\documentclass[12pt]{article}
\usepackage{ccfonts,euler}
\usepackage{fullpage,amsmath,amsfonts}

\usepackage{tikz}
\usetikzlibrary{calc,decorations.markings}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Z{{\mathbb Z}}
\def\N{{\mathbb N}}
\def\Re{\operatorname{Re}}
\def\Im{\operatorname{Im}}
\def\interior{\operatorname{int}}

\begin{document}

\title{Complex Analysis}
\author{Mock Midterm Exam}
\date{October 30, 2021}
\maketitle

% \vspace*{\fill}
% {\baselineskip=40pt
% \begin{tabbing}
% Student ID Number: \= \kill
% Last Name: \> \rule[-2pt]{10cm}{0.3pt} \\
% First Name:  \> \rule[-2pt]{10cm}{0.3pt} \\
% Signature: \> \rule[-2pt]{10cm}{0.3pt} \\
% %% Code: \> \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% %% \fbox{\rule{0pt}{1.5em}\rule{1.5ex}{0pt}}
% \end{tabbing}}


% \vspace*{\fill}
% \vspace*{\fill}

% \newpage

\begin{enumerate}

\item
\begin{enumerate}
\item Find all solutions to the equation
\[
  z^5 = 1-\i \,.
\]

\item List all possible values for $\i^\i$.  Which value corresponds
to the principal branch of the logarithm?
\end{enumerate}
\hfill(5+5)

%\newpage

\item What is the radius of convergence of the power series of
\[
  f(z) = \frac{(z-1)^2}{z^2 - 1}
\]
about the point $z_0=1+\i$?
\hfill(5)

%\newpage
\item Integrate the square root function
\[
  f(z) = \sqrt z
\]
around the unit circle $C=\{z \colon \lvert z \rvert = 1 \}$ with
standard orientation in two different ways:
\begin{enumerate}
\item by parameterizing the unit circle in the standard way,
\item by integrating around a contour as sketched below.


\begin{center}
%%% Contour integration picture adapted from
%%% https://tex.stackexchange.com/q/103176
\begin{tikzpicture}
% Configurable parameters
\def\gap{0.2}
\def\bigradius{3}
\def\littleradius{0.4}

% Axes
\draw [-latex] (-1.25*\bigradius, 0) -- (1.25*\bigradius,0);
\draw [-latex] (0, -1.25*\bigradius) -- (0, 1.25*\bigradius);
% Red path
\draw[line width=1pt,   decoration={ markings,
  mark=at position 0.2455 with {\arrow[line width=1.2pt]{>}},
  mark=at position 0.765 with {\arrow[line width=1.2pt]{>}},
  mark=at position 0.87 with {\arrow[line width=1.2pt]{>}},
  mark=at position 0.97 with {\arrow[line width=1.2pt]{>}}},
  postaction={decorate}]
  let
     \n1 = {asin(\gap/2/\bigradius)},
     \n2 = {asin(\gap/2/\littleradius)}
  in (\n1:\bigradius) arc (\n1:360-\n1:\bigradius)
  -- (-\n2:\littleradius) arc (-\n2:-360+\n2:\littleradius)
  -- cycle;

% The labels
\node at (3.6,-0.3){$\Re$};
\node at (-0.35,3.53) {$\Im$};
\node at (-0.65,0.48) {$\gamma_4$};
\node at (-1.85,2.85) {$\gamma_2$};
\node at (1.95,0.4) {$\gamma_1$};
\node at (1.555,-0.4) {$\gamma_3$};

\end{tikzpicture}
\end{center}
\hfill(5+10)
\end{enumerate}

%\newpage\hbox{}\newpage

\item Find
\[
  \int_C \frac1{z^2 \, (z+1)^2} \, \d z
\]
where $C=\{z \colon \lvert z \rvert = \frac12 \}$ with standard
orientation. 
\hfill(10)


%\newpage

\item Suppose $f(z)$ is entire and suppose there exist $c>0$ and
$n\in \N$ such that
\[
  \lvert f(z) \rvert \leq c \, \lvert z \rvert^n
\]
for all $z \in \C$.  Prove that $f$ is a polynomial.

\emph{Hint:} Consider $f^{(m)}$ for $m>n$.
\hfill(10)

%\newpage\hbox{}\newpage

\end{enumerate}

\end{document}
