\documentclass[a4paper,12pt]{article}
\title{Complex Analysis Questions}
\date{\today}
\author{Mark Allen}
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\begin{document}


\maketitle

\section{Introduction}
This is an attempt to create a list of all possible ``give definition'' or ``prove theorem'' type questions that can be asked based on the notes for the course so far. It includes all exercise/problem type questions given so far in lectures, but doesn't include any homework exercises (yet). The emphasis is on providing a complete overview of what has been covered in the course so far, in chronological order, as well as an easy way to test how well the material is understood, and so I make no attempt to mimic realistic `exam' questions, nor do I attempt to divide the questions into easy and hard. For the type of questions you'll actually see on an exam, see one of the several other lists of questions available.

I intend to make one of these for at least each of the Maths courses, and possibly for the Physics classes too.
\subsection{Week 1}
The first week covered the basic definitions of complex numbers and sets. Everything here should be pretty obvious, with the possible exception of one or two of the proofs.
\begin{enumerate}
\item
Define the field of complex numbers.
\item
Verify the field axioms for the field of complex numbers.
\item
Define the \emph{modulus}.
\item
Define the \emph{complex conjugate}.
\item
Let $ z,w \in \C $ Show that the following hold: 
\begin{enumerate}
\item
triangle inequality;
\item
reverse triangle inequality;
\item
parallelogram law.
\end{enumerate}
\item
Show how each $z = x + iy \in \C$ can be expressed in
\begin{enumerate}
\item
Trigonometric form;
\item
Polar form;
\end{enumerate}
\item
Show how the product of two complex numbers is equivalent to adding the angles and multiplying the lengths.
\item
Prove de Moivre's formula
\item
Solve the following:
\begin{enumerate}
\item
$z^3 = 1$
\item
$z^4 = 16i $
\end{enumerate}
\item
Define the \emph{open disk}.
\item
Define an \emph{interior point}.
\item
Define a \emph{boundary point}.
\item
Define an \emph{isolated point}.
\item
Describe the points in the following sets:
\begin{enumerate}
\item
$ A = \{z \in \C : \mbox{Im}(z) > 0\} $
\item
$ A = \{\frac{i}{n} : n \in \mathbb{N}\} $
\end{enumerate}
\item
Define an \emph{open set}
\item
Define a \emph{closed set}.
\item
Define a \emph{bounded set}.
\item
Define a \emph{compact set}.
\item
State whether the following are open, closed, bounded or compact.
\begin{enumerate}
\item
$ \{ z \in \C : 0 < \mbox{Re}(z) \leq 1, \mbox{Im}(z) = 0\} $
\item
$ \{ z \in \C : 0 \leq \mbox{Re}(z) \leq 1, \mbox{Im}(z) = 0\} $
\end{enumerate}
\item
Define a \emph{connected set}.
\item Is the set $\{z \in \C : |z| = 1 \mbox{ and Im}(z) \neq 0\} $ connected?
\item
Define a \emph{convergent sequence}.
\item
Define the \emph{Cauchy criterion}.
\item
What two statements are equivalent to ``$(z_n)$ is convergent''
\item
Prove this.
\item
Show that:
\begin{enumerate}
\item
$ \left( \frac{i^n}{n} \right)^{\infty}_{n=1} $ converges to $0$
\item
$ \left( \cos{\frac{1}{n}} + i\sin{\frac{1}{n}}\right)^{\infty}_{n=1}$ converges to $1$
\end{enumerate}
\item
Define a \emph{limit point}
\item
Prove the alternative definition for a closed set:
\begin{theorem}
Let $A \subseteq \C$
Then A is closed if and only if A contains all of its limit points.
\end{theorem}
\item
Define the \emph{limit} of a function.
\item
Prove the following theorem:
\begin{theorem}
Let $f,g : A \longrightarrow \C$ with
\[L = \lim_{z \to w} f(z) \mbox{ and } M = \lim_{z \to w} g(z)\]
then:
 \begin{eqnarray*}
 L & = & \lim_{z \to w} f(z) + g(z) = M+L \\
 L & = & \lim_{z \to w} f(z)g(z) = ML \\
 L & = & \lim_{z \to w} \frac{f}{g}(z) = \frac{M}{L}, L \neq 0 
\end{eqnarray*}
\end{theorem}
\item
Define \emph{continuity} at a point.
\end{enumerate}
\subsection{Week 2}
This week mostly focused on the continuity and differentiability of complex functions.

\begin{enumerate}
\item
What two statements are equivalent to `f is continuous on A' (for an open set A in $\C$)
\item
Prove this.
\item
Prove the following theorem:
\begin{theorem}
Let $A,B \in \C$ be open sets and $f:A\rightarrow\C$, $g:B\rightarrow\C$ both continuous with $f(a) \subset B$. Then $g \circ f : A \rightarrow \C$ is continuous.
\end{theorem}
\item
Define \emph{complex differentiability.}
\item
Show that the function \begin{eqnarray*} f : \C & \longrightarrow & \C \\
 z & \longmapsto & z \end{eqnarray*} is complex differentiable everywhere in $\C$.
 \item
 Show that the function \begin{eqnarray*} f: \C & \longrightarrow & \C \\
 z & \longmapsto & \bar{z} \end{eqnarray*} is not complex differentiable everywhere in $\C$.
 \item
 Prove that if $f : A \longrightarrow \C$ is complex differentiable at $w \in A$ then $f$ is continuous at $w$.
 \item
 Prove the sum, product and quotient rules for complex differentiable functions $f,g : A \longrightarrow \C$ at $w \in A$.
 \item
 Prove the following theorem:
 \begin{theorem}
 Let $A \subseteq \C$ be an open set,
 \[z_0 = x_0 + iy_0 \in A\]
 \[f:A\longrightarrow\C\mbox{ , }  f = u+iy \]
 Then $f$ is complex differentiable at $z_0$ if and only if:
 \begin{itemize}
 \item
 $u$ and $v$ are both real differentiable functions at $(x_0,y_0)$
 \item
 The Cauchy-Riemann equations:
 \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\mbox{ , }
  \frac{\partial u}{\partial y} = - \frac{\partial v}{\partial x}\]
  are satisfied at $(x_0,y_0)$
\end{itemize} 
\end{theorem} 
\end{enumerate}
\subsection{Week 3}
This week we proved the complex chain rule, defined analytic functions and started looking at the exponential, logarithmic, and trigonometric functions.
\begin{enumerate}
\item
Define and prove the complex chain rule.
\item
Define an \emph{analytic function}
\item
Show that \begin{eqnarray*}
\mbox{Exp}:\C & \longrightarrow & \C \\
z=x+iy & \longmapsto & e^x(\cos y + i\sin y)
\end{eqnarray*}
is analytic in $\C$.
\item
Why are polynomials analytic in $\C$?
\item
Why are rational functions analytic on their rational domain?
\item
Define the exponential function.
\item
How does the exponential function map complex numbers geometrically?
\item
Prove that the exponential function is analytic in $\C$.
\item
Show that $e^{z+w} = e^{z}e^w$
\item
Prove that $e^z \neq 0 \mbox{ }\forall z \in \C$.
\item
Show that the exponential function is periodic with period $2\pi i$.
\item
Define complex $\sin$ and $\cos$.
\item
Why are $\sin$ and $\cos$ analytic in $\C$?
\item
Show $\frac{d}{dz}\cos{z}=-\sin{z}$ and $\frac{d}{dz}\sin{z}=\cos{z}$.
\item
Prove the following theorem:
\begin{theorem}
Let $f:\longrightarrow\C$ be an open, one-to-one, complex differentiable function at $z_0 \in A$.\\
Let $g:f(A)\longrightarrow\C$ be the inverse of f, and suppose g is continuous.\\
If $f'(z_0) \neq 0$, then g is complex differentiable at $w_0 := f(z_0)$, with $g'(w_0) = \frac{1}{f'(z_0)}$.
\end{theorem}
\item
Define the complex \emph{logarithm function}.
\item
Define the \emph{principal logarithm}.
\item
Prove that the logarithm function Log is analytic on $\C \backslash (-\infty,0]$ (and state why it is not analytic on $\C \backslash \lbrace 0\rbrace$).
\item
Show that $\mbox{Log}'(w_0) = \frac{1}{w_0}$.
\item
Define the complex \emph{square root}.
\item
Give an expression for the various values of $\sqrt{w}$.
\item
Show that $\sqrt{w}$ is analytic on $\C \backslash(-\infty,0)$.
\item
Show that $\sqrt{w}$ is continuous on $\C \backslash(-\infty,0)$.
\item
At which points is \begin{eqnarray*}
 f:\C & \longrightarrow & \C \\
z=x+iy & \longmapsto & x^2 + iy^2
\end{eqnarray*}
complex differentiable?
\end{enumerate}
\end{document}
