\documentclass[12pt]{amsart}
\title[Problem Sheet 4]{Complex Analysis: Problem Sheet 4}
\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} Let $\phi \in (0,2\pi]$. Compute the length of the arc of the
circle of radius $R$, centered at $0$, between angles $0$ to $\phi$.
({\em Use the parametrization: $\gamma(\theta) = Re^{\iota\theta}$,
$0\leq \theta\leq \phi$.})\\

\problem{2} Let $\gamma : [0,1]\to \C$ be any piecewise smooth path
joining $0$ to $1+3\iota$. Compute the following:\\
\begin{tabular}{ccc}
(a) $\ds \int_{\gamma} \cos(z)\, dz$, &
(b) $\ds \int_{\gamma} \sin(z^2)z\, dz$, &
(c) $\ds \int_{\gamma} e^{2z}\, dz$, \\
(d) $\ds \int_{\gamma} z^3+2\, dz$, &
(e) $\ds \int_{\gamma} (z^2+1)^2\, dz$. &\\
&&
\end{tabular}

\problem{3} Let $k\in\Z$. Prove that:
\[
\int_0^{2\pi} e^{\iota k\theta}\, d\theta = \left\{
\begin{array}{cl}
0 & \text{if } k\neq 0,\\
2\pi & \text{if } k=0.
\end{array}
\right.
\]

\problem{4} Let $f(z)$ be a continuous function defined on an open, connected
subset $\Omega\subseteq \C$. Let $\gamma:[a,b]\to\Omega$ be a smooth path, and
let $\tau : [0,1] \to [a,b]$ be a smooth function. Define
$\ds \widetilde{\gamma} : [0,1]\to\Omega$ as $\widetilde{\gamma}(t)
= \gamma(\tau(t))$. Prove that
\[
\int_{\widetilde{\gamma}} f(z)\, dz = 
\int_{\gamma} f(z)\, dz.
\]

\begin{center}
\[
\boxed{
\text{\bf In Problems 5-12 below: (a) sketch $\gamma$, (b) compute $\ds \int_{\gamma}f(z)\, dz$.}
}
\]
\end{center}

\problem{5} $f(z) = \Re(z)$, and $\gamma(t) = Re^{\iota t}$, $0\leq t\leq 2\pi$.\\

\problem{6} $f(z) = \Im(z)$, and $\gamma(t) = Re^{\iota t}$, $0\leq t\leq 2\pi$.\\

\problem{7} $f(z) = y-x-(3x^2)\iota$, where $z=x+y\iota$, and
\[
\gamma(t) = \left\{
\begin{array}{lc}
t\iota, & \text{if } 0\leq t\leq 1,\\
(t-1)+\iota, & \text{if } 1\leq t\leq 2,\\
(3-t)(1+\iota), & \text{if } 2\leq t\leq 3.
\end{array}
\right.
\]

\problem{8} $f(z) = \overline{z}$ and $\gamma(\theta) = 2e^{\iota\theta}$,
$\ds -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$.\\

\problem{9} $f(z) = 2\Re(z)+\Im(z)$, and $\gamma(t) = t(1+\iota)$,
$0\leq t\leq 1$.\\

\problem{10} Let $n\in\Z$, $z_0\in\C$, $R\in\R_{>0}$.
\[
f(z) = (z-z_0)^{n-1}, \text{ and } \gamma(\theta) = z_0 + Re^{\iota\theta},
0\leq \theta\leq 2\pi.
\]

\problem{11} $f(z) = \frac{1}{z}$, and $\gamma(t) = 1+t\iota$, 
$-1\leq t\leq 1$.\\

\problem{12} Let $r\in\R_{>0}$ and $\phi\in (-\pi,\pi)$.
\[
f(z) = \frac{1}{z}, \text{ and } 
\gamma(t) = (1+t(r-1))e^{\iota t\phi},\ 0\leq t\leq 1.
\]


\hrule\hrule

\vspace*{0.2in}

\noindent {\bf Bonus.} Let $\gamma$ be a positively oriented contour. Let $S$
be the area of the interior of $\gamma$. Prove that
\[
\int_{\gamma} \Re(z)\, dz = \iota S,
\text{ and } \int_{\gamma} \Im(z)\, dz = -S.
\]

{\em This is a generalization of Problems 5,6 above. Hint: Green's theorem,
or recall how area enclosed by a curve was defind in Calculus I/II.}
\end{document}
