\documentclass[12pt]{amsart}
\title[]{Complex Analysis: \\ Problems for $e^z,\sin(z),\cos(z),\ln(z),z^{\alpha}.$}
\author{}
\usepackage[margin=1.2in]{geometry}
\usepackage{amsmath,amssymb}
\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}
\newcommand{\ds}{\displaystyle}

\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\R}{\mathbb{R}}

\newcommand{\nC}{\mathbb{C}^{\times}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Z}{\mathbb{Z}}
\renewcommand{\iota}{{\mathbf i}}
\newcommand{\ol}[1]{\overline{#1}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}

\begin{document}
\maketitle

\problem{1} Find all values of $z\in\C$ such that:\\
\begin{tabular}{ll}
(a) $\ds e^z = 1+\sqrt{3}\ \iota$. & (b) $\ds e^z = -2$.\\
(c) $\ds e^z = 1 + \iota$. & (d) $\ds e^{2z+1} =1$. \\
&
\end{tabular}

\problem{2} Prove the following:
\begin{enumerate}
\item $\ds \lp \frac{e}{2} \lp -1-\sqrt{3}\ \iota\rp\rp^{3\pi\iota}
 = -e^{2\pi^2}$.\\

\item $\ds (1-\iota)^{4\iota} = e^{\pi}\lp
\cos(\ln(4)) + \iota \sin(\ln(4))\rp$.\\
\end{enumerate}

\problem{3} Prove that $\ds\left| e^{-z}\right| < 1$ if, and
only if $\Re(z)>0$.\\

\problem{4} Prove that $\ds \left| e^{z^2} \right| \leq e^{|z|^2}$
for every $z\in \C$.\\

\problem{5} Compute the limit, as $z\to 0$, of the following
functions.
\[
\frac{\sin(z)}{z}, \qquad \frac{e^z-e^{-z}}{z}.
\]

\problem{6} Identify the domains of the following functions.
Then, compute their derivative with respect to $z$.\\
\begin{tabular}{ll}
(a) $\ds e^{z^2+2z}$. & (b) $\ds \ln(3z+1)$.\\
(c) $\ds z^{1-\iota}$. & (d) $\ds z\ln(z)$.\\
(e) $\ds \lp z^2+1\rp^{\frac{1}{2}}$. & (f) $\ln\lp z^2-1\rp$.\\
&
\end{tabular}

\problem{7} Use the definition of $\sin(z), \cos(z)$, for $z\in\C$,
to prove the following trigonometric identities.
\begin{enumerate}
\item $\ds \cos(2z) = \cos^2(z) - \sin^2(z)$.\\

\item $\ds 2\cos(z)\cos(w) = \cos(z+w)+\cos(z-w)$.\\

\item $\ds (1-\cos(z-w))(1-\cos(z+w)) = (\cos(z)-\cos(w))^2$.\\
\end{enumerate}

\problem{8} (Optional, but fun!) Let $\mathcal{C}$ be a circle
with center $O$. Let $A$ and $B$ two points on $\mathcal{C}$.
Prove that, for any point $C$ on $\mathcal{C}$ different
from $A$ and $B$, one of the following must hold (depending
on whether $ACB$ is the long, or the short arc):
\[
\text{Either } \angle{AOB} = 2\angle{ACB}, \text{ or }
\angle{AOB} = 2\lp \pi-\angle{ACB}\rp.
\]

\end{document}
