\documentclass{article}
\usepackage{amsmath, amssymb,verbatim, amsthm, graphicx}

%exercise environment
\newcounter{exercise}
\setcounter{exercise}{0}
\newenvironment{exercise}{\addtocounter{exercise}{1} \noindent{\bf{Exercise \theexercise.}}}{\vspace{.5cm}}

\newtheorem{thm}{Theorem}

\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\C}{\mathbb{C}}

\textwidth 6.5in
\textheight 9in
\oddsidemargin 0in
\evensidemargin 0in
\headsep -0.5in

\setlength{\parskip}{0.25cm}

\pagenumbering{gobble}

\begin{document}

\begin{center}
\includegraphics[width=0.5in]{../ltr-logo.jpg}
\end{center}

\noindent{\Large{\textsc{Complex Variables\hfill Assignment 8.3  \\ Spring 2020 \hfill Due April 1}}}

\large

\vspace{1cm}

\begin{exercise}
Recall that for $h:[a,b] \to \C$ we defined
\[
\int_a^b h(t) \, dt = \int_a^b \operatorname{Re} h(t) \, dt + i \int_a^b \operatorname{Im} h(t) \, dt.
\]
If $\alpha \in \C$, use this definition to prove that
\[
\int_a^b \alpha h(t) \, dt = \alpha \int_a^b h(t) \, dt.
\]
\end{exercise}

\begin{exercise}
Let $\Omega \subset \C$ be a domain, $f : \Omega \to \C$ be continuous, and set $u = \operatorname{Re} f$, $v = \operatorname{Im} f$.  If $\gamma(t)$, $t \in [a,b]$, is a $C^1$ path in $\Omega$, prove that
\[
\int_{\gamma} u \, dx - v \, dy + i \int_{\gamma} v \, dx + u \, dy = \int_a^b f(\gamma(t)) \, \gamma'(t) \, dt.
\]
\end{exercise}

\begin{exercise}
Textbook exercise IV.1.1.
\end{exercise}

\begin{exercise}
Textbook exercise IV.1.4. [{\em Suggestion:}  Use Green's theorem]
\end{exercise}

\begin{exercise}
Textbook exercise IV.1.5.
\end{exercise}

\end{document}
