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\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 3}
\date{Due in class Wednesday, October 6, 2021}
\maketitle

\begin{enumerate}

\item (Stein \& Shakarchi, Chapter 2 Exercise 3.)  Evaluate the integrals
\[
  \int_0^\infty \e^{-ax} \, \cos(bx) \, \d x
  \quad \text{and} \quad
  \int_0^\infty \e^{-ax} \, \sin(bx) \, \d x
\]
with $a>0$ by integrating $\e^{-Az}$, $A = \sqrt{a^2 + b^2}$, over an
appropriate sector with angle $\omega$, with $\cos \omega = a/A$.

\item (Cf.\ Stein \& Shakarchi, Chapter 2 Exercise 7.) Let $f \colon
D_1(0) \to \C$ be holomorphic, and let
\[
  d = \sup_{w,z \in D_1(0)} \lvert f(w)-f(z) \rvert
\]
denote the diameter of the image of $z$.  Show that
\[
  2 \, \lvert f'(0) \rvert \leq d
\]
and that equality holds if and only if $f$ is linear.

\item (Stein \& Shakarchi, Chapter 2 Exercise 10.) Weierstrass's
theorem states that a continuous function on $[0, 1]$ can be uniformly
approximated by polynomials. Can every continuous function on the
closed unit disc be approximated uniformly by polynomials in the
variable $z$?


\item (Stein \& Shakarchi, Chapter 2 Exercise 13.) Suppose $f \colon
\C \to \C$ is entire and such that for every $z_0 \in \C$ at least one
coefficient in the expansion
\[
  f(z) = \sum_{n=0}^\infty c_n \, (z-z_0)^n
\]
is equal to $0$.  Prove that $f$ is a polynomial.

\emph{Hint:} Use the fact that $c_n \, n! = f^{(n)}(z_0)$ and use a
countability argument.

\end{enumerate}

\end{document}
