\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Z{{\mathbb Z}}
\def\Re{\operatorname{Re}}
\def\Im{\operatorname{Im}}
\def\interior{\operatorname{int}}

\begin{document}

\title{Complex Analysis}
\author{Homework 2}
\date{Due in class Wednesday, September 29, 2021}
\maketitle

\begin{enumerate}

\item Let $\gamma$ be a piecewise smooth curve in $\C$ and let $f
\colon \C \to \C$ be continuous.  Show that
\begin{equation*}
  \int_\gamma f(z) \, \d z = - \int_{\gamma^-} f(z) \, \d z
\end{equation*}
where $\gamma^-$ denotes the curve $\gamma$ with reversed orientation.

\item Let $\gamma$ be piecewise smooth simple closed curve in $\C$
encircling the origin.  Compute
\begin{equation*}
  \int_\gamma z^n \, \d z
\end{equation*}
for every $n \in \Z$.

\item Show that
\[
  \int_0^\infty \sin(x^2) \, \d x
  = \int_0^\infty \cos(x^2) \, \d x
  = \frac{\sqrt{2\pi}}4 \,.
\]
\emph{Hint:} Integrate $\e^{-z^2}$ along the sector of the disk of
radius $R$ centered at the origin that lies in the first octant of the
complex plane, then let $R \to \infty$.  (This is Chapter~2,
Exercise~1 from Stein \& Shakarchi who have a sketch of this contour
in the book.)

\item Show that
\[
  \int_0^\infty \frac{\sin x}x \, \d x = \frac\pi2 \,.
\]
\emph{Hint:} Integrate along the indented semicircle contour as in the
example from class (or Stein \& Shakarchi, Chapter~2, Exercise~2).

\item Let $\Omega \subset \C$ be open.  Let $f$ be holomorphic on
$\Omega$ except perhaps at a single point $w$.  Show that if $f$ is
bounded near $w$, then
\[
  \int_\gamma f(z) \, \d z = 0
\]
for any closed piecewise smooth curve $\gamma \subset \Omega$.



\end{enumerate}



\end{document}
