\batchmode
\documentclass[12pt,a4paper]{article}
\makeatletter
\usepackage{html}
\usepackage{latexsym, amsfonts, amssymb, amsbsy}
\usepackage[dvips]{graphicx}





\setlength{\textheight}{247mm}\setlength{\textwidth}{200mm}


\newcommand {\Cite}[2]{\cite{#2},  #1}

\newcommand {\Ent}[1]{[\![#1]\!]}

\newcommand {\text}[1]{\mbox{\rm {#1}}}

\newcommand {\myarraystretch}{1.2}

\renewcommand {\arraystretch}{\myarraystretch}

\newcommand {\submitted}{{\rm Submitted}}

\newcommand {\homeier}[1]{}

\makeatother
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\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{\ell }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2358: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\@mathcal {K}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2360: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\theta =60$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2364: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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{\newpage
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}

\stepcounter{section}
\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2408: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:Uint}
U(\vec r)
= \int \frac{\rho(\vec r')}{\vert\vec r-\vec r'\vert}\,  \mbox{d}^3r'
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{\vert\vec r-\vec r'\vert}
=4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r) \,  Y_{\ell}^{m*}(\vec r'/r')}{2\ell+1}
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r=\vert \vec r\vert$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2410: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r'=\vert \vec r'\vert$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2412: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_{<}=\min(r, r')$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2414: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_{>}=\max(r, r')$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y_{\ell}^{m}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2418: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:exakt}
U(\vec r)
= 4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
\int \rho(\vec r')
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
     Y_{\ell}^{m*}(\vec r'/r')
\,  \mbox{d}^3r'\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec r'$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2426: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho(\vec r')\ne 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2428: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U(\vec r)
= 4\pi
\sum_{\ell m}
  r^{-\ell-1} \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
\int \rho(\vec r')
     r'^{\ell}
     Y_{\ell}^{m*}(\vec r'/r')
\,  \mbox{d}^3r'\>
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Q_{\ell}^{m}=
   \int
     r'^{\ell}
     Y_{\ell}^{m}(\vec r'/r')
     \rho(\vec r')
     \,  \mbox{d}^3r'  \>
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:Uq}
U(\vec r)=U_Q(\vec r)
= 4\pi
\sum_{\ell m}
  \frac{1}{r^{\ell+1}}   \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
Q_{\ell}^{m*}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2430: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec R$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2432: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eq:ql}
Q_{\ell}^{m*} = Y_{\ell}^{m*}(\vec R/R) \, q_{\ell}\>, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqYadd}
\frac{4\pi}{2\ell+1}
\sum_{m}
  Y_{\ell}^{m}(\vec r/r)
Y_{\ell}^{m*}(\vec R/R)
=
P_{\ell}\left(\frac{\vec r\cdot \vec R}{r\, R}\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_\ell$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2434: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqUqzyl}
U_Q(\vec r)
=
\sum_{\ell=0}^{\infty}
P_{\ell}\left(\frac{\vec r\cdot \vec R}{r\, R}\right)
\frac{q_{\ell}}{r^{\ell+1}}\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^2 U =0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2436: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r\to\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2438: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathcal{Z}_{\ell}^{m}(\vec r)=
r^{-\ell-1} Y_{\ell}^{m}(\vec r/r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2440: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec R/R$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2442: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r^{-\ell-1}P_{\ell}(\cos \alpha)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2444: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2448: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathbb{R}^3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2452: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U_Q$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2464: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U(\vec r)-U_Q(\vec r)
= 4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
\int \rho(\vec r')
\left\{     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
-
     \frac{r'^{\ell}}{r^{\ell+1}}
\right\}
Y_{\ell}^{m*}(\vec r'/r')
\,  \mbox{d}^3r'\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
-
     \frac{r'^{\ell}}{r^{\ell+1}} = 0\quad\mbox{for }r'\le r\>
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U-U_Q$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2466: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U(\vec r)-U_Q(\vec r)
= 4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
\int_{r'>r} \rho(\vec r')
\left\{     \frac{r^{\ell}}{r'^{\ell+1}}
-
     \frac{r'^{\ell}}{r^{\ell+1}}
\right\}
Y_{\ell}^{m*}(\vec r'/r')
\,  \mbox{d}^3r'\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\ell$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2470: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho(\vec r)=\exp(-\gamma r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2476: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma>0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2478: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\ell=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2480: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}U(\vec r)-U_Q(\vec r)
=  4\pi
\int_{r}^{\infty} \exp(-\gamma r')
\left\{     \frac{1}{r'}
-
     \frac{1}{r}
\right\}
r'^2\,  \mbox{d} r'
= -4\pi
{\frac {{\exp({-\gamma\,  r})}
\left (2+\gamma\,  r\right
)}{  r\, {\gamma}^{3}}}\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec r\ne 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2488: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^2 U_Q=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2490: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla^2 U(\vec r) = -4\pi \rho(\vec r)\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q_{\ell}^{m}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2498: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U_Q(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2504: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$U(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2506: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2508: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{Z}_\ell^{m}(r)=\int \rho(\vec r')
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
     Y_{\ell}^{m}(\vec r'/r')
\,  \mbox{d}^3r'\>, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{Z}_{\ell}^{m}(r) = Y_{\ell}^{m}(\vec R/R)
\, \tilde{z}_{\ell}(r)\>, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqUzyl}
U(\vec r)
= 4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
Y_{\ell}^{m*}(\vec R/R)
\, \tilde{z}_{\ell}(r)
=
\sum_{\ell=0}^{\infty}
P_{\ell}\left(\frac{\vec r\cdot \vec R}{r\, R}\right)
\, \tilde{z}_{\ell}(r)\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2520: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$rR\cos
\theta=\vec r\cdot\vec R$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2522: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\label{eq:Dint}
I[\phi_j, \phi_k](\vec r) = \int \frac{\phi_j^{*}(\vec r')\,  \phi_k(\vec r')}
{\vert\vec r-\vec r'\vert}\,  \mbox{d}^3r'
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_j(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2524: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi_k(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2526: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathbb{R}^3 \to
\mathbb{C}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2528: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi_{\alpha}(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2530: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\psi_{\alpha}(\vec r) = \sum_{j} c_{j, \alpha} \phi_j(\vec r)\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec R_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2534: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_j(\vec r) = \chi_j(\vec r-\vec R_j)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chi_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2536: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\chi_j(\vec r)=\rho_j(r)
\sigma_j(\theta, \phi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2538: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x^a y^b z^c$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2540: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Y_{\ell}^{m}(\theta, \phi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2542: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(r)\exp(-\alpha r^{\tau})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2544: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau=2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2548: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2550: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}G(\alpha, \vec r) = \exp(-\alpha r^2)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}e(\alpha, \vec r) = \exp(-\alpha r)\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\ell=m=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2554: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\chi_{n, \ell}^{m}(\alpha, \vec r) = (\alpha\, r)^{n-1} \, \exp(-\alpha r) \, 
Y_{\ell}^{m}(\theta, \phi)\>, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}B_{n, \ell}^m(\alpha, \vec r)= \frac{1}{2^{n+\ell}(n+\ell)!)}\, 
\hat{k}_{n-1/2}(\alpha r) \,  (\alpha r)^\ell \,  Y_{\ell}^{m}(\theta, \phi)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat
k_{\nu}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2558: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\hat k_\nu(z) = \left( \frac{2}{\pi}\right)^{1/2} z^\nu K_\nu(z)
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$
K_{\nu}(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2560: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec R_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2566: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$I[\phi_j, \phi_k](\vec
r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2568: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_{jk}(\vec r)=\phi_j(\vec
r)\phi_k(\vec r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2572: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2576: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\label{eqOrthogen}
s(x) = \sum_{j=0}^{\infty} c_j\, P_j(x)
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqOrthogenpartsums}
s_n(x) = \sum_{j=0}^{n} c_j\, P_j(x)\>, 
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d^{(m)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2582: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathcal{K}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2584: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathcal{I}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2588: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2592: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$t_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2596: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}s_n=s+t_n\>, \qquad t_n=-\sum_{j=n+1}^{\infty} c_j\, P_j(x) \>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_n(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2598: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}t_n= \alpha_{n+1}(x)
P_{n+1}(x) + \beta_{n+1}(x) P_{n+2}(x)  
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_{n+2}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2600: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_{n+1}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2602: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q_{n+1}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2604: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q_n(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2612: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$w_{n+1}(x)=(2n+1)xw_{n}(x)/(n+1)-nw_{n-1}/(n+1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2614: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_0(x)=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2616: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_1(x)=x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2618: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q_0(x)=log((1+x)/(1-x))/2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2620: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Q_1(x)=xQ_0(x)-1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2622: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\displaystyle P_n(\cos
\theta)-\mbox{i}\frac{2}{\pi}Q_n(\cos\theta)=\exp(\mbox{i}n\theta)
{k}{n^{-1/2}}\left(1+O(1/n)\right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2624: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\displaystyle
P_{n+1}(\cos\theta)=(1+1/n)^{-1/2}(\cos\theta\,P_n(\cos\theta)+2\sin
\theta\, Q_n(\cos \theta)/\pi)(1+O(1/n))$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2630: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}t_n= A_{n+1}(x)
P_{n+1}(x) + B_{n+1}(x)
Q_{n+1}(x)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{n}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2634: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B_{n}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2636: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2644: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqOrthomodseq}
s_n = s + \omega_n\, (c\,   \tilde P_n(x) + d\,\tilde Q_n(x))\>
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde P_n(x)=P_{n+1}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2646: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde
Q_n(x)=Q_{n+1}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2648: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqOrthorec}
\zeta_{n}^{(0)} v_{n} + \zeta_{n}^{(1)} v_{n+1} + \zeta_{n}^{(2)}
v_{n+2} = 0\>
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\zeta_n^{(j)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2650: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{s_n - s }{\omega_n} = c\,  \tilde P_n(x)+ d\,\tilde Q_n(x)\>
{}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\displaystyle
        \zeta_{n}^{(0)} \frac{s_{n}-s}{\omega_{n}} +
        \zeta_{n}^{(1)} \frac{s_{n+1}-s}{\omega_{n+1}} +
        \zeta_{n}^{(2)} \frac{s_{n+2}-s}{\omega_{n+2}}        
=0\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}s
= \frac{\displaystyle
        \zeta_{n}^{(0)} \frac{s_{n}}{\omega_{n}} +
        \zeta_{n}^{(1)} \frac{s_{n+1}}{\omega_{n+1}} +
        \zeta_{n}^{(2)} \frac{s_{n+2}}{\omega_{n+2}}
        }
       {\displaystyle
        \zeta_{n}^{(0)} \frac{1}{\omega_{n}} +
        \zeta_{n}^{(1)} \frac{1}{\omega_{n+1}} +
        \zeta_{n}^{(2)} \frac{1}{\omega_{n+2}}
        }\>, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}s_n^{(1)}
= \frac{\displaystyle
        \zeta_{n}^{(0)} \frac{s_{n}}{\omega_{n}} +
        \zeta_{n}^{(1)} \frac{s_{n+1}}{\omega_{n+1}} +
        \zeta_{n}^{(2)} \frac{s_{n+2}}{\omega_{n+2}}
        }
       {\displaystyle
        \zeta_{n}^{(0)} \frac{1}{\omega_{n}} +
        \zeta_{n}^{(1)} \frac{1}{\omega_{n+1}} +
        \zeta_{n}^{(2)} \frac{1}{\omega_{n+2}}
        }\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqKtrans}
\begin{array}{r@{}l}
 \displaystyle & \displaystyle \mbox{N}_n^{(0)} = s_n/\omega_{n}\>, 
\qquad
\mbox{D}_n^{(0)} = 1/\omega_{n}\>, \\ 
 \displaystyle & \displaystyle\mbox{N}_{n}^{(k+1)} =  
\left.\left(\zeta^{(0)}_{n+k}\mbox{N}_{n}^{(k)}
                    + \zeta^{(1)}_{n+k}
                      \, \mbox{N}_{n+1}^{(k)}
                    + \zeta^{(2)}_{n+k}\mbox{N}_{n+2}^{(k)}
                      \right)\right/\delta_n^{(k)}\>, \\ 
 \displaystyle & \displaystyle\mbox{D}_{n}^{(k+1)} =  
\left.\left(\zeta^{(0)}_{n+k}\mbox{D}_{n}^{(k)}
                    + \zeta^{(1)}_{n+k}
                      \, \mbox{D}_{n+1}^{(k)}
                    + \zeta^{(2)}_{n+k}\mbox{D}_{n+2}^{(k)}
                      \right)\right/\delta_n^{(k)}\>, \\ 
\displaystyle & \displaystyle
\mathcal{K}_n^{(k)}(\{\delta_{n}^{(k)}\}, \{\zeta_n^{(j)}\}, 
\{s_n\}, \{\omega_n\})= \mbox{N}_n^{(k)}/
\mbox{D}_n^{(k)}\>
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\delta_n^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2666: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega_n^{(k)}=1/D_n^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2674: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega_n^{(k)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2680: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega_n=\omega_n^{(0)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2682: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_{n+k}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2686: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqOrthotransformedSequence}
\{s_n\} \longrightarrow \{s_n'\} = \{
\mathcal{K}_{n-2[\![n/2]\!]}^{([\![n/2]\!])}
(\{\delta_{n}^{(k)}\}, \{\gamma_n^{(j)}\}, \{s_n\}, \{\omega_n\})\}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_n^{(0)}=n+2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2698: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_n^{(1)}=-(2n+5)x$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2700: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_n^{(2)}=n+3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2702: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_{n+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2704: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$[\![x]\!]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2706: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2710: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{2N}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2718: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{2N}'=\mathcal{K}_{0}^{(N)}
(\{\delta_{n}^{(k)}\}, \{\gamma_n^{(j)}\}, \{s_n\}, \{\omega_n\})\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2726: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}s_{2N}' = \frac{p_{2N}(x)}{q_N(x)}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{2N}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2730: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{N}(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2734: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\delta_{n}^{(k)}=1/(n+1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2738: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega_n=c_n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2740: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}s_4'=\lefteqn{ - {\displaystyle \frac {1}{2}} (x^{4}\,{c_{4}}\,{c_{1}}
\,{c_{3}}\,{c_{2}} + ( - 196\,{c_{1}}^{2}\,{c_{3}}\,{c_{4}} + 492
\,{c_{4}}\,{c_{1}}\,{c_{2}}^{2} - 300\,{c_{1}}\,{c_{2}}\,{c_{3}}
^{2})\,x^{3}} \\ 
 & & \mbox{} + ( - {c_{1}}\,{c_{2}}\,{c_{3}}\,{c_{4}} - 196\,{c_{
1}}\,{c_{3}}\,{c_{4}}\,{c_{0}} + 328\,{c_{4}}\,{c_{1}}^{2}\,{c_{2
}} - 180\,{c_{1}}\,{c_{2}}^{2}\,{c_{3}})\,x^{2} + ( \\ 
 & & 114\,{c_{2}}\,{c_{3}}\,{c_{4}}\,{c_{0}} - 164\,{c_{4}}\,{c_{
1}}\,{c_{2}}^{2} + 328\,{c_{4}}\,{c_{1}}\,{c_{2}}\,{c_{0}} + 180
\,{c_{1}}\,{c_{2}}\,{c_{3}}^{2} \\ 
 & & \mbox{} - 120\,{c_{1}}^{2}\,{c_{2}}\,{c_{3}} - 114\,{c_{1}}
^{2}\,{c_{3}}\,{c_{4}})x\mbox{} + 60\,{c_{1}}\,{c_{2}}^{2}\,{c_{3}} - 120\,{c_{1}
}\,{c_{2}}\,{c_{3}}\,{c_{0}} \\ 
 & & \mbox{} - 114\,{c_{1}}\,{c_{3}}\,{c_{4}}\,{c_{0}}){c_{0}}
 \left/ {\vrule height0.47em width0em depth0.47em} \right. \! 
 \! (98\,x^{2}\,{c_{4}}\,{c_{1}}\,{c_{3}}\,{c_{0}} \\ 
 & & \mbox{} + ( - 164\,{c_{4}}\,{c_{1}}\,{c_{2}}\,{c_{0}} - 57\,{c_{2}}\,{c_{3}}\,{c_{4}}\,{c_{0}})\,x + 6\,{c_{1}}\,
{c_{2}}\,{c_{3}}\,{c_{4}} + 60\,{c_{1}}
\,{c_{2}}\,{c_{3}}\,{c_{0}} \\ 
 & & \mbox{} + 57\,{c_{1}}\,{c_{3}}\,{c_{4}}\,{c_{0}})
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{displaymath}s_4={\displaystyle \frac {35}{8}} \,{c_{4}}\,x^{4} + {\displaystyle 
\frac {5}{2}} \,{c_{3}}\,x^{3} + ({\displaystyle \frac {3}{2}} \,
{c_{2}} - {\displaystyle \frac {15}{4}} \,{c_{4}})\,x^{2} + ({c_{1
}} - {\displaystyle \frac {3}{2}} \,{c_{3}})\,x + {c_{0}} - 
{\displaystyle \frac {1}{2}} \,{c_{2}} + {\displaystyle \frac {3
}{8}} \,{c_{4}}\>.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{2N}'$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2750: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{4}'$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2754: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2756: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x=\cos\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2762: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_n'$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2764: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$O(1/n^\alpha)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2770: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau-$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2774: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2776: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\{s_{\tau n}\}_{n=0}^{\infty}=\{s_0,s_\tau,s_{2\tau},\dots\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2778: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau>1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2780: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{}^\tau s_{n}=\mathcal{K}_{n-2[\![n/2]\!]}^{([\![n/2]\!])}
(\{\delta_n^{(k)}\},
\{\gamma_n^{(j)}\},
\{s_{\tau\,n}\},
\{{}^{\tau}\omega_{n}\})
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_n^{(1)}=-(2n+5)\,x_\tau$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2786: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqorthotau}
x_{\tau} = \cos(\tau \mbox{\,arccos}(x))\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${}^{\tau}\omega_{n}=c_{\tau n}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2790: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\label{eqrhobsp}
\rho(\vec r) = \exp(-\alpha r) \exp(-\beta \vert\vec r-\vec R\vert)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\begin{array}{r@{}l}
U(\vec r)
\displaystyle & \displaystyle{}= \int
\frac{\exp(-\alpha r') \exp(-\beta \vert\vec r'-\vec R\vert)}
{\vert\vec r-\vec r'\vert}\,  \mbox{d}^3r'\\ 
\displaystyle & \displaystyle{}=
4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
  \int
     \frac{r_{<}^{\ell}}{r_{>}^{l+1}}
     Y_{\ell}^{m*}(\vec r'/r')
\\ 
\displaystyle & \displaystyle{}
\times\exp(-\alpha r')
\exp(-\beta \vert\vec r'-\vec R\vert)
\,  \mbox{d}^3r'
\\ 
\displaystyle & \displaystyle{}=
4\pi
\sum_{\ell m}
  \frac{Y_{\ell}^{m}(\vec r/r)}{2\ell+1}
  \int
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
     Y_{\ell}^{m*}(\vec r'/r')
     \exp(-\alpha r')
\\ 
\displaystyle & \displaystyle\times \frac{-4\pi}{\sqrt{r' R}} \sum_{\ell'm'}
  Y_{\ell'}^{m'}(\vec r'/r')
     Y_{\ell'}^{m'*}(\vec R/R)
\\ 
\displaystyle & \displaystyle{}\times  \left(
    \frac{\partial}{\partial \beta}
  \right)
  I_{\ell'+1/2}(\beta r'_{<}) K_{\ell'+1/2}(\beta r'_{>})
\,  \mbox{d}^3r'
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r'_{<}=\min(r', R)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2794: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r'_{>}=\max(r', R)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2796: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\exp(-\lambda \sqrt{r^2+\rho^2-2r\rho\cos\theta})
=-\sum_{\ell=0}^{\infty} (2\ell+1)\,  P_{\ell}(\cos\theta)
\frac{1}{\sqrt{r\rho}} \frac{\partial}{\partial
\lambda}[I_{\ell+1/2}(\lambda r) K_{\ell+1/2}(\lambda\rho)]
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r<\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2798: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqUbsp}
\begin{array}{r@{}l}
U(\vec r)
\displaystyle & \displaystyle{}=
\frac{-4\pi}{\sqrt{R}}
\sum_{\ell=0}^{\infty}
P_{\ell}\left(\vec r\cdot\vec R/(rR)\right)
  \int_0^{\infty} r'^{3/2}
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
     \exp(-\alpha r')
\\ 
\displaystyle & \displaystyle {}\times
  \left(
    \frac{\partial}{\partial \beta}
  \right)
  I_{\ell+1/2}(\beta r'_{<}) K_{\ell+1/2}(\beta r'_{>})
\,  \mbox{d} r'\>.
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{z}_{\ell}(r)=\frac{-4\pi}{\sqrt{R}}
  \int_0^{\infty} r'^{3/2}
     \frac{r_{<}^{\ell}}{r_{>}^{\ell+1}}
     \exp(-\alpha r')
  \left(
    \frac{\partial}{\partial \beta}
  \right)
  I_{\ell+1/2}(\beta r'_{<}) K_{\ell+1/2}(\beta r'_{>})
\,  \mbox{d} r'\>.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\label{eqUqbsp}
\begin{array}{r@{}l}
U_Q(\vec r)
\displaystyle & \displaystyle{}=
\frac{-4\pi}{\sqrt{R}}
\sum_{\ell=0}^{\infty}
P_{\ell}\left(\vec r\cdot\vec R/(rR)\right)
  \int_0^{\infty} r'^{3/2}
     \frac{r'^{\ell}}{r^{\ell+1}}
     \exp(-\alpha r')
\\ 
  \displaystyle & \displaystyle{}\times \left(
    \frac{\partial}{\partial \beta}
  \right)
  [I_{\ell+1/2}(\beta r'_{<}) K_{\ell+1/2}(\beta r'_{>})]
\,  \mbox{d} r'\>, 
\end{array}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}q_{\ell}=
\frac{-4\pi}{\sqrt{R}}
  \int_0^{\infty} r'^{3/2+\ell}
     \exp(-\alpha r')
  \left(
    \frac{\partial}{\partial \beta}
  \right)
  [I_{\ell+1/2}(\beta r'_{<}) K_{\ell+1/2}(\beta r'_{>})]
\,  \mbox{d} r'\>
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho(\vec r)=\rho(x, y, z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2800: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\alpha=3/10$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2802: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\beta=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2804: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec R=2\vec e_z$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2806: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec e_z$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2808: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sqrt{x^2+y^2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2824: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r=\sqrt{x^2+y^2+z^2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2826: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\cos\theta=z/r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2830: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}s_{\ell} = \sum_{j=0}^{\ell}
P_{j}\left(\cos\theta\right)
\frac{q_{j}}{r^{j+1}}\>
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}s'_{\ell} = \mathcal{K}_{\ell-2[\![\ell/2]\!]}^{([\![\ell/2]\!])}
(\{(\ell+1)^{-1}\}, \{\gamma_\ell^{(j)}\}, \{s_\ell\}, \{q_\ell/r^{\ell+1}\})\}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_\ell^{(0)}=\ell+2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2840: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_\ell^{(1)}=-(2\ell+5)
\cos\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2842: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_\ell^{(2)}=\ell+3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2844: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P_{\ell+1}(\cos\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2846: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s_{\ell}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2850: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$s'_{\ell}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2852: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\ell=20$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2856: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$x=\cos(\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2860: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}s_{\ell} = \sum_{j=0}^{\ell}
P_{j}\left(\cos\theta\right)
\tilde{z}_j(r)\>
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}s'_{\ell} = \mathcal{K}_{\ell-2[\![\ell/2]\!]}^{([\![\ell/2]\!])}
(\{(\ell+1)^{-1}\}, \{\gamma_\ell^{(j)}\}, \{s_\ell\}, \{\tilde{z}_\ell(r)\})\}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2904: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$\cdot 10^{4}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2906: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2908: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{5}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2910: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2924: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{6}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2926: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{3}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2940: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{7}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2942: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cdot 10^{8}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2954: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-\lg\vert 1-s_\ell/s\vert$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2976: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-\lg\vert 1-s'_\ell/s\vert$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2978: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2980: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\htmlimage{thumbnail=0.5,scale=4}
    \includegraphics{rewr2s.eps}    
\label{fig:sn}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\htmlimage{thumbnail=0.5,scale=4}
    \includegraphics{rewr2sd.eps}    
\label{fig:snd}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\htmlimage{thumbnail=0.5,scale=4}
    \includegraphics{rewr2.eps}    
\label{fig:gain}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\mathcal{Y}}_{\ell_1}^{m_1} (\nabla) F_{\ell_2}^{m_2} (r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3744: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^{2 n} {\mathcal{Y}}_{\ell_1}^{m_1} (\nabla) F_{\ell_2}^{m_2} (r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3746: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${H}_{2}^{+}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3756: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3760: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3768: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}


\end{document}