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\begin{document}

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\noindent
\parbox{2.25in}{\bf Math 5160}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Do after Gamelin I.2}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
%\vspace{0.2in}

\begin{description}
\item[\#1 Basics] Initial Calculations
\begin{enumerate}[(a)]
\item Let $z=3-4i$ and $w=-1+2i$. Compute $z^2,\conj{z},|z|,1/z,wz,w+z,z/w$.
\item Solve $z^4-5z^2+6=0$. \quad {\it Note:} Find \underline{all} complex solutions.
\item Assume $z_1=x_1+y_1i$ has positive real and imaginary parts and let $z_2 = \conj{z_1}$. What are $z_1+z_2$ and $z_1-z_2$? Plot $z_1,z_2,z_1+z_2,z_1-z_2$ together.
\item Plot the region $|z-2i| \leq 2$.
\item Write $i$, $1-\sqrt{3}i$, and $\sqrt{3}+i$ in polar form, multiply them together (in polar), then convert back from polar to standard form. 
\item Find the $3^\mathrm{rd}$ roots of unity (in polar and standard form). Then solve $z^3=-1+i$.
\end{enumerate}

\item[\#2 Plotting Against You] Let $z=x+yi$ be as shown below:

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    scale=5,
    axis/.style={very thick, ->, >=stealth'},
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    >=2pt},
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    \draw[axis] (-0.65,0)  -- (0.65,0) node(xline)[right]  {Real};
    \draw[axis] (0,-0.65) -- (0,0.65) node(yline)[above] {Imaginary};
       
    \draw[dashed,color=gray] (0,0) circle (0.403/2); 
    
    \draw[thick] (0.403/2,0.02)--(0.403/2,-0.02) node[below right] {$2$};
    \draw[thick] (-0.02,0.403/2)--(0.02,0.403/2) node[above right] {$2i$};

    \fill[red]  (-0.2/2,0.35/2) coordinate (out) circle (.4pt) node[above left] {$z=x+yi$};
\end{tikzpicture}

\begin{enumerate}[(a)]
\item Plot and label $-z$
\item Plot and label $\bar{z}$
\item Plot and label $1/z$
\item Plot and label $z^2$
%\item Is $\mathrm{Log}(z^2)=2\mathrm{Log}(z)$? Explain why or why not.
%\item {\bf [Grad. Problem]} What must be true about $z$ so that $\mathrm{Log}(z^2)=2\mathrm{Log}(z)$? Justify your answer.
\end{enumerate}

\item[\#3 Transformers Go!] In class, we showed that $w=(z-(1+2i))e^{i\pi/4}+(1+2i)$ rotates the complex plane $45^\circ$ about the point $(1,2)=1+2i$. Also, $w=-\bar{z}+2$ reflects the complex plane across the line $\mathrm{Re}(z)=1$. Give similar formulas for the following transformations:

\begin{enumerate}[(a)]
\item Rotate the plane $30^\circ$ about the point $-1+i$.
\item Reflect across the line through $1+\sqrt{3}i$ and $3-\sqrt{3}i$. 
\end{enumerate}

\end{description}

\end{document}