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\begin{document}

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\noindent
\parbox{2.25in}{\bf Math 5160}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Mon., Feb. $3^\mathrm{rd}$, 2020}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
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\begin{description}
\item[\#1 Basics] More ``Basic'' Calculations
\begin{enumerate}[(a)]
\item Compute $e^z$ for $z=-\dfrac{i\pi}{3}$, $\dfrac{1}{2}-\dfrac{i\pi}{4}$, and $-1+\dfrac{i3\pi}{2}$.
\item Compute $\mathrm{Log}(4+4i)$ and $\mathrm{Log}(-10i)$. Then also compute $\log(4+4i)$ and $\log(-10i)$.
\item Compute $(-1)^i$ and indicate what the principal branch yields.
\item Compute $\left(\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i\right)^{1+i}$ and indicate what the principal branch yields.
\item Compute $\sin(2i)$ and $\tan(2i)$. 
\item Compute $\cosh(i \pi/3)$ and $\sinh(i \pi/3)$.
\end{enumerate}

\item[\#2 Exponentially Difficult Graphs] Sketch each of the following figures and its image under the exponential map $w=e^z$. Indicate images of vertical and horizontal lines in your sketch.

\begin{enumerate}[(a)]
\item The vertical strip $0<\mathrm{Re}(z)<1$.
\item The horizontal strip $5\pi/3 < \mathrm{Im}(z) < 8\pi/3$.
\item The rectangle $0<x<1$, $0<y<\pi/4$.
\item {}[{\bf Extra Credit}] The disk $|z| \leq \pi/2$.
\item {}[{\bf Extra Credit}] The disk $|z| \leq \pi$.
\item {}[{\bf Extra Credit}] The disk $|z| \leq 3\pi/2$.
\end{enumerate}

\item[\#3 Log Your Complaint] Let $z=x+yi$ be as shown below:

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    >=2pt},
    every node/.style={color=black}
    ]
    % axis
    \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 Is $\mathrm{Log}(z^2)=2\mathrm{Log}(z)$? Explain why or why not.
\item In general, for which $z$ do we have $\mathrm{Log}(z^2)=2\mathrm{Log}(z)$? Justify your answer.
\item For $z \not=0$, it is always that case that $e^{\mathrm{log}(z)}=z$ (and $e^{\mathrm{Log}(z)}=z$). But obviously, ``$\mathrm{log}(e^z)=z$'' cannot hold because $\mathrm{log}$ is multivalued. When is it the case that $\mathrm{Log}(e^z)=z$?
\end{enumerate}

\item[\#4 A Tangential Issue] One can show that $\mathrm{arctan}(z) = \dfrac{i}{2}\mathrm{log}\left(\dfrac{1-iz}{1+iz}\right)$ when $z \not= \pm i$. The principal branch of inverse tangent is $\mathrm{Arctan}(z) =  \dfrac{i}{2}\mathrm{Log}\left(\dfrac{1-iz}{1+iz}\right)$ when $z \not= \pm i$.

\begin{enumerate}[(a)]
\item Derive the above formula for $\mathrm{arctan}(z)$.
\item Show that it is never the case that $\tan(z)= \pm i$. [Taking into account our formula for $\mathrm{arctan}(z)$, which is defined for all $z\not= \pm i$, and the fact that $\tan(\mathrm{arctan}(z))=z$, we get that the range of $\tan(z)$ is $\mathbb{C}-\{ \pm i \}$.]
\end{enumerate}

\end{description}

\end{document}