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

\begin{document}

\title{Complex Analysis}
\author{Homework 9}
\date{Due in class Thursday, November 25, 2021}
\maketitle

\begin{enumerate}

\item (Stein \& Shakarchi, Chapter 8 Exercise 12.) A complex number $w
\in \D$ is a fixed point for the map $f \colon \D \to \D$ if $f (w) =
w$.
\begin{enumerate}
\item Prove that if $f \colon \D \to \D$ is holomorphic and has two
distinct fixed points, then $f$ is the identity.

\emph{Hint:} Consider the function $g = \psi_w \circ f \circ \psi_w$
where $w$ is one of the fixed points and $\psi_w$ is the automorphism
of the disc that exchanges $0$ and $w$.

\item  Must every holomorphic function $f \colon \D \to \D$ have a
fixed point?

\emph{Hint:} Consider the upper half-plane.
\end{enumerate}




\item We say that a power series with coefficients $a_n$ is
\emph{asymptotic} to a function $f \colon \R \to \R$, written
\[
  f(x) \sim \sum_{n=0}^\infty a_n \, x^n \,,
\]
if, for every $N \in \mathbb N$,
\[
  f(x) - \sum_{n=0}^N a_n \, x^n = O(x^{N+1})
  \quad \text{as } x \searrow 0 \,.
  \tag{*}
\]

\emph{Notes:} An asymptotic series may not be convergent.  The concept
carries over to functions on $\C$; in that case, the validity of an
asymptotic expansions is typically restricted to a sector of the
complex plane.

\begin{enumerate}
\item Show that
\[
  \frac1{1-x} \sim \sum_{n=0}^\infty x^n
  \quad \text{as } x \searrow 0
\]
and 
\[
  \frac1{1-x} + \e^{-1/x} \, \sin (\e^{1/x})
  \sim \sum_{n=0}^\infty x^n
  \quad \text{as } x \searrow 0 \,.
\]
In other words, an asymptotic series does not uniquely characterize a
function.

\item Demonstrate that term-by-term differentiation of the asymptotic
series, in the second case, does not yield an asymptotic series for
the derivative of the left-hand function.
\end{enumerate}

\item Let 
\[
  I(s) = \int_0^b \e^{-sx} \, f(x) \, \d x
\]
where $b>0$ and $f \colon [0,1] \to \R$ is continuous and has an
asymptotic expansion of the form (*).  Prove that
\begin{equation*}
  I (s) \sim \sum_{n=0}^\infty a_n \, \Gamma(n+1) \, s^{-n-1}
  \quad \text{as } s \to \infty \,.
\end{equation*}
(This is a simplified version of what is known as \emph{Watson's
lemma}.)



\end{enumerate}



\end{document}
