\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts,fullpage}

\def\C{{\mathbb C}}
\def\e{{\mathrm e}}
\def\d{{\mathrm d}}
\def\Ei{\operatorname{Ei}}

\begin{document}

\title{Applied Analysis}
\author{Homework 1}
\date{due September 14, 2007}
\maketitle

\begin{enumerate}
\item Let $D \subset \C$, $z_0 \in \overline D$, and $f,g \colon D \to
  \C$.  Show that 
\[
  f(z) = O(g(z)) \quad \text{as } z \to z_0 \text{ from } D \,,
\]
according to the definition given in class, if and only if
\[
  \limsup_{z \to z_0} \left| \frac{f(z)}{g(z)} \right| < \infty \,.
\]

\item (Cf.\ Miller, p.\ 18.) Suppose $p>0$ and $z>0$.  Show that
\[
  \ln z = o(z^p) \quad \text{as } z \to \infty \,.
\]

\item (Cf.\ Miller, p.\ 19.) Suppose $q>0$ and $z>0$.  Show that
\[
  z^q = O(\e^z)
\]

\item (Cf.\ Miller, p.\ 20.) Give an example to show that in general
\[
  a(z) \, (1 + O(g(z))) + b(z) \, (1 + O(g(z)))
  \neq (a(z) + b(z)) \, (1 + O(g(z))) \,.
\]

\item Show, by example, that the statement
\[
  f(z) = O(g(z)) \quad \text{as } z \to z_0 \text{ from } D 
\]
contains absolutely no information about the magnitude of $f'(z)$.

\item Recall the asymptotic series for the exponential integral
\[
  \Ei(z) = \int_z^\infty \frac{\e^{-t}}t \, \d t \,,
\]
where
\[
  \Ei(z) = \e^{-z} \, 
  \biggl[
    \frac1z
    - \frac1{z^2}
    + \frac{2!}{z^3}
    - \frac{3!}{z^4}
    + \dots +
    \frac{(-1)^{n+1} \, (n-1)!}{z^n}
  \biggr]
  + R_n (z)
\]
with
\[
  R_n(z) = (-1)^n \, n! \int_z^\infty \e^{-1} \, t^{-n-1} \, \d t \,.
\]
\begin{enumerate}
\item Find a (near-)optimal truncation $n(z)$ as a function of $z$.

\emph{Note:} It is sufficient to derive the leading order behavior of
$n(z)$.  You will find Stirling's approximation useful.

\item Give an estimate for $\lvert R_{n(z)}(z) \rvert$ and describe
  its behavior.

\end{enumerate}

\end{enumerate}

\end{document}
