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\lhead{ Math 115 Spring \the\year}
\rhead{Prof.~Bernoff}
\chead{\bf Problem Set 1}
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\begin{center}
\large Complex Arithmetic 
\end{center}
{\bf Due date:} Tuesday, January 29th in class.

\begin{problems}
\item Simplify each of the following as much as possible.  (In other
  words, write each in the form $a+bi$.)
\begin{enumerate}
\item $i^{2019}+i^{-2019}$
\item $(1+2i)^3$
\item $\dfrac{1}{1+2i}$
\item $\dfrac{1+i}{1-i}$
\item $\dfrac{3}{i}+\dfrac{i}{3}$
\item $\Im\left(\dfrac{a+bi}{c+di}\right)$ \qquad (assume $a$, $b$, $c$ and $d$ are
  real)
\item $\left|\overline{(1-2i)^3}-(1+i)^3\right|$ \smallskip \\
  \textbf{Note:} To save some effort, think about whether
  $\overline{z_1z_2}$ is the same as $\overline{z_1}\,\overline{z_2}$
  for any two complex numbers $z_1$ and $z_2$.
\end{enumerate}

\vfill

  
\item \begin{enumerate}
\item What is the graphical relationship between $z$ and its
  multiplicative inverse, $z^{-1}$, if $|z|=1$?
\item What is the graphical relationship between $z$ and $-z$?
\item What is the graphical relationship between $z$ and $-iz$?
\item What is the graphical relationship between $z_1$, $z_2$, and
  $(z_1+z_2)/2$?
\item Describe the set of all points in the complex plane that satisfy
  $z=\overline{z}$.
\item Describe the set of all points in the complex plane that satisfy
  $1\leq |z|<2$.
\end{enumerate}

\vfill

\item What are all possible solutions of $z^4+4=0$?  From this
  information, write out a complete factorization of $z^4+4$.
  
 \vfill


\item Compute all possible values of $(i-1)^{1/3}$.

\vfill
\item We introduced the polar form for complex numbers $a+bi=re^{i\theta}$
 where {\bf Euler's Formula} 
\begin{equation*} 
 e^{i\theta} \equiv\cos\theta +i\sin{\theta}
\end{equation*} 
introduces the exponential notation for describing the polar angle $\theta$.  Fortunately, this exponential function obeys all the rules
we learned for the real exponential function, but strictly speaking we should prove them all.  Prove the product rule
$$e^{i\theta_1}e^{i\theta_2}= e^{i(\theta_1+\theta_2)}.$$
You will need to remember your trigonometric identities for $\cos (\theta_1+\theta_2)$ and $\sin (\theta_1+\theta_2)$.
\vfill
\end{problems}

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