\begin{frame}
\frametitle{Radial Wave Equation}
\scriptsize
\begin{block}{\bf Overview}
\begin{itemize}
\item The Macdonald function $\color{eqncolor}K_\nu (z)$ is a solution to the modified Bessel equation:
$$\color{eqncolor} 
z^2 w'' + zw' - (z^2 + \nu^2)w = 0
$$
\item For half-integer order, 
$$\color{eqncolor} 
K_{\ell + 1/2}(z) = \sqrt{\frac{\pi}{2z}} e^{-z} W_\ell(z), \qquad
W_\ell(z) = \sum_{k=0}^\ell \frac{c_{\ell k}}{z^k},\qquad
c_{\ell k} = \frac{1}{2^k k!}\frac{(\ell + k)!}{(\ell - k)!}
$$
\item $\color{eqncolor} K_{\ell+1/2}(z)$ can also be expressed using the monic \emph{Bessel polynomial} $\color{eqncolor}p_{\ell}(z)$: 
$$\color{eqncolor}
K_{\ell+1/2}(z) = \sqrt{\frac{\pi}{2z}}\frac{e^{-z}}{z^\ell}p_{\ell}(z), \hspace{2em} p_{\ell}(z) = \sum_{k=0}^{\ell}c_{\ell k}z^{\ell-k}
$$
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Radial Wave Equation}
\scriptsize
\begin{block}{\bf Overview}
\begin{itemize}
\item The set $\color{eqncolor}\{b_{\ell j}/(\ell+1/2): j = 1,\dots,\ell\}$ is the collection of roots scaled by the Bessel order $\color{eqncolor}\nu = \ell + 1/2$.   
\item These roots accumulate on a fixed transcendental curve in the left-half plane, a parametrization of which is given by:
$$\color{eqncolor}
z(\lambda) = -\sqrt{\lambda^2 - \lambda\tanh\lambda}\pm
\mathrm{i}\sqrt{\lambda\coth\lambda-\lambda^2}
$$
for $\color{eqncolor}\lambda \in [0,\lambda_0]$, where $\color{eqncolor}\lambda_0 \simeq 1.19967864025773$
solves $\color{eqncolor}\tanh\lambda_0 = 1/\lambda_0$.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}
\frametitle{Radial Wave Equation}
\small
\begin{columns}[c]
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\column{3.1in}
Scaled zeros $\displaystyle
\color{eqncolor}\frac{b_{\ell j}}{(\ell+1/2)}$ 
of $\color{eqncolor}K_{\ell+1/2}(z)$ and $\color{eqncolor}W_\ell(z)$:
\\[2mm]
${\color{red}+}$ $\displaystyle \color{eqncolor} 
K_{3/2}(z) = \sqrt{\frac{\pi}{2 z}} e^{-z}\left(1+\frac{1}{z}\right)$
\\[2mm]
$\color{blue}\diamond$ $\displaystyle \color{eqncolor}
K_{5/2}(z) = \sqrt{\frac{\pi}{2 z}}e^{-z}\left(1+\frac{3}{z}+\frac{3}{z^2}\right)$
\\[2mm]
$\color{black}\circ$ $\displaystyle \color{eqncolor}
K_{7/2}(z) = \sqrt{\frac{\pi}{2 z}} e^{-z}\left(
1+\frac{6}{z} + \frac{15}{z^2} + \frac{15}{z^3}\right)$
\\[2mm]
$\color{magenta}*$ $\displaystyle \color{eqncolor}
K_{9/2}(z) = \sqrt{\frac{\pi}{2 z}} e^{-z}\left(
1+\frac{10}{z} + \frac{45}{z^2} + \frac{105}{z^3} + \frac{105}{z^4}\right)$
\column{1.4in}
\begin{picture}(-1.0,-2.0)
     \put(-0.3,-0.65){\includegraphics[width=11cm]{PDFfigs/zerosofK.pdf}}
\end{picture}
\end{columns}
\end{frame}