\documentclass[12pt]{article}

\usepackage{times}
\usepackage{graphics}
\usepackage{amssymb}

\def\C{{\mathbb C}}
\def\R{{\mathbb R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\ol}{\overline}
\newcommand{\scalar}[2]{\langle #1,#2\rangle}
\newcommand{\et}{\tilde{e}}
\renewcommand{\Re}{\mathrm{Re}}
\renewcommand{\Im}{\mathrm{Im}}

\begin{document}

\vspace{2cm}

\begin{center}

{\bf \Large  185A Homework 1}\\[10mm]
\end{center}

\noindent

%{\small Due Friday January 13}\\[2mm]

\noindent
\textbf{Question 1}
In class we defined the product $z_1z_2$ of a pair of complex numbers $(z_1,z_2)$ by 
(i) viewing these as vectors, (ii) multiplying their lengths to find the length of the vector $z_1z_2$ and (iii) adding the angles relative to some axis to find the direction of $z_1z_2$.
Show that this rule obeys all the field axioms. Also show that if $z_1=(x_1,y_1)$ and $z_2=(x_2,y_2)$ in
Cartesian coordinates, then the above rules imply
$$
z_1z_2=(x_1x_2-y_1y_2,x_1y_2+x_2y_1)\, .
$$  

\noindent
\textbf{Question 2}
Verify the following properties for $z,w\in \C$:
\begin{enumerate}
\item $\overline{z+w}=\overline{z}+\overline{w}$ and $\overline{zw}=\overline{z} \; \overline{w}$
\item $\Re(z)=\frac{z+\overline{z}}{2}$ and $\Im(z)=\frac{z-\overline{z}}{2i}$
\item
$z=\overline{z}$ if and only if $z\in \R$ is purely real.\newline
$z=-\overline{z}$ if and only if $z\in i\R$ is purely imaginary.\newline
Find the maximal subset of $\C$ which is left fixed (as a set and then also pointwise) by complex conjugation.
\end{enumerate}

\noindent
\textbf{Question 3}
Let $\scalar{z}{w}\equiv(\ol z w+\ol w z)/2$.
Prove the following identities:
\begin{enumerate}
\item $\scalar{z}{w}^2 + \scalar{iz}{w}^2 = |z|^2 |w|^2$
\item $|\scalar{z}{w}| \le |z|\cdot|w|$ (Cauchy-Schwarz inequality)
\item $|z+w|^2=|z|^2+|w|^2+2\scalar{z}{w}$ (cosine law)
\item $|z|\ge 0$ and $|z|=0$ if and only if $z=0$
\item $|z|^2=\scalar{z}{z}$
\item $|zw|=|z|\cdot |w|$
\item $|z+w|\le |z|+|w|$ (triangle inequality)
\item $|\frac{z}{w}|=\frac{|z|}{|w|}$.
\end{enumerate}

\noindent
\textbf{Question 4}
Define the function $\et(\theta)=\cos(\theta)+i\sin(\theta)$.
Show that 
$$
\frac{1}{\et(\theta)}=\et(-\theta)
$$
and use this fact to extend de Moivre's theorem to all $n\in \Z$.

\noindent
{\bf Question 5}
Prove Lagrange's trigonometric identity:
$$
1+\cos\theta+\cos 2\theta+\cdots+\cos n\theta = \frac{1}{2}
+\frac{\sin(n+\frac{1}{2})\theta}{2\sin\frac{\theta}{2}}
$$
where we assume that $\sin\theta/2 \neq 0$.
%(\textit{Hint:} Transform the left-hand side into the real
%part of sums of exponentials and use
%$1+x+x^2+\cdots+x^n=\frac{x^{n+1}-1}{x-1}$.)

%\newpage

\noindent
\noindent
{\bf Question 6}
Let $x,y,u,v\in\R$ and study matrices of the form
$$
q=\left(
\begin{array}{rrrr}
x&v&u&-y\\
-v&x&y&u\\
-u&-y&x&-v\\
y&-u&v&x
\end{array}
\right)
$$
Do they form a field? (If not, characterize their failure to do so.)
[{\it Hint; Look up the ``quaternions''.}]

\noindent
\noindent
{\bf Question 7}
Define $u:{\mathbb R}^2\to{\mathbb R}$ via ${\mathbb R}^2\ni(x,y)\mapsto u(x,y)$ and 
$$
\Delta:=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\, .
$$
Suppose $v:{\mathbb R}^2\to{\mathbb R}$ obeys
$$
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}\mbox{ and } 
\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}\, .
$$
and that $u$ is ``smooth'' ({\it i.e.} equality of mixed partials). Show that both $u$ and $v$ 
solve the Laplace equation
$$
\Delta u = 0 = \Delta v\, .
$$

\noindent
\noindent
{\bf Question 8}
Show that the set of matrices 
$$
\left\{
\left(
\begin{array}{rr}
x&-y\\
y&x
\end{array}\right)
\colon
x,y\in{\mathbb R}
\right\}
$$
form a field ({\it hint}: you need to define both multiplication and addition rules).

\end{document}
