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

\def\Rl{\mathbb{R}}
\def\Cx{\mathbb{C}}
\def\Nl{\mathbb{N}}

\begin{document}

\centerline{{\sc Problem Set 8}}
\centerline{Math 207B, Winter 2012}
\centerline{Due: Mon., Mar. 12}



\bigskip\noindent
{\bf 1.} Let $G(x,\xi)$ be the Green's function for the Sturm-Liouville problem
\[
-u'' = \lambda u,\qquad u(0) = u(1) = 0,
\]
given by
\[
G(x,\xi) = x_< (1- x_>).
\]
(a) What are the eigenvalues $\mu_n$ and eigenfunctions $\phi_n$ of $G$, where $n=1,2,\dots$? (Find them from the corresponding
eigenvalues and eigenfunctions of the Sturm-Liouville problem.)

\smallskip\noindent
(b) Compute
\[
\int_0^1 \int_0^1 G(x,\xi)^2 \, dx d\xi.
\]

\smallskip\noindent
(c) Use the identity
\[
\int_0^1 \int_0^1 G(x,\xi)^2 \, dx d\xi = \sum_{n=1}^\infty \mu_n^2
\]
to deduce that
\[
\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}.
\]



\newpage\noindent
{\bf 2.} Define the Abel integral operator $K$, acting on continuous functions $u(x)$ where $0\le x \le 1$, by
\[
(K u)(x) = \int_0^x \frac{u(y)}{(x-y)^{1/2}} \, dy,\qquad 0\le x \le 1.
\]
(a) Is $K$ a Hilbert-Schmidt operator?

\smallskip\noindent
(b) Show that $K^2 = \pi L$ where $L$ is the integration operator
\[
L u(x) = \int_0^x u(y)\, dy. 
\]
\textsc{Hint.} The substitution $t = x\sin^2\theta + y\cos^2\theta$ shows that
\[
\int_y^x \frac{dt}{(x-t)^{1/2} (t-y)^{1/2}} = \pi.
\] 
(c) Suppose that $f : [0,1]\to \Rl$ is a smooth function with $f(0) = 0$. Deduce that the solution
of the Abel integral equation
\[
\int_0^x \frac{u(y)}{(x-y)^{1/2}} \, dy = f(x),\qquad 0\le x \le 1
\]
is given by
\[
u(x) = \frac{1}{\pi} \int_0^x \frac{f'(y)}{(x-y)^{1/2}}\, dy.
\]
\textsc{Hint.} Solve the equation $Lu = (1/\pi) Kf$.

\end{document}             % End of document.


