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\title{Analysis II}
\author{Homework 2}
\date{Due in class Monday, February 18, 2019}
\maketitle

\begin{enumerate}

\item Suppose $f = a \, (x-1)^2 + \psi(x) \, (x-1)^3$ satisfies the
requirements of the simple version of Laplace method from class (also
see the handout for a precise statement).  Suppose $g$ is smooth,
bounded, with $g(1)>0$.  Show that, as $s \to \infty$,
\[
  \int_0^\infty \e^{-sf(x)} \, g(x) \, \d x
  \sim g(1) \, \sqrt{\frac\pi{as}} \,.
\]
\emph{Hint}: Set
\[
  G(x) = \int_1^x g(z) \, \d z \,,
\]
then change variables $y = G(x)$ in the first integral.


\item Suppose $f \colon [0, \infty) \to \R$ is continuous and
\begin{itemize}
\item[(i)] $f(0)>0$,
\item[(ii)] there exists $\delta,M>0$ such that $f$ is differentiable
on $[0,\delta]$ with $\lvert f'(x) \rvert \leq M$ for every $x \in [0,
\delta]$,
\item[(iii)] there exist $b,C>0$ such that $\lvert f(x) \rvert \leq C
\, \e^{bx}$ for all $x \geq 0$.
\end{itemize}
In this setting,
\begin{enumerate}
\item Show that, as $s \to \infty$,
\[
  \int_0^\infty \e^{-sx} \, f(x) \, \d x \sim \frac{f(0)}s \,,
\]
\item Formulate a theorem which provides higher order corrections to
the statement from (a); you will need to adjust your assumptions on
$f$ accordingly.

\item Test your formula from part (b) on the integral
\[
  \int_0^\infty \e^{-sx} \, (1-\cos x) \, \d x \,.
\]
(Note: for this integral, you can find the leading order asymptotics
by direct repeated integration by parts to compare.)

\end{enumerate}

\item (Rudin, Theorem 7.9.) Let $f,f_n \colon E \to \R$ with pointwise
limit
\[
  \lim_{n\to\infty} f_n(x) = f(x)
\]
for every $x \in E$.  Set
\[
  M_n = \sup_{x \in E} \, \lvert f_n(x) - f(x) \rvert \,.
\]
Show that $f_n \to f$ uniformly in $E$ if and only if
$\lim_{n\to\infty} M_n = 0$.

\item (Rudin, Example 7.4.) Let
\[
  f_n(x) = \lim_{m\to\infty} (\cos n! \pi x)^{2m} \,.
\]
\begin{enumerate}
\item Argue that
\[
  f_n(x) = 
  \begin{cases}
    1 & \text{if } n!x \in \mathbb Z \,, \\
    0 & \text{otherwise} \,.
  \end{cases}
\]

\item Show that the pointwise limit
\[
  f(x) = \lim_{n\to\infty} f_n(x) =
  \begin{cases}
    0 & \text{if } x \in \mathbb I \,, \\
    1 & \text{if } x \in \mathbb Q \,.
  \end{cases}
\]
\item Argue that $f$ is not Riemann integrable.

\end{enumerate}

\item (Rudin, Exercise 7.9.) Let $\{f_n\}$ be a sequence of continuous
functions which converges uniformly to a function $f$ on a set $E$.
Prove that
\[
  \lim_{n \to \infty} f_n(x_n) = f(x)
\]
for every sequence of points $x_n \in E$ such that $x_n \to x$, and $x
\in E$.

\end{enumerate}



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