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\def\e{{\mathrm e}}
\def\i{{\mathrm i}}
\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 5}
\date{Due in class Thursday, October 21, 2021}
\maketitle

\begin{enumerate}

\item (Stein \& Shakarchi, Chapter 3 Exercise 7.)  Prove that, for
$a>1$, 
\[
  \int_{0}^{2\pi} \frac1{(a + \cos \theta)^2} \, \d \theta
  = 2 \pi \, \frac{a}{(a^2-1)^{3/2}} \,.
\]
\emph{Hint:} Recognize this expression as a parameterization of a line
integral in the complex plane, then use the residue theorem.

\item (Stein \& Shakarchi, Chapter 3 Exercise 9.)  Show that
\[
  \int_{0}^1 \log(\sin \pi x) \, \d x
  = - \log 2 \,.
\]
\emph{Hint:} Integrate along the contour $\gamma = \i \, (\infty, 0])
\cup [0,1] \cup 1+\i \, [0,\infty)$.


\item (Stein \& Shakarchi, Chapter 3 Exercise 10.)  Show that, for
$a>0$,
\[
  \int_{0}^\infty \frac{\log x}{a^2+x^2} \, \d x
  = \frac\pi{2a} \, \log a \,.
\]
\emph{Hint:} Integrate along the contour bounded by the semicircle in
the upper halfplane of radius $r$, the semicircle in the upper
halfplane of radius $R$, both centered at the origin, plus the line
segments along the real axis connecting the two semicircles.  Then let
$r\to 0$ and $R \to \infty$.


\item (Stein \& Shakarchi, Chapter 3 Exercise 13.)
Suppose $f$ is holomorphic in a punctured disc $D_r(z_0) \setminus \{
z_0 \}$ and that
\[
  \lvert f(z) \rvert \leq A \, \lvert z - z_0 \rvert^{\varepsilon -1}
\]
for some $\varepsilon>0$ and $A>0$ and all $z$ near $z_0$.  Show that
the singularity of $f$ at $z_0$ is removable.
\end{enumerate}



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