% LaTeX file -- PM 352, Winter 2010, Assignment 2

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\begin{document}
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%%%%%%%   TITLE   %%%%%%
{\noindent\textbf{\large PM 352 \hfill Assignment 2 \hfill Due Friday January 29.}} 

\vskip2em 

\begin{Ex}

\item Let $f(z) = e^{1/z}$ for $z \ne 0$.
Let $r > 0$, and set $\bA_r = \{ z \in \bC : 0 < |z| < r\}$. 
Determine the range $f(\bA_r)$.
\hint Solve $e^{1/z} = a$.


\item Define $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$ and $\cos z = \frac{e^{iz} + e^{-iz}}{2}$ for $z \in \bC$.
\begin{parts}
\item Show that $\sin(w+z) = \sin z \cos w + \cos z \sin w$ for all $w,z \in \bC$.

\item Notice that $\sin(iy) = i\sinh y$ and $\cos(iy) = \cosh y$ for $y\in\bR$.
Show that  for $x,y \in \bR$, 
\[ |\sin(x+iy)|^2 = \sin^2 x + \sinh^2 y \qand
 |\cos(x+iy)|^2 = \cos^2 x + \sinh^2 y .
\]
\item Solve $\sin z = 2$.
\end{parts}


\item Prove that if $f$ is analytic, then the real-valued function $\log |f(z)|$ is harmonic.\\
\hint write $f(x+iy) = u(x,y) + i v(x,y)$.


\item \begin{parts}
\item Let $f$ be analytic on an open set $U$ containing the unit circle \vspace{.4ex}
$\bT=\{ e^{i\theta} : 0 \le \theta \le 2\pi\}$.  Show that if $f$ has an analytic primitive
defined on all of $U$, then $\int_0^{2\pi} f(e^{it}) e^{it}\,dt = 0$.

\item Let $f(z) = 1/z$ on $\bA = \{ z : \frac12 < |z| < 2 \}$.  
Compute $\int_0^{2\pi} f(e^{it}) e^{it}\,dt$.
Use this to show that there is no branch of the logarithm defined on $\bA$.
\end{parts}

\item Let $g(z) = \sum_{n\ge1} (n^2+n) (z+1)^n$. 
Find the radius of convergence and a closed form formula for $g(z)$.  
Justify your steps.  


\item 
Let $f(z) = \sum_{n\ge0} a_n (z-z_0)^n$ be a power series which has a finite
radius of convergence $R>0$.  Suppose that there is some $z_1$ with $|z_1 - z_0| = R$
such that the series converges absolutely at $z_1$.  Prove that the series converges
uniformly on the closed disk $\ol{B_R(z_0)}$.


\item Let $f(z) = \sum_{n\ge0} a_n (z-z_0)^n$ be a power series which has a
positive radius of convergence $R$.  Prove that if $a_1 \ne 0$, then $f$ is one-to-one on
a small ball, $B_r(z_0)$, around $z_0$.\\
\hint WLOG, $z_0=0$.  Look at $\frac{f(z)-f(w)}{z-w}$.
\\


\item[A.\ \  \textbf{Bonus Problem. \ Please hand in separately.}]  \strut \newline
You are given two power series, $f(z) = \sum_{n\ge0} a_nz^n$, which has
radius of convergence $R_1>0$, and $g(z) = \sum_{n\ge0} b_nz^n$, which has
radius of convergence $R_2>0$. Define a new power series by setting
$c_n = \sum_{k=0}^n a_k b_{n-k}$ for $n \ge0$, and $h(z) = \sum_{n\ge0} c_nz^n$. 
\begin{parts}
\item Show that $h(z)$ has radius of convergence \textit{at least} $R = \min\{R_1,R_2\}$. 

\item Prove that $h(z) = f(z) g(z)$ for $|z| < \min\{R_1,R_2\}$. 

\item Find an example for which the radius of convergence of the series
for $h$ is greater than $R$.
\end{parts}

\end{Ex}
\end{document}