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\centerline{{\sc Problem Set 4}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Fri., Feb. 10}


\bigskip\noindent
{\bf 1.} (a) Consider the $2\pi$-periodic function $f(x;\epsilon)$ defined for $\epsilon >0$ by
\[
f(x;\epsilon) = \begin{cases} 1/\epsilon & \mbox{if $0<x<\epsilon$,} \\ 0 & \mbox{if $\epsilon < x< 2\pi$.} \end{cases},
\qquad f(x+2\pi;\epsilon) = f(x;\epsilon).
\]
Sketch the graph of $f(x,\epsilon)$ on the real line. Compute its Fourier series
\[
f(x;\epsilon) =\sum_{n=-\infty}^\infty f_n(\epsilon) e^{inx}.
\]
(b) Define the $2\pi$-periodic $\delta$-function $\delta_p$, or `$\delta$-comb', by
\[
\delta_p(x) = \sum_{n=-\infty}^\infty \delta(x-2\pi n)
\]
where $\delta(x)$ is $\delta$-function on the real line
supported at $x=0$. Draw a picture of $\delta_p$. Show that a formal
computation of the Fourier coefficients of $\delta_p$ gives
\[
\delta_p(x) = \frac{1}{2\pi} \sum_{n=-\infty}^\infty e^{inx}.
\]
(c) Show that you recover the Fourier series in (b) by taking the limit as $\epsilon\to 0^+$ of the Fourier series in (a).


\bigskip\noindent
{\bf 2.} Consider the $2\pi$-periodic square wave $S$ defined by
\[
S(x) = \begin{cases} 1 & \mbox{if $0<x<\pi$,} \\ 0 & \mbox{if $-\pi < x< 0$.} \end{cases},\qquad S(x+2\pi) = S(x).
\]
(a) Sketch the graph of $S$ on the real line and explain why the (distributional) derivative of $S$ is given by
\[
S'(x) = \delta_p(x) - \delta_p(x-\pi)
\]
where $\delta_p$ is the periodic $\delta$-function from Problem 1.

\smallskip\noindent
(b) Compute the Fourier series of $S$
\[
S(x) = \sum_{n=-\infty}^\infty S_n e^{inx}.
\]
(c) Show that the formal term-by-term derivative of this series agrees with the Fourier series of
$\delta_p(x) - \delta_p(x-\pi)$ from Problem 1.

\bigskip\noindent
{\bf 3.}
Consider the non-homogeneous Sturm-Liouville problem
\begin{align}
\begin{split}
&-u'' = \lambda u + f(x) \qquad 0 <x<1,
\\
&u'(0) = 0, \qquad u(1) = 0
\end{split}
\label{slprob1}
\end{align}
where $\lambda = k^2$ with $k >0$ is a strictly positive constant.

\smallskip\noindent
(a) Give the eigenvalues $\lambda_n$  and eigenfunctions $\phi_n(x)$ of the homogeneous problem, $n=1,2,3,\dots$, which satisfy
\begin{align*}
&-\phi_n'' = \lambda_n \phi_n \qquad 0 <x<1,
\\
&\phi_n'(0) = 0, \qquad \phi_n(1) = 0.
\end{align*}
(b) Write out the eigenfunction expansion for the Green's function $G(x,\xi;\lambda)$.

\smallskip\noindent
(b) Find an explicit expression for the Green's function $G(x,\xi;\lambda)$
in terms of appropriate solutions of the homogeneous equation. Show that your solution has
poles at the eigenvalues $\lambda = \lambda_n$ and that its residues give the eigenfunctions.


\bigskip\noindent
{\bf 4.} Consider the non-homogeneous Sturm-Liouville problem on the real line
\begin{align}
\begin{split}
&-u'' + u = f(x) \qquad -\infty<x<\infty,
\\
&u(x) \to 0 \qquad \mbox{as $x\to \pm\infty$}
\end{split}
\label{slprob}
\end{align}
where $f(x)$ is compactly supported or decays to zero sufficiently rapidly as $x\to \pm \infty$.

\smallskip\noindent
(a) Show that the Greens function $G(x,\xi)$ satisfying
\begin{align*}
&-\frac{d^2 G}{dx^2} + G = \delta(x-\xi) \qquad -\infty<x<\infty,
\\
&G(x,\xi) \to 0 \qquad \mbox{as $x\to \pm\infty$}
\end{align*}
is given by
\[
G(x,\xi) = \frac{1}{2} e^{-|x-\xi|}.
\]
(b) Write down the Green's function representation of the solution of (\ref{slprob}). Verify explicitly that
it is a solution.


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