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\centerline{\Large \bf Homework Set 1}
\begin{tabbing}
{\bf ASSIGNED: Fri Jan 23} \` {\bf DUE: Fri Jan 30}\\
\end{tabbing}

\begin{enumerate}
\item Which of the points $i$, $2-i$ and $-3$ is farthest from the origin? Which is closest?
\item Draw an Argand diagram containing the following vectors
\begin{description}
\item[(a)] $7 \mbox{cis}({\displaystyle \frac{3 \pi}{4}})$
\item[(b)] $4 \mbox{cis}({\displaystyle \frac{- \pi}{6}})$
\end{description}
\item Find the argument of the following complex numbers and write in polar form.
\begin{description}
\item[(a)] $-1/2$
\item[(b)] $- \pi i$
\item[(c)] $(\sqrt{3} - i)^2$
\end{description}
\item Show that, for arbitrary $z_1$, $z_2$ and $z_3,$ the following is true: $\overline{ z_1 z_2 \overline{z_3}} = \overline{z_1 z_2} z_3$
%\item Show that $\mbox{arg} z_1 \overline{z_2} = \mbox{arg} z_1 - \mbox{arg} z_2$
\item Find a counterexample which proves that the "theorem" \\
$\Arg (z_1 z_2) = \Arg z_1 + \Arg z_2$ is {\bf false}. 
(You may also want to think about why the "theorem" becomes true when $\Arg$ is replaced with $\arg$.) Show that your counterexample is just another example that $\arg (z_1 z_2) = \arg z_1 + \arg z_2$ is {\bf true} for all $z$.
%\item Write $(1+i)^24$ in polar form and then find its modulus.
\item Compute the following
\begin{description}
\item[(a)] $|(3+4i)(1+2i)(i)|$
\item[(b)] ${\displaystyle \left| \frac{i (2+i)^3}{(1-i)^2} \right|}$
\item[(c)] ${\displaystyle \left| \frac{(\pi + i)^{100}}{(\pi-i)^{100}}| \right|}$
\end{description}
\item Compute the following roots and state which is the principal root (i.e. $k=0$) in each case.
\begin{description}
\item[(a)] $(-16)^{1/4}$
\item[(b)] $\sqrt{i-1}$
\item[(c)] $(1-i\sqrt{3})^{1/3}$
\end{description}

\item Find the four roots of the equation $z^4+4=0$ and use them to factor $z^4+4$ into two quadratic factors with purely real coefficients.

\newpage
\item Solve the following equations
\begin{description}
\item[(a)] ${\displaystyle \frac{z}{1-z}} = 1 - 5 i$
\item[(b)] $iz = 4 - z i$
\item[(c)] $ 8 z^2 + (2 - i)z = 0$
\item[(d)] $z^2 + 2 z + (1-i) = 0$
\end{description}
\item Find $z_1$ and $z_2$ and write the answers in {\it exponential form}
\begin{eqnarray}
(1 -i)z_1 + 3 z_2 & = & 2 - 3 i \nonumber \\
i z_1 + (1 + 2i) z_2 & = & 1 \nonumber 
\end{eqnarray}
\item Find all complex numbers $z$ such that $(z+1)^5 = z^5$.
\item Give sketches representing the following regions in the complex plane
\begin{description}
\item[(a)] $| z - 2 + i| \le 1$
\item[(b)] $| 2 z + 3| > 4$
%\item[(c)] $\mbox{Im } z > 1$
\item[(c)] $\mbox{Im } z = 1$
%\item[(e)] $0 \le \mbox{arg} z \le \pi/4 \qquad (z \ne 0)$
\item[(d)] $| z - 4| \ge |z|$
\item[(e)] $(\mbox{Re } z)^2 > 1$
%\item[(h)] $0 < | z - 2| < 3$
\end{description}
\item Using your answers to the previous question, answer the following.
\begin{description}
\item[(a)] Which of the sets are {\it open}?
\item[(b)] Which of the sets are {\it bounded}?
\item[(c)] Which of the sets are {\it domains}?
\item[(d)] Describe the boundary of each set.
\end{description}

\end{enumerate}

\centerline{NOTES}
All homework sets will be due in class one week from the class they are given out in. You are strongly encouraged to work collaboratively on the homework, though each person must hand in indvidually-written work.
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