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\author{Math 25}

\title{Homework 7 --- due Nov 7th}

\begin{document}

\maketitle
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There are two sections: A and B.
Please return each part
to the proper CA.

\renewcommand{\thesection}{\Alph{section}}

\section{Problems graded by Benjamin}

\subsection{} Prove that the function 
\[ f(x) = \begin{cases} 
0 & \text{if $x=0$} \\
x^2 \sin(1/x) & \text{if $x \neq 0$}
\end{cases} \]
is differentiable everywhere on $\RR$ but that $f$ is not $C^1$.

\subsection{} Suppose that $\{f_n\}_{n\geq 1}$ is a sequence of $C^1$
functions $f_n : [0,1] \ra \RR$, and that $f,g : [0,1] \ra \RR$ are
two functions such that: for all $x \in [0,1]$, $f_n(x) \ra f(x)$
(i.e. $f_n \ra f$ pointwise), and  $f_n' \ra g$ uniformly on
$[0,1]$.

Prove that $f$ is $C^1$, that $f'=g$ and that $f_n \ra f$ uniformly. 

\section{Problems graded by Inna}

\subsection{} Let $f:[0,1] \ra \RR$ be a continuous function. Find the
limit as $n \ra \infty$ of \[ I_n = n \int_0^1 x^n f(x)dx \] and then find
the next term in the asymptotic development (in order to do this, you
can assume that $f$ is as regular as you want, for example that $f$ is
$C^\infty$). 

\subsection{} For $n \geq 0$, let $J_n: \RR \ra \RR$ be the $n$-th
Bessel function, defined by
\[ J_n(z) = \frac{1}{\pi} \int_0^\pi \cos(n \theta - z \sin \theta) d
\theta. \]
The following questions are largely independant:
\begin{enumerate}
\item Explain why $J_n(z)$ is a $C^\infty$ function of $z$.
\item Prove that $J_1(z)=-J_0'(z)$
and that $J_{n-1} (z) - J_{n+1} (z) =  2 J_n'(z)$.
\item (extra credit) Prove that \[ J_n''(z) + \frac{1}{z} J_n'(z) + \left( 1 -
  \frac{n^2}{z^2} \right) J_n(z) = 0.\] 
\item (extra credit) Prove that $J_n(x) \ra 0$ as $x \ra +\infty$.
\end{enumerate}
\end{document}