%%%%		-*- 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:   Hermite Equation
%%%%

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


\author % (optional, use only with lots of authors)
{J~Robert~Buchanan}
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%   affiliation.

\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.

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Fall
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\the\year}

\subject{Talks}
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% Since this a solution template for a generic talk, very little can
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%\section{Introduction}

\begin{frame}\frametitle{Objectives}

In this lesson we will:
\begin{itemize}
\item explore the Hermite ordinary differential equation,
\item find two linearly independent solutions to Hermite's equation,
\item define the Hermite polynomials, and
\item determine some properties of the Hermite polynomials.
\end{itemize}

\end{frame}
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\begin{frame}\frametitle{Hermite's Ordinary Differential Equation}

\[
%\label{eq:spc:hermite}
y'' - 2 x y' + \lambda y
	= 0
\mbox{ for $-\infty < x < \infty$,}
\]
where $\lambda$ is a constant, is known as the
\textbf{Hermite equation}.
This is an important equation in mathematical physics and is often
encountered in applications involving the quantum harmonic oscillator.

Two linearly independent solutions to Hermite's differential equation are
\begin{align*}
y_{1}(x)
	&= 1 + \sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n} (4k - 4 -
          \lambda)}{(2n)!} x^{2n}	\\
y_{2}(x)
	&= x + \sum_{n=1}^{\infty} \frac{\prod_{k=1}^{n} (4k - 2 -
          \lambda)}{(2n+1)!} x^{2n+1}	.
\end{align*}

\end{frame}
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\begin{frame}\frametitle{Hermite Polynomials}

If $\lambda = 2 m$ for some $m \in \mathbb{N}$ then one of the
infinite series solutions terminates after a finite number of terms
and can be written as a polynomial.
In this case the ordinary differential equation
is called the
\textbf{Hermite equation of order $m$}.

\vspace*{0.10in}

If $m$ is even (or equivalently if $\lambda = 4 N$ for some $N \in
\{ 0, 1, 2, \ldots \}$) then
\[
%\label{eq:spc:hermitepoly1}
y_{1}(x)
	= 1 + \sum_{n=1}^{N} \frac{\prod_{k=1}^{n} (4k - 4 - 4N)}{(2n)!} x^{2n}	.
\]
If $m$ is odd (or equivalently if $\lambda = 4N - 2$ for some $N \in
\mathbb{N}$) then
\[
%\label{eq:spc:hermitepoly2}
y_{2}(x)
	= x + \sum_{n=1}^{N} \frac{\prod_{k=1}^{n} (4k - 4N)}{(2n+1)!} x^{2n+1}	.
\]

\end{frame}
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\begin{frame}\frametitle{Examples}

The \textbf{Hermite polynomial}
denoted as $H_{n}(x)$ is defined to be
the polynomial solution to the Hermite equation with $\lambda = 2n$
for which the coefficient of $x^{n}$ is $2^{n}$.
The even- and odd-degree Hermite polynomials can be represented by the
single formula:
\[
%\label{eq:spc:hermiteformula}
H_{n}(x)
	= n! \sum_{k=0}^{\lfloor n/2\rfloor} \frac{(-1)^{k}}{k! (n-2k)!} (2x)^{n-2k}	 .
\]
\begin{center}
\begin{tabular*}{\columnwidth}{@{}rcl@{}}
$H_{0}(x)$	& = &
$\displaystyle 1$	\\
$H_{1}(x)$	& = &
$\displaystyle 2x$	\\
$H_{2}(x)$	& = &
$\displaystyle 4x^{2} - 2$	\\
$H_{3}(x)$	& = &
$\displaystyle 8 x^{3} - 12 x$	\\
$H_{4}(x)$	& = &
$\displaystyle 16 x^{4} - 48 x^{2} + 12$	\\
$H_{5}(x)$	& = &
$\displaystyle 32 x^{5} - 160 x^{3} + 120 x$	\\
\end{tabular*}
\end{center}

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\begin{frame}\frametitle{Illustrations}

\begin{center}
\resizebox{1.00\textwidth}{!}{\includegraphics{hermiteps}}
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\begin{frame}\frametitle{Properties}

\[
%\label{eq:spc:hermiterodrigues}
H_{n}(x)
	= (-1)^{n} e^{x^{2}} \frac{d^{n}}{dx^{n}} \left[ e^{-x^{2}} \right]
\]

\begin{theorem}
%\label{thm:spc:hermiteorthog}
The Hermite polynomials $H_{n}(x)$ are orthogonal on the interval
$(-\infty, \infty)$ with respect to the weighting function
$e^{-x^{2}}$.
For $m, \, n \in \{ 0, 1, 2, \ldots \}$,
\[
%\label{eq:spc:hermiteorthog}
\int_{-\infty}^{\infty} H_{m}(x) H_{n}(x) e^{-x^{2}} \, dx
	= 2^{n} n! \sqrt{\pi} \delta_{mn}	,
\]
where $\delta_{mn}$ is the Kronecker delta function.
\end{theorem}

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

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

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