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\begin{document}

Helsinki University of Technology \hfill Somersalo/Bingham \\
Mathematics

\section*{Mat-1.198 Scattering Theory}

\subsection*{9$^\textrm{th}$ set of exercises, 9.4.2003}

\vfill

\emph{%
The exam will be on Tuesday, the 20th of May from 10.00 to 13.00 in U322.
}

% \emph{%
% By doing the exercises, you can earn points that will be taken into account 
% when determining your grade for the course:
% Do these exercises in advance, mark them as solved
% on the list at the exercise session and be prepared to present
% your solution on the blackboard.
% }

\begin{enumerate}

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\item Consider the one-dimensional
potential scattering of the previous week. 
Derive the PML equation in the one-dimensional case: The
stretching function 
%$F:\R\to \C$, $x\mapsto x + i\tau(x)$
\[
  F: \R\to \C, \quad x\mapsto 
  \begin{cases}
    x - i\tau(a - x), & x < a \\
    x, & a \le |x| \le b \\
    x + i\tau(x - b), & x > b
  \end{cases}
\]
maps the real axis to a 
curve in the complex plane.
Here $(a, b) \supset [-M, M]$ is the truncated computation domain.

\vfill

\item Consider the PML equation in radial coordinates, i.e., the
computational region $B$ is a disc,
\[
 \overline D\subset B=\{x\in\R^2\mid |x|<R\}\subset\R^2.
\]
Given a stretching function $\tau:[0,\infty)\to[0,\infty)$, calculate
the PML equation using the polar coordinate representation for the
Laplacian.

\vfill

\item The previous example allows us to write the PML equation as
\[
 \nabla\cdot A\nabla u + a k^2 u =0,
\]
where $A = A(x)\in\C^{2\times 2}$, $a = a(x)\in \C$. Find this
representation. In particular, what is $A$ in Cartesian coordinates?

\vfill
\vfill

\end{enumerate}
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