%%---------------------------------------------------------------%%
%%-----------------------Packages-----------------------%%
%%---------------------------------------------------------------%%
%Some packages are essential for compiling your document!
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{lastpage}
\usepackage{fancyhdr}
\usepackage[margin=1.25in]{geometry}
\usepackage{mathrsfs}
\usepackage{mathtools}
\usepackage[shortlabels]{enumitem}
%\usepackage{mathpazo} %A different font
%\usepackage{eulervm} %A different font
\usepackage{hyperref}
\hypersetup{
	colorlinks=true, %set true if you want colored links
	linktoc=all,     %set to all if you want both sections and subsections linked
	linkcolor=red,  %choose some color if you want links to stand out
}
\usepackage{color}
\definecolor{darkblue}{rgb}{0.0, 0.0, 0.55}


%%---------------------------------------------------------------%%
%%------------------------Custom Commands------------------------%%
%%---------------------------------------------------------------%%
%Some LaTeX commands, like \mathbb{} are used so often that its useful to define a shorter command. Here are some useful examples, but you can add your own. 

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\e}{\varepsilon}
\renewcommand{\S}{{\mathbb{S}}}
\renewcommand{\phi}{\varphi}
\providecommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\defeq}{\overset{\text{def}}{=}}
\newcommand{\ds}{\displaystyle}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\providecommand{\Log}{\operatorname{Log}}
\providecommand{\Arg}{\operatorname{Arg}}
\providecommand{\arg}{\operatorname{arg}}

%%---------------------------------------------------------------%%
%%---------------------------------------------------------------%%



%%---------------------------------------------------------------%%
%%-----------------------Enivoronments---------------------------%%
%%---------------------------------------------------------------%%

\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{remark}[2][Remark]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}
	{\end{trivlist}}
\newenvironment{definition}[2][Definition]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}
	{\end{trivlist}}
\newenvironment{principle}[2][Principle]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}
	{\end{trivlist}}
\newenvironment{problem}[2][Exercise]{\begin{trivlist}
		\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}
	{\end{trivlist}}











%%---------------------------------------------------------------%%
%%-----------------------Header and Footer-----------------------%%
%%---------------------------------------------------------------%%
\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\newcommand{\Hrule}{\rule{\linewidth}{0.3mm}}
\renewcommand{\headrulewidth}{.5pt}
\renewcommand{\footrulewidth}{.5pt}

\pagestyle{fancy}
\fancyhf{}
\fancyhead[C]{%
	\textbf{\color{darkblue}MAT 103A: Complex Analysis (W21)}\quad
	\hfill \ {\itshape}\quad
	{\hfill \ \textbf{\color{darkblue} Problem Set 1}}
	\fancyfoot[R]{\leftmark}
	\fancyfoot[R]{\footnotesize\textit{Page \thepage \ of \pageref{LastPage}}}
	\fancyfoot[L]{\footnotesize{Created by \href{http://jadynbreland.com}{J.V. Breland} (Updated: \today)}}
}
%%---------------------------------------------------------------%%
%%---------------------------------------------------------------%%



%%---------------------------------------------------------------%%
%----------You can safely ignore anything above this line--------%%
%%---------------------------------------------------------------%%


%%---------------------------------------------------------------%%
%%------------------Document Begins Here-------------------------%%
\begin{document}
	%%---------------------------------------------------------------%%
	%%---------------------------------------------------------------%%
	
	
	%%---------------------------------------------------------------%%
	%%--------------------------Title--------------------------------%%
	%%---------------------------------------------------------------%%
	\begin{center}
		{\Large\textbf{\color{darkblue}Problem Set 1 (Due 1/15/20 11:59PM)}}
	\end{center}
	
	\bigskip
	
	\fbox{\parbox{5.5in}{
			\vspace{10pt}
			\textbf{\large  Your Name: } 
			\vspace{10pt}
	}}
	
	\bigskip
	
	\fbox{\parbox{5.5in}{
			\vspace{10pt}
			\textbf{\large Names of Any Collaborators: }
			\vspace{10pt}
	}}
	
	\bigskip
	
	\noindent Please review the problem set guidelines in the \href{https://people.ucsc.edu/~jbreland/teaching/W21_MATH103A/build/W21_103A_syllabus.pdf}{syllabus} before turning in your work. Additionally:
	\begin{enumerate}[(a)]
		\item you may use freely any results that we discussed during the lecture;
		
		\item you may use any results that are proved in Chapter 1 of the textbook;
		
		\item you may use any exercises from Chapter 1 of the textbook if and only if they do not render the problem trivial. 
		
	\end{enumerate}
	In any case, you should make it clear when a statement in your proof follows from another result, and which result you are using. 
		
	%%---------------------------------------------------------------%%
	%%---------------------------------------------------------------%% 
	
	%The enumerate enviroment produces numbered lists. The command \item specifies a list item.
	\begin{enumerate}[\bfseries P1.]
		
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 1-----------------------------%%
		%%---------------------------------------------------------------%%
		\item Let $p(z) = az^2 + bz + c$ be a polynomial with complex coefficients ($a\neq 0$). 
		
		\begin{enumerate}[(a)]
			\item By completing the square, show that the solution to $p(z) = 0$ is\footnote{Note that the expression $(b^2-4ac)^{1/2}$ has multiple values!}
			\[
			z = \frac{-b + (b^2-4ac)^{1/2}}{2a}.
			\]
			
			\item Find the roots of the polynomial $p(z) = iz^2 -1$ in the form $z = x+iy$. 
		\end{enumerate}
		%%---------------------------------------------------------------%%
		%--------------------------Problem 2-----------------------------%%
		%%---------------------------------------------------------------%%
		\item Consider the set of matrices
		\[
		X \defeq \left \{ \begin{pmatrix}x & -y\\ y & x \end{pmatrix} : x,y \in \R \right \}.
		\]
		It is easy to see that $X$ is closed under matrix addition and matrix multiplication, that is, if $A,B \in X$, then $A+B, AB \in X$.\footnote{Parts (a) and (b) of this problem shows that $X$ and $\C$ are algebraically indistinguishable. In the language of abstract algebra, $\phi$ is an \emph{isomorphism of rings}. The subset $\left\{\begin{pmatrix}x & 0\\ 0 & x\end{pmatrix} : x \in \R \right\} \subset X$ is algebraically indistinguishable from the real numbers in the same sense. } 
		\begin{enumerate}[(a)]
			
			\item Let $\C$ denote the set of complex numbers. Show that the map $\phi: X \to \C$ defined by 
			\[
			\phi \begin{pmatrix}x & -y\\ y & x \end{pmatrix}  = x + iy
			\]
			is a bijection. 
			
			\item Let $A,B \in X$ and let $I = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$ be the identity matrix. Show that $\phi$ has the following properties:
			
			\begin{enumerate}[(i)]
				
				\item $\phi(A+B) = \phi(A)+\phi(B)$;
				
				\item $\phi(AB) = \phi(A)\phi(B)$;
				
				\item $\phi(I) = 1$. 
			\end{enumerate}
			
			\item Find a matrix $J$ satisfying $J^2 = -I$ and show that $\phi(J) = i$. 
			
		\end{enumerate}
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 3-----------------------------%%
		%%---------------------------------------------------------------%%
		\item \begin{enumerate}[(a)]
			
			\item Let $z\in \C$. Using the principle of mathematical induction, show that the formula holds for all $n\in \N$:
			\[
			1 + z + z^2 + \cdots + z^n = \frac{1-z^{n+1}}{1-z}.
			\]
			
			\item If $\omega_1,\ldots,\omega_n$ are the \emph{distinct} $n$-th roots of unity, show that $\ds \sum_{i=1}^n \omega_i = 0$.
			
			\item Compute the following sum of real numbers:
			\[
			\cos \frac{\pi}{7} + \cos \frac{3\pi}{7} + \cos \frac{5\pi}{7} +\cos \frac{7\pi}{7}+ \cos \frac{9\pi}{7} + \cos \frac{11\pi}{7} + \cos \frac{13\pi}{7} + \cos\frac{15\pi}{7}.
			\]
			You must justify your computation. 
		\end{enumerate}
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 4-----------------------------%%
		%%---------------------------------------------------------------%%
		\item Let $z,w \in \C$. 
		\begin{enumerate}[(a)]
			\item Prove the formula
			%The equation environment displays math and labels it with a number. You can refer to this equation by using \label{label-name} to give the equation a name and \eqref{label-name} to reference it. 
			\begin{equation}\label{eq:P3}
			\abs{z+w}^2 = \abs{z}^2 + 2 \Re z\conj{w} + \abs{w}^2.\footnote{Notice that $\Re z \conj{w} = \Re w\conj{z}$.}
			\end{equation}
			
			\item From \eqref{eq:P3}, deduce the parallelogram law 
			\[
			\abs{z+w}^2 + \abs{z-w}^2 = 2\abs{z}^2 + 2\abs{w}^2 
			\]
			Give a geometric interpretation of this formula. 
		\end{enumerate}
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 5-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Let $z_0\neq z_1 \in \C$ and let $\lambda >0$. Show that if $\lambda \neq 1$, then the set of points
		%The equation environment displays math and labels it with a number. You can refer to this equation by using \label{label-name} to give the equation a name and \eqref{label-name} to reference it. 
		\begin{equation}\label{eq:P4} 
		|z-z_0| = \lambda |z-z_1| 
		\end{equation}
		is a circle of radius {\color{blue}$\ds R = \frac{\lambda}{\abs{1-\lambda^2}} \abs{z_0 - z_1}$ centered at $\ds w = \frac{z_0 - \lambda^2 z_1}{1-\lambda^2}$ }.\footnote{Suggestion: Try to show that $|z-w|^2 = R^2$. Start by squaring both sides of \eqref{eq:P4} using \eqref{eq:P3}. Manipulate the resulting equation so that $\frac{\lambda^2|z_1|^2-|z_0|^2}{1-\lambda^2}$ is isolated on one side. Using \eqref{eq:P3} again, add something to both sides to complete the square.} If $\lambda = 1$, show that \eqref{eq:P4} defines a line.\footnote{Hint: When $\lambda = 1$, argue geometrically!}
		
		Show that every circle in the complex plane can be written in the form of equation \eqref{eq:P4} for some $\lambda >0 , \lambda \neq 1$ and $z_0 \neq z_1 \in \C$. 
		
		
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 6-----------------------------%%
		%%---------------------------------------------------------------%%
		\item 
		\begin{enumerate}[(a)]
			\item  Find a necessary and sufficient condition on $z \in \C$ such that $\abs{z -i} = \abs{z + i}$. 
			
			\item Repeat with the condition $\abs{z - 1} = \abs{z+1}$. 
			
		\end{enumerate}
	
		%%---------------------------------------------------------------%%
		%--------------------------Problem 7-----------------------------%%
		%%---------------------------------------------------------------%%
	
		\item Consider the map $\ds M(z) = \frac{z-3}{1-2z}$, $z \in \C$. For which values of $c \in \R$ is the image of the circle $|z-1| = c$ under $M$ a line? What is the equation of the line when considered as a subset of the plane $\R^2$? 
		
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 8-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Let $U \subseteq \C$. We say that $U$ is \emph{open} if it does not contain any of its boundary points. Prove that $U$ is open if and only if every $z \in U$ is an interior point. 
		
		%%---------------------------------------------------------------%%
		%--------------------------Problem 9-----------------------------%%
		%%---------------------------------------------------------------%%
		
		\item Let $\e>0$. Prove that the open disk $D_\e(z_0) \defeq \{z \in \C : \abs{z-z_0} < \e\}$ is a domain.
		
	\end{enumerate}
	%%---------------------------------------------------------------%%
	%%------------------Document Ends Here---------------------------%%
\end{document}
%%-------------------------------------------------------------%%