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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 6}
\date{Due in class Thursday, October 28, 2021}
\maketitle

\begin{enumerate}

\item (Stein \& Shakarchi, Chapter 2 Exercise 8.)  

If $f$ is a holomorphic function on the strip $-1 < y < 1$, $x \in \R$
with $\lvert f (z) \rvert \leq A \, (1 + \lvert z \rvert )^\eta$,
$\eta$ a fixed real number, for all $z$ in that strip, show that for
each integer $n \geq 0$ there exists $A_n \geq 0$ so that, for all
$x \in \R$,
\[
  \lvert f^{(n)} (x) \rvert \leq A_n \, (1 + \lvert x \rvert)^\eta \,.
\]

\emph{Hint:} Cauchy inequalities.

\item (Stein \& Shakarchi, Chapter 4 Exercise 2.)
If $f \in \mathcal{F}_a$ with $a > 0$, then for any positive integer
$n$ one has $f^{(n)} \in \mathcal{F}_b$ whenever
$0 \leq  b < a$.

\emph{Hint:} Modify the solution to the previous exercise.

\item (Stein \& Shakarchi, Chapter 4 Exercise 3.) Show, by contour
integration, that
\[
  \frac1\pi \int_{-\infty}^\infty
  \frac{a}{a^2 + x^2} \, \e^{-2\pi \i x \xi} \, \d x
  = \e^{-2 \pi a \lvert \xi \rvert}
\]
for $a > 0$ and $\xi \in \R$.  Then check explicitly that
\[
  \int_{-\infty}^\infty
  \e^{- 2 \pi a \lvert \xi \rvert} \, \e^{2 \pi \i \xi x} \,
  \d \xi
  = \frac1\pi \frac{a}{a^2 + x^2} \,.
\]


\item (Laurent series expansion, cf.\ Stein \& Shakarchi, Chapter~3
Problem~3.)
Let $f$ be holomorphic on an open set $\Omega \subset \C$ containing
the closed annulus $\{ z \colon r_1 \leq \lvert z \rvert \leq r_2 \}$
where $0<r_1<r_2$.  Prove that $f$ has a series representation
\[
  f(z) = \sum_{n=-\infty}^\infty a_n \, z^n
\]
which converges absolutely in the interior of the annulus.

\emph{Hint:} If $z$ is in the interior of the annulus, you can use the
Cauchy integral formula to write
\[
  f(z) = \frac1{2\pi\i} \int_{C_{r_2}(0)}
    \frac{f(\zeta)}{\zeta-z} \, \d \zeta
    - \frac1{2\pi\i} \int_{C_{r_1}(0)}
    \frac{f(\zeta)}{\zeta-z} \, \d \zeta \,.
\]
Then, use an appropriate geometric series expansion for $1/(\zeta-z)$.
(Similar tricks are used to prove that holomorphic implies analytic
and appear in the proof of the Poisson summation formula.)

\end{enumerate}



\end{document}
