\begin{frame}
\frametitle{Introduction}
\small
\begin{block}{\bf Setting}
\begin{itemize}
\item Outgoing solutions to the ``radial wave equation"
\item Transparent radiation boundary conditions on spherical domain
\item \alert{Dirichlet-to-Neumann map}, \alert{Macdonald function} 
\end{itemize}
\end{block}
\begin{block}{\bf Outline}
\begin{itemize}
\item Radial wave equation and the Macdonald function
\item Non-reflecting boundary conditions
\item Identities for roots of the Macdonald function
\item Other work
\end{itemize}
\end{block}
\end{frame}


\begin{frame}
\begin{center}
\Huge Radial Wave Equation
\end{center}
\end{frame}

\begin{frame}
\frametitle{Radial Wave Equation}
\scriptsize
\begin{block}{\bf Overview}
\begin{itemize}
\item
Ordinary wave equation on $\color{ec}\mathbb{R}^{3}$, ($\color{ec}c=1$ here): 
\hspace{2em} 
$\color{ec} \Delta\psi - \frac{1}{c^{2}}\frac{\partial^{2}\psi}{\partial t^{2}} = 0$
\item
Laplace operator in spherical coordinates: 
$$\color{ec}
\Delta\psi = \frac{1}{r^{2}}\frac{\partial}{\partial r} r^{2} \frac{\partial}{\partial r} \psi + \frac {1}{r^{2}}\Delta_{S^2} \psi, \hspace{2em} \Delta_{S^2} = \frac{1}{\sin \theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\partial\theta} + \frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial \phi^{2}}
$$
\item
Generic solution in terms of $\color{ec} Y_{\ell m}(\theta,\phi)$ of $\color{ec} \Delta_{S^2}$,  where
$$\color{ec} \Delta_{S^2}Y_{\ell m}(\theta,\phi) = -\ell(\ell+1)Y_{\ell m}(\theta,\phi)$$
is:
$$\color{ec}
\psi = \sum_{\ell =0}^\infty \sum_{m = - \ell}^\ell \psi_{\ell m}(t,r) Y_{\ell m}(\theta,\phi) = \sum_{\ell =0}^\infty \sum_{m = - \ell}^\ell \frac{\Psi_{\ell m}(t,r)}{r}Y_{\ell m}(\theta,\phi).
$$
\end{itemize}
\end{block}
\end{frame}


\begin{frame}
\frametitle{Radial Wave Equation}
\scriptsize
\begin{block}{\bf Overview}
\begin{itemize}
\item "multipole" solution $\color{ec}
\psi(t,r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$:
$$\color{ec} 
\psi_{\ell m}(t,r)Y_{\ell m}(\theta,\phi) = r^{-1}\Psi_{\ell m}(t,r) Y_{\ell m}(\theta,\phi)
$$
\item Eliminating angular dependence, multipole coefficients $\color{ec} \psi_{\ell m}$ and $\color{ec} \Psi_{\ell m}$ satisfy:
$$\color{ec}
 - \partial^2_t \psi_{\ell m}+ \partial^2_r \psi_{\ell m}
+ \frac{2}{r}\partial_r \psi_{\ell m}- \frac{\ell(\ell+1)}{r^2}\psi_{\ell m} = 0,
$$
$$\color{ec}
-\partial^2_t\Psi_{\ell m} + \partial^2_r\Psi_{\ell m} - \frac{\ell(\ell+1)}{r^2}\Psi_{\ell m} = 0.
$$
\end{itemize}
\end{block}
\end{frame}