%%%%		-*- 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:   Spherical Harmonics
%%%%

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\title{Spherical Harmonics}
\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.

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

\subject{Talks}
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\begin{document}

\begin{frame}
  \titlepage
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%\begin{frame}
%  \frametitle{Outline}
%  \tableofcontents
  % You might wish to add the option [pausesections]
%\end{frame}


% 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 the eigenfunctions and eigenvalues of Laplace's equation in
  spherical coordinates,
\item the properties of the spherical harmonic functions.
\end{itemize}

\end{frame}
%%

\begin{frame}\frametitle{Laplace's Equation in Spherical Coordinates}

Consider Laplace's equation in spherical coordinates $(\rho, \varphi,
\theta)$, (note that coordinate $\varphi$ is
the zenith angle between the positive $z$-axis and the line formed by
connecting a point in three-dimensional space to the origin.
As such, $0 \le \varphi \le \pi$.
The coordinate $\theta$ is the azimuthal coordinate as in the polar
coordinate system.
This is conventional notation for mathematicians though physicists and
engineers often reverse the interpretations of $\varphi$ and
$\theta$).
\[
%\label{eq:spc:laplacianspherical}
\frac{\partial^{2} u}{\partial \rho^{2}} +
\frac{2}{\rho} \frac{\partial u}{\partial \rho} +
\frac{1}{\rho^{2}} \left(
\frac{\partial^{2} u}{\partial \varphi^{2}} +
\cot \varphi \frac{\partial u}{\partial \varphi} +
\csc^{2} \varphi \frac{\partial^{2} u}{\partial \theta^{2}}
\right)
	= 0	.
\]
Assume a product solution of the form $u(\rho,\varphi,\theta) =
R(\rho) \Phi(\varphi) \Theta(\theta)$.
\begin{align*}
0
	= {} & R''(\rho) \Phi(\varphi) \Theta(\theta) +
\frac{2}{\rho} R'(\rho) \Phi(\varphi) \Theta(\theta) +
\frac{1}{\rho^{2}} R(\rho) \Phi''(\varphi) \Theta(\theta)	\\
	&+ \frac{1}{\rho^{2}} \left(
\cot \varphi R(\rho) \Phi'(\varphi) \Theta(\theta) +
\csc^{2} \varphi R(\rho) \Phi(\varphi) \Theta''(\theta)
\right)	\\
-\frac{\Theta''(\theta)}{\Theta(\theta)}
	= {} & \sin^{2} \varphi \left( \rho^{2} \frac{R''(\rho)}{R(\rho)} +
2 \rho \frac{R'(\rho)}{R(\rho)} +
\frac{\Phi''(\varphi)}{\Phi(\varphi)} +
\cot \varphi \frac{\Phi'(\varphi)}{\Phi(\varphi)}
\right)
\end{align*}

\end{frame}
%%

\begin{frame}\frametitle{Solutions (1 of 3)}

\[
-\frac{\Theta''(\theta)}{\Theta(\theta)}
	= \sin^{2} \varphi \left( \rho^{2} \frac{R''(\rho)}{R(\rho)} +
2 \rho \frac{R'(\rho)}{R(\rho)} +
\frac{\Phi''(\varphi)}{\Phi(\varphi)} +
\cot \varphi \frac{\Phi'(\varphi)}{\Phi(\varphi)}
\right)
\]
The right-hand side of the equation depends on $\rho$ and $\varphi$ and
the left-hand side depends on $\theta$, thus both sides equal a
constant, denoted as $\sigma$.
Consequently one of the ordinary differential equations implied by the
separation of variables procedure is
\[
\Theta''(\theta) + \sigma \Theta(\theta) = 0	.
\]
The solution to Laplace's equation should be $2
\pi$-periodic in variable $\theta$, thus the needed constant is
$\sigma = m^{2}$ where $m = 0, 1, 2, \ldots$.
For convenience express the general solution as a linear combination of
the complex exponentials $\Theta(\theta) = c_{1} e^{i \, m \, \theta} +
c_{2} e^{-i \, m \, \theta}$ where $i = \sqrt{-1}$.

\end{frame}
%%

\begin{frame}\frametitle{Solutions (2 of 3)}

\[
\sin^{2} \varphi \left( \rho^{2} \frac{R''(\rho)}{R(\rho)} +
2 \rho \frac{R'(\rho)}{R(\rho)} +
\frac{\Phi''(\varphi)}{\Phi(\varphi)} +
\cot \varphi \frac{\Phi'(\varphi)}{\Phi(\varphi)}
\right)
	= m^{2}	,
\]
the variables $\rho$ and $\varphi$ can be separated as follows
\[
\rho^{2} \frac{R''(\rho)}{R(\rho)} + 2 \rho \frac{R'(\rho)}{R(\rho)}
	= -\frac{\Phi''(\varphi)}{\Phi(\varphi)} - \cot \varphi
\frac{\Phi'(\varphi)}{\Phi(\varphi)} + m^{2} \csc^{2} \varphi	.
\]
The right-hand side of the equation is a
function of $\varphi$ only and the left-hand -side is a function of
$\rho$ only, thus both sides are constant, denoted as $\lambda$.

\end{frame}
%%

\begin{frame}\frametitle{Solutions (3 of 3)}

Consider the equation:
\begin{align*}
-\frac{\Phi''(\varphi)}{\Phi(\varphi)} - \cot \varphi
\frac{\Phi'(\varphi)}{\Phi(\varphi)} + m^{2} \csc^{2} \varphi
	&= \lambda	\\
\Phi''(\varphi) + \cot \varphi \Phi'(\varphi) + ( \lambda
  - m^{2} \csc^{2} \varphi ) \Phi(\varphi)
	&= 0	,
\end{align*}
where $\lambda$ is a constant and make the change of variable $x =
\cos \varphi$.
The equation above can
be rewritten as
\[
(1 - x^{2}) \frac{d^{2} \Phi}{d x^{2}} - 2 x \frac{d \Phi}{d x} +
\left( \lambda - \frac{m^{2}}{1-x^{2}} \right) \Phi
	= 0	.
\]
If $\lambda = n (n+1)$ where $n = 0, 1, 2, \ldots$, this becomes the
associated Legendre differential equation.

Define
\[
f(\varphi, \theta)
	= \Phi(\varphi) \Theta(\theta)
	= e^{i \, m \, \theta} P_{n}^{m}(\cos \varphi)
\]
with $n = 0, 1, 2, \ldots$ and $-n \le m \le n$.

\end{frame}
%%

\begin{frame}\frametitle{Spherical Harmonic Functions}

\begin{center}
\resizebox{1.00\textwidth}{!}{%
\begin{tabular}{@{}ccccc@{}}
{}	& $m=0$ & $m=1$ & $m=2$ & $m=3$	\\
$n=0$	& $\frac{1}{2 \sqrt{\pi}}$ & {} & {} & {}	\\
$n=1$	& $\frac{1}{2} \sqrt{\frac{3}{\pi}} \cos \varphi$
                &  $-\frac{1}{2} \sqrt{\frac{3}{2 \pi}} e^{i \,
                  \theta} \sin \varphi$& {} & {}	\\
$n=2$	& $\frac{1}{4} \sqrt{\frac{5}{\pi}} (3 \cos^{2} \varphi - 1)$
		& $-\frac{1}{2} \sqrt{\frac{15}{2 \pi}} e^{i \, \theta}
                  \cos \varphi \sin \varphi$
		& $\frac{1}{4} \sqrt{\frac{15}{2 \pi}} e^{2 i \,
                  \theta} \sin^{2} \varphi$ & {}	\\
$n=3$	& $\frac{1}{4} \sqrt{\frac{7}{\pi}} (5 \cos^{3} \varphi - 3
          \cos \varphi)$
		& $-\frac{1}{8} \sqrt{\frac{21}{\pi}} e^{i \, \theta}
                  (5 \cos^{2} \varphi - 1) \sin \varphi$
		& $\frac{1}{4} \sqrt{\frac{105}{2 \pi}} e^{2 i \,
                  \theta} \cos \varphi \sin^{2} \varphi$
		& $-\frac{1}{8} \sqrt{\frac{35}{\pi}} e^{3 i \,
                  \theta} \sin^{3} \varphi$	\\
\end{tabular}
}
\end{center}

\end{frame}
%%

\begin{frame}\frametitle{Properties}

\begin{theorem}
%\label{thm:spc:orthogsphericalhamonic}
Let $n = 0, 1, 2, \ldots$ and let $m$ be an integer with $-n \le m \le
n$.
Likewise let $\hat{n} = 0, 1, 2, \ldots$ and let $\hat{m}$ be an
integer with $-\hat{n} \le \hat{m} \le \hat{n}$ then
\[
%\label{eq:spc:sphericalharmonicorthog}
\int_{-\pi}^{\pi} \int_{0}^{\pi}
e^{i \, m \, \theta} P_{n}^{m}(\cos \varphi)
e^{-i \, \hat{m} \, \theta} P_{\hat{n}}^{\hat{m}}(\cos \varphi)
\sin \varphi \, d\varphi \, d\theta
	= 0
\]
if $n \ne \hat{n}$ or $m \ne \hat{m}$.
On the other hand if $n = \hat{n}$ and $m = \hat{m}$, then
\[
%\label{eq:spc:sphericalharmonicorthonorm}
\int_{-\pi}^{\pi} \int_{0}^{\pi}
\left( P_{n}^{m}(\cos \varphi) \right)^{2}
\sin \varphi \, d\varphi \, d\theta
	= \frac{4 \pi}{2n+1} \, \frac{(n+m)!}{(n-m)!}	.
\]
\end{theorem}

\end{frame}
%%

\begin{frame}\frametitle{Spherical Harmonic Functions}

For integers $n = 0, 1, 2, \ldots$ and $-n \le m \le n$ the
normalized function $f(\varphi, \theta)$ is denoted as
$Y_{n}^{m}(\varphi, \theta)$ where
\[
%\label{eq:spc:shdefn}
Y_{n}^{m}(\varphi, \theta)
	= \sqrt{\frac{2n+1}{4 \pi} \, \frac{(n-m)!}{(n+m)!}} e^{i \, m \,
  \theta} P_{n}^{m}(\cos \varphi)
\]
These functions are called the
\textbf{spherical harmonic functions}.
There are $2n+1$ orthonormal functions corresponding to the eigenvalue $n
(n+1)$.

\end{frame}
%%

%\begin{frame}\frametitle{Homework}

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

%\end{frame}
%%

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