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\begin{document}

\title{Complex Analysis}
\author{Homework 4}
\date{Due in class Thursday, October 14, 2021}
\maketitle

\begin{enumerate}

\item (Stein \& Shakarchi, Chapter 3 Exercise 2.)  Evaluate the integral
\[
  \int_{-\infty}^\infty \frac1{1+x^4} \, \d x \,.
\]

\item (Stein \& Shakarchi, Chapter 3 Exercise 4.)  Show that, for
$a>0$,
\[
  \int_{-\infty}^\infty \frac{x \, \sin x}{x^2+a^2} \, \d x
  = \pi \, \e^{-a} \,.
\]

\item (Stein \& Shakarchi, Chapter 3 Exercise 6.)  Show that, for
$n \in \mathbb N$,
\[
  \int_{-\infty}^\infty \frac{1}{(1+x^2)^{n+1}} \, \d x
  = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)} \,.
\]

\item (Stein \& Shakarchi, Chapter 3 Exercise 16.) Suppose $f$ and $g$
are holomorphic in a region containing the disc $\lvert z \rvert \leq
1$.  Suppose that $f$ has a simple zero at $z=0$ and vanishes nowhere
else in $\lvert z \rvert \leq 1$.  Let
\[
  f_\varepsilon(z) = f(z) + \varepsilon \, g(z) \,.
\]
Show that, if $\varepsilon$ is sufficiently small, then
\begin{enumerate}
\item $f_\varepsilon$ has a unique zero in $\lvert z \rvert \leq 1$,
and
\item if $z_\varepsilon$ denotes this zero, the mapping $\varepsilon
\mapsto z_\varepsilon$ is continuous.
\end{enumerate}


\end{enumerate}

\end{document}
