\documentclass[12pt]{amsart}
\title[Problem Sheet 2]{Complex Analysis: Problem Sheet 2}
\author{}
\usepackage[margin=1.2in]{geometry}
\usepackage{amsmath,amssymb}
\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}
\newcommand{\ds}{\displaystyle}

\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}

\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\renewcommand{\iota}{{\mathbf i}}
\newcommand{\ol}[1]{\overline{#1}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}

\begin{document}
\maketitle


\problem{1} For each of the following subsets of $\C$,
determine whether it is open, closed, or neither.

\begin{tabular}{ll}
(a) $\ds |z-2+\iota| \leq 1$ & (b) $\ds |2z+3|>4$ \\
(c) $\ds \Im\lp\frac{1}{z}\rp > 1$ ($z\neq 0$) & (d) $\Im(z) \neq 0$.\\
(e) $0\leq \arg(z)\leq \frac{\pi}{2}$ ($z\neq 0$) & 
(f) $\C \setminus \Z\ \  
(\ = \{z\in\C : z\text{ is not an integer}\})$ \\
&
\end{tabular}

\problem{2} For the subsets given in Problem 1 above,
determine whether they are bounded or not.\\

\problem{3} Sketch the following subset of $\C$ and
determine whether it is connected or not: $\Re(z^2)>1$.\\

\problem{4} In the following problems,
a subset $S\subset \C$ and a point $z_0\in\C$ are given.
Determine whether $z_0$ is an accumulation point of $S$
or not.
\begin{itemize}
\item[(a)] $\ds S = \left\{ \frac{1}{n} : n=1,2,3\ldots\right\}$, $z_0=0$.\\

\item[(b)] $\ds S = \{0\leq \arg(z)\leq \frac{\pi}{2}, z\neq 0\}$, $z_0=0$.\\

\item[(c)] $S=\Z$, $z_0=\iota$.\\

\item[(d)] $S = \{\Im(z)\neq 0\}$, $z_0 = 1$.\\
\end{itemize} 

\problem{5} Let $S\subset\C$ be such that every accumulation
point of $S$ is in $S$. Prove that $S$ is closed.\\

\problem{6} Let $S\subset \C$ be a finite subset:
$S = \{z_1,z_2,\ldots, z_n\}$. Prove that $S$ has no
accumulation points.\\

\problem{7} Compute the following limits.
({\em Use the results of Section 5.3, Lecture 5}).
\begin{tabular}{ll}
(a) $\ds \lim_{z\to 0} \frac{z^2-2}{z+\iota} \qquad\qquad $ &
(b) $\ds \lim_{z\to 1+\iota} \frac{z+\iota}{z^2+2}$ \\
(c) $\ds \lim_{z\to -2} \frac{1}{z^3} \qquad \qquad $ &
(d) $\ds \lim_{z\to\iota} \frac{\iota z^3-1}{z+\iota}$\\
&
\end{tabular}

\problem{8} Verify that the following limits do not exist:
({\em See the argument in Section 5.2, Lecture 5}).
(a) $\ds \lim_{z\to 0} \frac{\ol{z}}{|z|}$,
(b) $\ds \lim_{z\to 0} \frac{\Im(z^2)}{|z|^2}$.\\

\begin{center}
{\em In problems 9 and 10, use Definition (4.3) from
Lecture 4.}
\end{center}

\problem{9} Let $f(z)$ and $g(z)$ be two functions of a complex
variable. Assume that the following two statements are true:
\begin{enumerate}
\item $\ds \lim_{z\to z_0} f(z) = 0$.\\

\item There exists $M>0$ and $r>0$ such that 
$|g(z)|<M$ for every $z \in D^{\times}(z_0;r)$. Recall:
$D^{\times}(\alpha; r) = \{w\in\C : 0<|w-\alpha|<r\}$
is the punctured disc.
\end{enumerate}

Prove that $\ds \lim_{z\to z_0} f(z)g(z) = 0$.\\

\problem{10} Let $f(z)$ and $g(z)$ be two functions of a complex
variable. Assume that the following two limits exist:
\[
\lim_{z\to z_0} f(z) = A,\ \qquad \lim_{z\to z_0} g(z) = B.
\]
Prove the following:
\begin{enumerate}
\item $\ds \lim_{z\to z_0} (f(z)+g(z)) = A+B$.\\

\item $\ds \lim_{z\to z_0} f(z)g(z) = AB$.\\

\end{enumerate}

\problem{11} Let $f(z)$ be a function of a complex
variable. Assume that the following limit exists:
$\ds\lim_{z\to z_0} f(z) = A$.
Prove the following:
\begin{enumerate}
\item $\ds \lim_{z\to z_0} \ol{f(z)} = \ol{A}$.\\

\item $\ds \lim_{z\to z_0} |f(z)|^2 = |A|^2$.\\
\end{enumerate}

\problem{12} For each of the following functions, check if
the Cauchy--Riemann equations hold. Here $x=\Re(z)$
and $y=\Im(z)$.
\begin{enumerate}
\item $\ds f(z) = x^2 + y^2 + 2xy\iota$.\\

\item $\ds f(z) = e^x\cos(y) + e^x\sin(y)\iota$.\\
\end{enumerate}

\problem{13} For each function $f(z)$ below, write $f(z)
= u(x,y) + v(x,y)\iota$, where $x=\Re(z)$ and $y=\Im(z)$.
Check the Cauchy--Riemann equations.

\begin{enumerate}
\item $\ds f(z) = z^2 + \ol{z}$.\\

\item $\ds f(z) = \frac{1}{z}$.
\end{enumerate}
\end{document}
