalize each term so that the denominator is positive, and then solve for $b$.
\end{parts}


\item Find the Laurent expansion for $f(z) = \dfrac1{(z-1)(z-2)}$ is the regions
(i) $\bD = \{z : |z|<1\}$,\\ (ii) $\bA = \{ z : 1 < |z| < 2 \}$ and (iii) $U = \{ z : |z| > 2 \}$. 

\item Classify the singularities, including orders of poles, for these functions:
\begin{parts}
\item $\tan z$ \hspace{5cm}   (b) $z \sin(1/z)$,
\stepcounter{enumi} 
\item $f(z) = \dfrac{\log z}{(z-1)^3}$ where $\log z$ is the principle branch of the logarithm defined in $\bH = \{z : \re z > 0 \}$ (so that $\log 1 = 0$).
\end{parts}

\bigbreak\bigbreak
\noindent\hspace*{-17pt}\textbf{Bonus Problems. \ Please hand in separately.}

\item[A.]
Let  $f_n(z)$ be analytic functions on a connected open set $U$.
Suppose that $f_n$ never vanishes for $n\ge1$, and that 
they converge u.c.c. to an (analytic) function $f(z)$.
Prove that either $f=0$ or $f$ has no zeros at all.\\
\hint count the zeros of $f_n$ in a small disk around an isolated zero of $f$.

\item[B.]
The curve in the figure is homologous to zero in the plane with the two points removed.
Show that it is not homotopic to a point.\vspace{-3.7mm}
\begin{figure}[h]
\centering
\includegraphics[height = 1.5in]{fig4}
\end{figure}


\end{Ex}
\end{document}