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\begin{frame}
\begin{center}
\Huge Non-reflecting Boundary Conditions
\end{center}
\end{frame}

\begin{frame}
\scriptsize
\frametitle{Lemma [needed to compute LT of $\Psi_\ell(t,r)$]}
\begin{figure}
 \includegraphics[width=10cm]{PDFfigs/rline.pdf}
 \caption{
 $\color{ec} D$ and open region $\color{ec}r>r_B-\delta$}
\end{figure}
Assume $\color{ec} f(u)$ supported on $ \color{ec} [-r_B+\delta, -r_0-\delta]=D$, 
$\color{ec} f\in C_0^\infty (D)$.
\begin{block}{}
For $\color{eqncolor}r > r_B - \delta$, Fourier-Laplace transform of $\color{eqncolor}f^{(\ell - k)}(t-r)$ is
$$\color{eqncolor}
s^{\ell-k}e^{-sr}a(s), \hspace{2em} a(s) = \int_{-r_B+ \delta}^{-r_0-\delta} e^{-su} f(u) du.
$$
\end{block}
\begin{block}{proof}
$$\color{eqncolor}
\int_0^\infty e^{-st} f^{(\ell-k)}(t-r)dt = e^{-sr} \int_{-r}^\infty e^{-su}f^{(\ell-k)}(u) du
$$
$$\color{eqncolor}
= e^{-sr} \int_{-r_B+\delta}^{-r_0-\delta} e^{-su}f^{(\ell-k)}(u) du, \hspace{2em} -r<-r_B+\delta
$$
\end{block}
\end{frame}