\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{Legendre polynomial}

The \textbf{Legendre polynomial} $P_l$ (for $l =0,1,2,\ldots$) is the [[polynomial]] in one variable given by the formula

\begin{displaymath}
P_l(x) = \frac{1}{2^l l!} \frac{d^l}{d x^l}(x^2-1)^l
\end{displaymath}
Alternatively they can be defined via a generating function:

\begin{displaymath}
\frac{1}{\sqrt{1-2tx+t^2}} = \sum_{n\geq 0} P_n(x) t^n
\end{displaymath}
The Legendre polynomials satisfy:

\begin{itemize}%
\item the following [[differential equation]] of the second order\begin{displaymath}
\frac{d}{d x}[(1-x^2)\frac{d P_l}{d x}] + l(l+1) P_l = 0
\end{displaymath}

\item the recursion relations\begin{displaymath}
(l+1)P_{l+1}-(2l+1)x P_l+l P_{l-1}=0
\end{displaymath}

\item the mixed differential recursion relations\begin{displaymath}
\array {
P'_{l+1}-P'_{l-1} = (2l+1)P_l
\\
P'_{l+1}-x P'_l = (l+1)P_l
\\
(x^2-1)P_l'-l x P_l+l P_{l-1} = 0
}
\end{displaymath}


\end{itemize}
The Legendre polynomials form a complete system of orthogonal polynomials on the interval $[-1,1]$, that is an unnormalised basis of the [[Hilbert space]] $L^2[-1,1]$.

The first few Legendre polynomials are $P_0(x) = 1$, $P_1(x) = x$, $P_2(x)=\frac{1}{2}(3x^2-1)$, $P_3(x)=\frac{1}{2}(5x^2-3)$, $P_4(x)=\frac{1}{8}(35x^4-30x^2+3)$. Their values at $0$ are

\begin{displaymath}
P_{2n+1}(0)=0,\,\,\,\,\,P_{2n}(0)=(-1)^n\frac{(2n-1)!!}{(2n)!} = \frac{(-1)^n (2n)!}{2^{2n}(n!)^2}
\end{displaymath}
and $P_l(\pm 1)= (\pm 1)^l$.

One also has the integral formulas

\begin{displaymath}
\array {
\int^1_0 P_{2k+1}(x) d x = \frac{(-1)^k (2k)!}{2^{2k+1} k! (k+1)!}
\\
\int^1_0 P_{2k}(x) d x = \delta_{k0}
\\
\int^1_{-1} x P_l P_k = \left\lbrace
\itexarray{  \frac{2(l+1)}{(2l+1)(2l+3)},&k=l+1 \\
\frac{(2l)}{(2l-1)(2l+1)},&k=l-1\\
0,& otherwise}
\right.
}
\end{displaymath}
A generalization of Legendre polynomials are the \textbf{Legendre functions} $P_\nu$ where $\nu$ is not necessarily an integer and $P^m_l$ which are given by

\begin{displaymath}
P^m_l(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac{d^{l+m}}{d x^{l+m}} (x^2-1)^l = (-1)^m (1-x^2)^{m/2} \frac{d^{l}}{d x^{l}} P_l(x)
\end{displaymath}
for $m\geq 0$ and also

\begin{displaymath}
P^{-m}_l(x)= (-1)^m\frac{(l-m)!}{(l+m)!} P^m_l(x)
\end{displaymath}
These $P^m_l$ are satisfying the orthogonality relations

\begin{displaymath}
\int_{-1}^1 P^m_l(x) P^m_k(x) d x = \frac{2}{2l+1}\frac{(l+m)!}{(l-m)!}\delta_{lk}
\end{displaymath}
recursion relation

\begin{displaymath}
x P^m_l(x) = \frac{l+|m|}{2l+1}P^m_{l-1}(x)+\frac{l-|m|+1}{2l+1} P^m_{l+1}(x)
\end{displaymath}
and the differential equation

\begin{displaymath}
\frac{d}{d x}[(1-x^2)\frac{d P^m_l}{d x}]+[(l(l+1)-\frac{m^2}{1-x^2}]P^m_l = 0
\end{displaymath}
$P_\nu(x)$ is a special case of a [[hypergeometric function]], namely

\begin{displaymath}
P_\nu(x)={}_2 F_1(-\nu,\nu+1;1;\frac{1-x}{2})
\end{displaymath}
Legendre polynomials enter the expressions for the [[spherical function]]s for sphere $S^2$ in 3d:

\begin{displaymath}
Y_{l m}(\theta,\phi) = \sqrt{ \frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!} } P^m_l(cos\theta) e^{im\phi}
\end{displaymath}
(If it is not clear from mathML rendering -- both fractions are under square root -- including both the numerators and denominators).

If $(\theta,\phi)$ and $(\theta',\phi')$ are two points of the [[unit sphere]] in spherical coordinates (polar angle, azimuth), and $\gamma$ is the angle between the two corresponding rays from the origin then

\begin{displaymath}
P_l(cos \gamma) = \sum_{m=-l}^l (-1)^m P_l^m(cos \theta)P_l^{-m}(cos \theta') cos(m(\phi-\phi')) = \frac{4\pi}{2l+1}\sum_{m=-l}^l Y_{lm}(\theta,\phi) Y_{lm}^*(\theta',\phi')
\end{displaymath}
what for $l=1$ reduces to the \textbf{spherical law of cosine} from \href{http://en.wikipedia.org/wiki/Spherical_trigonometry}{spherical trigonometry}:

\begin{displaymath}
cos \gamma = P_1(cos \gamma) = cos\theta cos\theta' + sin\theta sin\theta' cos(\phi-\phi')
\end{displaymath}
The orthogonality relation for Legendre polynomials gives Laplace's formula

\begin{displaymath}
\int_{S^2} d\Omega_{\hat{k}} Y_l(\hat{k}) P_l(\hat{k}\hat{p}) = \frac{4\pi}{2l+1} \delta_{ll'} Y_l(\hat{p})
\end{displaymath}
where $\hat{k},\hat{p}$ are unit vectors and $Y_l = \sum a_m Y_{lm}$ is some spherical function.

The following orthogonality integral relation is over product of unit spheres in $\mathbb{R}^3$:

\begin{displaymath}
\int_{\hat{k}}\int_{\hat{p}} d\Omega_{\hat{k}}d\Omega_{\hat{p}} P_l(\hat{k}\hat{p}) P_{l'}(\hat{k}\hat{q}) P_{l''}(\hat{p}\hat{q}) = \left(\frac{4\pi}{2l+1}\right)\delta_{ll'}\delta_{ll''}, \,\,\,\,\,(*)
\end{displaymath}
where the arguments of Legendre polynomials are the [[inner product]]s of the unit vectors.

\emph{Zoran}: the last formula $(*)$ is in my own formula notes which I have written as a student many years ago and used hundreds of times, but it now looks to me suspicious; I have no time to check it right now.

[[!redirects Legendre polynomials]]



\end{document}