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\title{Applied Analysis}
\author{Homework 4}
\date{due October 19, 2007}
\maketitle

\begin{enumerate}

\item Use the method of steepest descent to determine the first two
  terms in an asymptotic expansion of
\[
  I(\lambda) = \int_0^{\pi/4} \cos(\lambda z^2) \, \tan^2 z \, \d z
\]
as $\lambda \to \infty$ from $\mathbb{R}_+$.

\item Consider the integral
\[
  I(\lambda) = \int_0^1 \e^{-4 \lambda z^2} \, 
  \cos(5 \lambda z - \lambda z^3) \, \d z 
\]
as $\lambda \to \infty$ from $\mathbb{R}_+$.
\begin{enumerate}
\item Explain why not even the leading order of its asymptotic
  expansion can be found by Laplace's method with $\Phi(z) = -4 z^2$
  and $g(z) = \cos(5 \lambda z - \lambda z^3)$.
\item Show that 
\[
  I(\lambda) = \frac12 \, \e^{-2\lambda} \, 
  \biggl(\sqrt{\frac\pi{\lambda}} + O(\lambda^{-3/2}) \biggr) 
\]
by using a steepest-descent path of integration in the complex plane. 
  
\end{enumerate}
.

\item Miller, p.\ 110, Exercise 4.3.

\item Miller, p.\ 110, Exercise 4.4.


\end{enumerate}
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