%% This LaTeX-file was created by <jl> Wed Feb 19 13:11:45 1997
%% LyX 0.10 (C) 1995 1996 by Matthias Ettrich and the LyX Team

%% Don't edit this file unless you are sure what you are doing.
\documentclass[12pt,letterpaper,oneside,onecolumn]{article}
\usepackage[]{fontenc}
\usepackage[dvips]{epsfig}

%%
%% BEGIN The lyx specific LaTeX commands.
%%

\makeatletter
\def\LyX{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\spacefactor1000}%
\newcommand{\lyxtitle}[1] {\thispagestyle{empty}
\global\@topnum\z@
\section*{\LARGE \centering \sffamily \bfseries \protect#1 }
}
\newcommand{\lyxline}[1]{
{#1 \vspace{1ex} \hrule width \columnwidth \vspace{1ex}}
}
\newenvironment{lyxbibliography}
{
\begin{thebibliography}{99}}
{\end{thebibliography}}
\def\lxq{"}
\newenvironment{lyxcode}
{\list{}{
\rightmargin\leftmargin
\raggedright
\itemsep 0pt
\parsep 0pt
\ttfamily
}%
\item[]
}
{\endlist}
\newcommand{\lyxlabel}[1]{#1 \hfill}
\newenvironment{lyxlist}[1]
{\begin{list}{}
{\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{\labelwidth}
\addtolength{\leftmargin}{\labelsep}
\renewcommand{\makelabel}{\lyxlabel}}}
{\end{list}}
\newcommand{\lyxletterstyle}{
\setlength\parskip{0.7em}
\setlength\parindent{0pt}
}
\newcommand{\lyxaddress}[1]{
\par {\raggedright #1 
\vspace{1.4em}
\noindent\par}
}
\newcommand{\lyxrightaddress}[1]{
\par {\raggedleft \begin{tabular}{l}\ignorespaces
#1
\end{tabular}
\vspace{1.4em}
\par}
}
\newcommand{\lyxformula}[1]{
\begin{eqnarray*}
#1
\end{eqnarray*}
}
\newcommand{\lyxnumberedformula}[1]{
\begin{eqnarray}
#1
\end{eqnarray}
}
\makeatother

%%
%% END The lyx specific LaTeX commands.
%%

\pagestyle{plain}
\setcounter{secnumdepth}{3}
\setcounter{tocdepth}{3}
\begin{document}

In spherical coordinates, you get \( \nabla ^{i}\nabla ^{i}\Phi =0 \) as \( \frac{1}{r^{2}}\frac{\partial }{\partial r}(r^{2}\frac{\partial \Phi }{\partial r})+\frac{1}{r^{2}sin(\theta )}\frac{\partial }{\partial \theta }sin(\theta )\frac{\partial \Phi }{\partial \theta }+\frac{1}{r^{2}sin^{2}(\theta )}\frac{\partial ^{2}\Phi }{\partial \phi ^{2}}=0 \). This seperates via \( \phi =\frac{U(r)}{r}V(\theta )W(\phi ) \) with
\( W(\phi )=e^{\pm im\phi } \) and associated legendre polynomials in \( \theta  \). The radial equation is
\( U(r)=Cr^{l}+Br^{-(l+1)} \). The \( \theta  \)equation transforms via x=cos(\( \theta  \)) to \( \frac{d}{dx}(1-x^{2})\frac{dV}{dx}+[l(l+1)-\frac{m^{2}}{1-x^{2}}]V=0 \). l has to be integer
to get a convergent power series. This solution is legendre polynomials
for m=0 and associated legendre polynomials for other m. The solution
is typically written in terms of the spherical harmonics as: \( \Phi (r,\theta ,\phi )=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}\frac{1}{r}(A_{lm}r^{l}+B_{lm}r^{-(l+1)})Y_{lm}(\theta ,\phi ) \)



\end{document}
