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

\section*{Mat-1.198 Scattering Theory}

\subsection*{1$^\textrm{st}$ set of exercises, 29.1.2003}

\vfill

\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.
}

\vfill
\begin{enumerate}
\item Show that in the two dimensional case, the fundamental solution
of the Helmholtz equation is
\[
 \Phi(x)=\frac i4 \, H^{(1)}_0(k|x|),
\]
i.e., verify the multiplicative constant. 
Further, find the explicit form of the asymptotic
expansion 
\[
 \Phi(x-y)\sim \frac{e^{ik|x|}}{\sqrt{|x|}}
\left[f(\hat x,y)+{\cal O}\left(\frac
 1{|x|}\right)\right],
\]
when $|x|\rightarrow\infty$ and $|y|<R$.

\vfill
\item Harmonic polynomials in $\R^2$: A polynomial $P_n(x,y)$ of degree
$n \in \N = \{0, 1, 2, \ldots\}$ in $\R^2$ is {\em homogenous}, if
\[
 P_n(\lambda x,\lambda y)=\lambda^n P_n(x,y)
 \quad
 \textrm{for all } \lambda>0,\, (x, y) \in \R^2
\]
and {\em harmonic}, if
\[
 \Delta P_n(x,y) = 0.
\]
The two-dimensional equivalent to spherical harmonics of $\R^3$ are
defined as the restrictions of the homogenous harmonic polynomials to
the unit circle $S^1$, or simply
\[
 Y_n(\theta)=P_n(\cos\theta,\sin\theta),\quad 0\leq\theta<2\pi.
\]
% What are these functions?
Show that a basis for these functions is given by
$\{e^{\pm i n \theta}\}$.


\item 
% By using the series expansion of the Bessel functions,
Verify the recurrence relation
\[
 J_{p-1}(z)-J_{p+1}(z)=2J_p'(z).
\]
In particular, deduce the useful formula
\[
 J_0'(z)=-J_1(z).
\]
(Note: This formula holds for the Neumann function $N_p$ as well,
and thus also for the Hankel functions $H^{(j)}_p = J_p + (-1)^{j-1} N_p$,
$j \in \{1, 2\}$.)


\vfill

\item The {\em Hankel transform} ${\cal H}_p$ is defined for functions
$f:\R_+\to\C$ as
\[
 {\cal H}_p(f)(k) =\int_0^\infty J_p(kr)f(r) rdr,\quad p=0,1,\ldots
\]
whenever this integral is convergent. 
For functions $f:\R^2\to \C$, we define the Fourier transform as
\[
 \widehat f(\xi)=\int_{\R^2} e^{-i\xi\cdot x}f(x)dx, \quad \xi\in\R^2.
\]
% By using the representation formula
% for Bessel functions, 
Find an expression for the two-dimensional Fourier
transform in terms of the Hankel transforms and the trigonometric
Fourier series representation
\[
  f(x) = \sum_{m=-\infty}^\infty f_m(r) e^{i m \theta},
  \qquad
  x = (r \cos \theta, r \sin \theta).
\]
(Assume here that all the necessary integrals and series are convergent, 
i.e., a formal derivation suffices.)

\end{enumerate}


\vfill

\emph{Useful formulae which you may assume when doing the excercises:}

\[
  J_p(z)
  = \frac 1{2\pi}\int_0^{2\pi}e^{i(z\sin\theta - p\theta)}d\theta
  = \sum_{j=0}^\infty\frac{(-1)^j}{j!(p+j)!}\left(\frac z2\right)^{p+2j}
\]
The series expansion also holds for $p \in \Z_-$ with the 
following interpretation for $j \in \Z_-$:
\[
  \frac{1}{j!} 
  = \frac{j + 1}{(j + 1)!} 
  = \frac{(j+1)(j+2)}{(j+2)!}
  = \cdots
  = \frac{(j+1)(j+2) \cdots (-1)0}{0!}
  = \frac{0}{1}
  = 0
  .
\]


\[
  N_0(z) = \frac{2}{\pi} \left( \ln \frac{z}{2} + \gamma \right)
  + \mathcal{O}(z^2)
  ,
  \qquad
  N_p(z) = \left(\frac{z}{2}\right)^{-p} \left[ 
    - \frac{(p - 1)!}{\pi} + {o}(1)
  \right]
  ,
  \qquad
  z \to 0,\, \re z > 0
\]

\[
  H^{(j)}_p(z) 
%  = J_p(z) + (-1)^{j-1} N_p(z)
  = \sqrt{\frac{2}{\pi z}}
  \; e^{i (-1)^{j} [(2 p + 1)\frac{\pi}{4} - z]}
  \left[1 + \mathcal{O}\left(\frac{1}{|z|}\right)\right],
  \qquad
  |z| \to \infty, -\pi < \arg z < 2\pi, j \in \{1, 2\}
\]

The notation $Y_p$ is also often used for the Neumann function $N_p$
in literature, for instance in the book by Colton and Kress.

\vfill

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