From 1ff8774bb6669b11e8599641f0acd7066913da57 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 28 Apr 2022 15:48:30 +0200
Subject: Add draftwatermark and associated legendre polynomials

---
 notes/FourierOnS2.tex       |  26 ++++++++++++++++++++++++++
 notes/build/FourierOnS2.pdf | Bin 56389 -> 61780 bytes
 2 files changed, 26 insertions(+)

diff --git a/notes/FourierOnS2.tex b/notes/FourierOnS2.tex
index edd4813..ced6b8d 100644
--- a/notes/FourierOnS2.tex
+++ b/notes/FourierOnS2.tex
@@ -7,6 +7,7 @@
 
 %% styling for this document
 \usepackage{tex/docstyle}
+\usepackage[firstpageonly, color={[gray]{0.9}}]{draftwatermark}
 
 %% Theorems, TODO: move to docstyle
 \usepackage{amsthm}
@@ -101,6 +102,7 @@ The answer has to do with solving other problems. That is because originally Fou
                        + \frac{\partial^2 f}{\partial \nu^2} = 0.
 \end{equation}
 
+\texttt{[FIXME: Should be an eigenvalue problem \(\nabla_2^2 f = \lambda f\)]}
 This PDE is known as Laplace's equation, and can be solved by separation using the ansatz
 \[
   f(\mu, \nu) = M(\mu)N(\nu),
@@ -242,6 +244,7 @@ Thus we first need examine the solutions to this equation before constructing th
   \end{equation}
   are solutions to Legendre's equation \eqref{eqn:legendre} when \(n > 0\).
 \end{proposition}
+\begin{proof} See appendix. \end{proof}
 
 The proof for this proposition is quite algebraically involved and is thus left in the appendix. Since this is a power series \eqref{eqn:legendre-poly} can also be rewritten using Gauss' Hypergeometric function.
 
@@ -306,6 +309,28 @@ In some applications, such as in quantum mechanics, it is more common to see it
   \end{align*}
 \end{proof}
 
+Now, using the solutions to the Legendre equation we can construct the solution to the more general problem:
+\begin{align}
+  \left( 1 - x^2 \right) \frac{d^2 y}{dx^2} &- 2x \frac{dy}{dx} \nonumber \\
+    & + \left[ n(n+1) - \frac{m^2}{1 - x^2} \right] y(x) = 0.
+    \label{eqn:legendre-as}
+\end{align}
+
+This equation is considerably more difficult, and again, we will just analyze the solution.
+
+\begin{definition}[Associated Legendre Polynomials]
+  Let \(m \in \mathbb{N}_0\). The polynomials
+  \begin{equation} \label{eqn:legendre-poly-as}
+    P_{m, n} (x) = \left( 1 - x^2 \right)^{m/2} \frac{d^m}{dx^m} P_n (x),
+  \end{equation}
+  are called the associated Legendre polynomials.
+\end{definition}
+
+\begin{lemma}
+  The associated Legendre polynomials \eqref{eqn:legendre-poly-as} are solutions to the associated Legendre differential equation \eqref{eqn:legendre-as}.
+\end{lemma}
+\begin{proof} See appendix. \end{proof}
+
 \subsection{Spherical harmonics}
 
 \clearpage
@@ -374,6 +399,7 @@ from which we can extract the recurrence relation
 \end{gather*}
 
 % TODO: finish reviewing proof
+  \texttt{[TODO: finish copying proof]}
 \if 0
 
 We can derive a recursion formula for $a_{k+2}$ from Eq.\eqref{eq:condition_2}, which can be expressed as
diff --git a/notes/build/FourierOnS2.pdf b/notes/build/FourierOnS2.pdf
index d40262c..d60d9bf 100644
Binary files a/notes/build/FourierOnS2.pdf and b/notes/build/FourierOnS2.pdf differ
-- 
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