%%%%		-*- Mode: LaTeX -*-
%%%% Author:    J. Robert Buchanan
%%%% Address:   Department of Mathematics
%%%%		Millersville University
%%%%            P.O. Box 1002
%%%%            Millersville, PA 17551-0302
%%%% Phone:     717-871-7305
%%%% FAX:	717-871-7948
%%%%
%%%% Email:     Robert.Buchanan@millersville.edu
%%%%
%%%% Date:      03/04/2025
%%%%
%%%% Purpose:   Laguerre Equation
%%%%

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\title{Laguerre Equation}
\subtitle{\textit{Partial Differential Equations}} % (optional)


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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\institute[Millersville University of Pennsylvania]
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{
  Department of Mathematics
}
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% - Keep it simple, no one is interested in your street address.

\date{ % (optional)
Fall
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\the\year}

\subject{Talks}
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\begin{document}

\begin{frame}
  \titlepage
\end{frame}

%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
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% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will learn:
\begin{itemize}
\item to solve the Laguerre ordinary differential equation,
\item to define the Laguerre polynomials,
\item the properties of the Laguerre polynomials,
\item to solve the associated Laguerre ordinary differential equation, and
\item the associated Laguerre polynomials.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Laguerre Ordinary Differential Equation}

\[
%\label{eq:spc:laguerre}
x y'' + (1-x) y' + \lambda y
	= 0
\mbox{ for $x > 0$.}
\]
The solutions to the Laguerre equation find applications in quantum
mechanics, numerical integration, and applied mathematics.

\vspace*{0.10in}

Using the method of Frobenius one solution to Laguerre's equation is
\[
%\label{eq:spc:laguerresoln1}
y_{1}(x)
	= 1 + \sum_{n=1}^{\infty} \frac{\prod_{k=0}^{n-1}
            (k-\lambda)}{(n!)^{2}} x^{n}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Laguerre Polynomials}

When $\lambda = m \in \{ 0, 1, 2, \ldots \}$ then $y_{1}(x)$ contains
powers of $x$ less than or equal to $m$.
The solution $y_{1}(x)$ is traditionally denoted as $L_{m}(x)$ and
\begin{align*}
L_{m}(x)
	&= 1 + \sum_{n=1}^{m} \frac{\prod_{k=0}^{n-1} (k-m)}{(n!)^{2}} x^{n}	\\
	&= 1 + \sum_{n=1}^{m} \frac{(-1)^{n} m(m-1)(m-2)\cdots(m-(n-1))}{(n!)^{2}}
x^{n}	\\
{}	&= m! \sum_{n=0}^{m} \frac{(-1)^{n}}{(n!)^{2}(m-n)!} x^{n}	.
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Examples}

\begin{center}
\begin{tabular*}{\columnwidth}{@{}rcl@{}}
$L_{0}(x)$	& = & $\displaystyle 1$	\\
$L_{1}(x)$	& = & $\displaystyle 1 - x$	\\
$L_{2}(x)$	& = & $\displaystyle 1 - 2x + \frac{1}{2} x^{2}$	\\
$L_{3}(x)$	& = & $\displaystyle 1 - 3x + \frac{3}{2} x^{2} -
                      \frac{1}{6} x^{3}$	\\
$L_{4}(x)$	& = & $\displaystyle 1 - 4x + 3 x^{2} - \frac{2}{3}
                      x^{3} + \frac{1}{24} x^{4}$	\\
\end{tabular*}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Illustrations}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{laguerres}}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Properties}

\begin{lemma}
%\label{lma:spc:laguerrerodrigues}
For $n = 0, 1, 2, \ldots$ the Rodrigues' form of the $n$th Laguerre
polynomial is
\[
%\label{eq:spc:laguerrerodrigues}
L_{n}(x)
	= \frac{e^{x}}{n!} \frac{d^{n}}{d x^{n}} [e^{-x} x^{n}]	.
\]
\end{lemma}

\begin{theorem}
%\label{thm:spc:laguerreorthog}
The Laguerre polynomials are orthogonal on $[0, \infty)$ with respect
to the weighting function $e^{-x}$.
For all nonnegative integers $m$ and $n$, then
\[
\int_{0}^{\infty} L_{m}(x) L_{n}(x) e^{-x} \, dx
	= \delta_{mn}	,
\]
where $\delta_{mn}$ is the Kronecker delta function.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Generalized Fourier Series}

If $f(x)$ is sufficiently smooth on the interval $[0, \infty)$ and if
\[
\int_{0}^{\infty} \left( f(x) \right)^{2} e^{-x} \, dx
	< \infty	,
\]
then $f(x)$ may be represented as a generalized Fourier series (a
Laguerre series) in terms of the Laguerre polynomials:
\[
f(x)
	= \sum_{n=0}^{\infty} a_{n} L_{n}(x)	.
\]
The coefficients $a_{n}$ of the series may be found from the integral
formula
\[
%\label{eq:spc:laguerrecoeff}
a_{n}
	= \int_{0}^{\infty} f(x) L_{n}(x) e^{-x} \, dx
\]
for $n = 0, 1, \ldots$.

\end{frame}
%%

\begin{frame}\frametitle{Associated Laguerre Differential Equation}

The \textbf{associated Laguerre differential
  equation} is
\[
%\label{eq:spc:genlaguerre}
x y'' + (\alpha + 1 - x) y' + \lambda y = 0
\mbox{ for $x \ge 0$,}
\]
where $\alpha > -1$ is a real number and $\lambda$ is also a real number.
The associated Laguerre equation often appears when solving the
Schr\"{o}dinger equation of quantum mechanics.

\vspace*{0.10in}

A general infinite series solution to the associated Laguerre
equation is
\[
y_{1}(x)
	= \frac{\Gamma(1+\lambda+\alpha)}{\Gamma(1+\lambda)
          \Gamma(1+\alpha)} \sum_{k=1}^{\infty} \frac{(-1)^{k}
          (\lambda-k+1)_{k}}{k! \, (1+\alpha)_{k}} x^{k}	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Associated Laguerre Polynomials}

When $\lambda = n \in \mathbb{N} \cup \{ 0 \}$, the series coefficients
simplify to
\[
a_{k}
	= \frac{(-1)^{k} \Gamma(1+n+\alpha)}{k! (n-k)!
          \Gamma(1+k+\alpha)}
\mbox{ for $k = 0, 1, \ldots, n$,}
\]
and $a_{k} = 0$ for $k \ge n+1$.
Thus one solution to the
associated Laguerre equation is the polynomial
\[
%\label{eq:spc:genlaguerrepoly}
L_{n}^{\alpha}(x)
	= \sum_{k=0}^{n} \frac{(-1)^{k} \Gamma(1+n+\alpha) x^{k}}{k!
          (n-k)! \Gamma(1+k+\alpha)}	.
\]
It should be noted that when $\alpha = 0$ the associated Laguerre
equation simplifies to the original Laguerre equation and
\[
L_{n}^{0}(x)
	= L_{n}(x)	.
\]

\end{frame}
%%

\begin{frame}\frametitle{Examples}

\begin{center}
\resizebox{1.00\textwidth}{!}{%
\begin{tabular}{@{}rcl@{}}
$L_{0}^{\alpha}(x)$	& = & $1$	\\
$L_{1}^{\alpha}(x)$	& = & $1+\alpha-x$	\\
$L_{2}^{\alpha}(x)$	& = & $1 + \frac{3}{2} \alpha + \frac{1}{2}
                              \alpha^{2} - (2+\alpha) x + \frac{1}{2} x^{2}$	\\
$L_{3}^{\alpha}(x)$	& = & $1 + \frac{11}{6} \alpha + \alpha^{2} +
                              \frac{1}{6} \alpha^{3} - \left( 3 +
                              \frac{5}{2} \alpha \right) x + \left(
                              \frac{3}{2} + \frac{1}{2} \alpha \right)
                              x^{2} - \frac{1}{6} x^{3}$
\end{tabular}
}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Properties}

\begin{lemma}
%\label{lma:spc:genlaguerrerodrigues}
The associated Laguerre
polynomial can be expressed as
\[
L_{n}^{\alpha}(x)
	= \frac{x^{-\alpha} e^{x}}{n!} \frac{d^{n}}{d x^{n}} [e^{-x} x^{n+\alpha}]	.
\]
\end{lemma}

\begin{theorem}
%\label{thm:spc:genlaguerreorthog}
The associated Laguerre polynomials
$L_{n}^{\alpha}(x)$, are orthogonal on $[0, \infty)$
with respect to the weighting function $e^{-x} x^{\alpha}$.
For all nonnegative integers $m$ and $n$,
\[
%\label{eq:spc:genlaguerreorthog}
\int_{0}^{\infty} L_{m}^{\alpha}(x) L_{n}^{\alpha}(x) e^{-x}
x^{\alpha} \, dx
	= \frac{\Gamma(n+\alpha+1)}{n!} \delta_{mn}	,
\]
where $\delta_{mn}$ is the Kronecker delta function.
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Generalized Fourier Series}

Suitably smooth functions can be represented as
\textbf{generalized Laguerre series},
\[
f(x)
	= \sum_{n=0}^{\infty} a_{n} L_{n}^{\alpha}(x)	,
\]
where
\[
%\label{eq:spc:genlaguerrecoeff}
a_{n}
	= \frac{n!}{\Gamma(n+\alpha+1)} \int_{0}^{\infty} f(x)
        L_{n}^{\alpha}(x) e^{-x} x^{\alpha} \, dx
\]
is the
\textbf{$n$th generalized Laguerre coefficient}.

\end{frame}
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%\begin{frame}\frametitle{Homework}

%\begin{itemize}
%\item Read Section $x.y$
%\item Exercises: 
%\end{itemize}

%\end{frame}
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