%--------------------------------------------------------------
\begin{frame}
\frametitle{Radial Wave Equation}
\small
\begin{itemize}
\item
Radial Wave Equation:
$$
\color{eqncolor}
- \partial^2_t \Psi_\ell+ \partial^2_r \Psi_\ell
- \frac{\ell(\ell+1)}{r^2}\Psi_\ell = 0
$$
\item
For $\color{eqncolor} \ell=0$, generic solutions from factoring wave equation:
$$\color{eqncolor}
\underrightarrow{f(t-r)}, \hspace{2em} \underleftarrow{g(t+r)}
$$
\end{itemize}
\begin{block}{\bf Lemma}
For $\color{eqncolor} \ell>0$, generic outgoing solution:
$$
\color{eqncolor}
\Psi_{\ell}(t,r)=\sum_{k=0}^\ell \frac{c_{\ell k}}{r^k}f^{(\ell-k)}(t-r)
\hspace{2em}
c_{\ell k} = \frac{1}{2^k k!}\frac{(\ell + k)!}{(\ell - k)!}
$$
\end{block}
\end{frame}
