From 95d6d5a46854e79d7b410a1fd4253ee4548e936e Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 18 Aug 2022 14:46:51 +0200
Subject: kugel: Orthogonality

---
 buch/papers/kugel/spherical-harmonics.tex | 203 ++++++++++++++++++++++++++++--
 1 file changed, 194 insertions(+), 9 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 2ded50b..2a00754 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -178,7 +178,7 @@ write the solutions
 The restriction that the separation constant $m$ needs to be an integer arises
 from the fact that we require a $2\pi$-periodicity in $\varphi$ since the
 coordinate systems requires that $\Phi(\varphi + 2\pi) = \Phi(\varphi)$.
-Unfortunately, solving \eqref{kugel:eqn:ode-theta} is as straightforward,
+Unfortunately, solving \eqref{kugel:eqn:ode-theta} is not as straightforward,
 actually, it is quite difficult, and the process is so involved that it will
 require a dedicated section of its own.
 
@@ -250,7 +250,7 @@ case of the former that is known known as the \emph{Legendre polynomials}, since
 we only need a solution between $-1$ and $1$.
 
 \begin{lemma}[Legendre polynomials]
-  \label{kugel:lem:legendre-poly}
+  \label{kugel:thm:legendre-poly}
   The polynomial function
   \[
     P_n(z) = \sum^{\lfloor n/2 \rfloor}_{k=0}
@@ -287,7 +287,7 @@ Legendre equation, we can make use of the following lemma patch the solutions
 such that they also become solutions of the associated Legendre equation
 \eqref{kugel:eqn:associated-legendre}.
 
-\begin{lemma} \label{kugel:lem:extend-legendre}
+\begin{lemma} \label{kugel:thm:extend-legendre}
   If $Z_n(z)$ is a solution of the Legendre equation \eqref{kugel:eqn:legendre},
   then
   \begin{equation*}
@@ -300,7 +300,7 @@ such that they also become solutions of the associated Legendre equation
   See section \ref{kugel:sec:proofs:legendre}.
 \end{proof}
 
-What is happening in lemma \ref{kugel:lem:extend-legendre}, is that we are
+What is happening in lemma \ref{kugel:thm:extend-legendre}, is that we are
 essentially inserting a square root function in the solution in order to be able
 to reach the parts of the domain near the poles at $\pm 1$ of the associated
 Legendre equation, which is not possible only using power series
@@ -356,9 +356,10 @@ $Y^m_n(\vartheta, \varphi)$.
   \label{kugel:def:spherical-harmonics}
   The functions
   \begin{equation*}
-    Y_{m,n}(\vartheta, \varphi) = P^m_n(\cos \vartheta) e^{im\varphi},
+    Y^m_n (\vartheta, \varphi) = P^m_n(\cos \vartheta) e^{im\varphi},
   \end{equation*}
-  where $m, n \in \mathbb{Z}$ and $|m| < n$ are called spherical harmonics.
+  where $m, n \in \mathbb{Z}$ and $|m| < n$ are called (unnormalized) spherical
+  harmonics.
 \end{definition}
 
 \begin{figure}
@@ -366,9 +367,195 @@ $Y^m_n(\vartheta, \varphi)$.
   \kugelplaceholderfig{\textwidth}{.8\paperheight}
   \caption{
     \kugeltodo{Big picture with the first few spherical harmonics.}
+    \label{kugel:fig:spherical-harmonics}
   }
 \end{figure}
 
+\kugeltodo{Describe how they look like with fig.
+\ref{kugel:fig:spherical-harmonics}}
+
+\subsection{Orthogonality of $P_n$, $P^m_n$ and $Y^m_n$}
+
+We shall now discuss an important property of the spherical harmonics: they form
+an orthogonal system. And since the spherical harmonics contain the Ferrers or
+associated Legendre functions, we need to discuss their orthogonality first.
+But the Ferrers functions themselves depend on the Legendre polynomials, so that
+will be our starting point.
+
+\begin{lemma} For the Legendre polynomials $P_n(z)$ and $P_k(z)$ it holds that
+  \label{kugel:thm:legendre-poly-ortho}
+  \begin{equation*}
+    \int_{-1}^1 P_n(z) P_k(z) \, dz
+    = \frac{2}{2n + 1} \delta_{nk}
+    = \begin{cases}
+      \frac{2}{2n + 1} & \text{if } n = k, \\
+      0 & \text{otherwise}.
+    \end{cases}
+  \end{equation*}
+\end{lemma}
+\begin{proof}
+  To start, consider the fact that that the Legendre equation
+  \eqref{kugel:eqn:legendre}, of which two distinct Legendre polynomials
+  $P_n(z)$ and $P_k(z)$ are a solution ($n \neq k$), can be rewritten in the
+  following form:
+  \begin{equation}
+    \frac{d}{dz} \left[ 
+      \left( 1 - z^2 \right) \frac{dZ}{dz}
+    \right] + n(n+1) Z(z) = 0.
+  \end{equation}
+  So we rewrite the Legendre equations for $P_n(z)$ and $P_k(z)$:
+  \begin{align*}
+    \frac{d}{dz} \left[ 
+      \left( 1 - z^2 \right) \frac{dP_n}{dz}
+    \right] + n(n+1) P_n(z) &= 0,
+    &
+    \frac{d}{dz} \left[ 
+      \left( 1 - z^2 \right) \frac{dP_k}{dz}
+    \right] + k(k+1) P_k(z) &= 0,
+  \end{align*}
+  then we multiply the former by $P_k(z)$ and the latter by $P_n(z)$ and
+  subtract the two to get
+  \begin{equation*}
+    \frac{d}{dz} \left[ 
+      \left( 1 - z^2 \right) \frac{dP_n}{dz}
+    \right] P_k(z) + n(n+1) P_n(z) P_k(z)
+    -
+    \frac{d}{dz} \left[ 
+      \left( 1 - z^2 \right) \frac{dP_k}{dz}
+    \right] P_n(z) - k(k+1) P_k(z) P_n(z) = 0.
+  \end{equation*}
+  By grouping terms, making order and integrating with respect to $z$ from $-1$
+  to 1 we obtain
+  \begin{gather}
+    \int_{-1}^1 \left\{
+      \frac{d}{dz} \left[ 
+        \left( 1 - z^2 \right) \frac{dP_n}{dz}
+      \right] P_k(z) 
+      -
+      \frac{d}{dz} \left[ 
+        \left( 1 - z^2 \right) \frac{dP_k}{dz}
+      \right] P_n(z) - k(k+1) P_k(z) P_n(z)
+    \right\} \,dz \nonumber \\
+    + \left[ n(n+1) - k(k+1) \right] \int_{-1}^1 P_k(z) P_n(z) \, dz = 0.
+    \label{kugel:thm:legendre-poly-ortho:proof:1}
+  \end{gather}
+  Since by the product rule
+  \begin{equation*}
+    \frac{d}{dz} \left[ (1 - z^2) \frac{dP_k}{dz} P_n(z) \right]
+    =
+    \frac{d}{dz} \left[ (1 - z^2) \frac{dP_n}{dz} \right] P_k(z)
+      + (1 - z^2) \frac{dP_n}{dz} \frac{dP_k}{dz},
+  \end{equation*}
+  we can simplify the first term in
+  \eqref{kugel:thm:legendre-poly-ortho:proof:1} to get
+  \begin{gather*}
+    \int_{-1}^1 \left\{
+      \frac{d}{dz} \left[ (1 - z^2) \frac{dP_k}{dz} P_n(z) \right]
+        - \cancel{(1 - z^2) \frac{dP_n}{dz} \frac{dP_k}{dz}}
+        - \frac{d}{dz} \left[ (1 - z^2) \frac{dP_n}{dz} P_k(z) \right]
+        + \cancel{(1 - z^2) \frac{dP_k}{dz} \frac{dP_n}{dz}}
+    \right\} \, dz \\
+    = \int_{-1}^1 \frac{d}{dz} \left\{ (1 - z^2) \left[
+      \frac{dP_k}{dz} P_n(z) - \frac{dP_n}{dz} P_k(z)
+    \right] \right\} \, dz
+    = (1 - z^2) \left[
+      \frac{dP_k}{dz} P_n(z) - \frac{dP_n}{dz} P_k(z)
+    \right] \Bigg|_{-1}^1,
+  \end{gather*}
+  which always equals 0 because the product contains $1 - z^2$ and the bounds
+  are at $\pm 1$. Thus, of \eqref{kugel:thm:legendre-poly-ortho:proof:1} only
+  the second term remains and the equation becomes
+  \begin{equation*}
+    \left[ n(n+1) - k(k+1) \right] \int_{-1}^1 P_k(z) P_n(z) \, dz = 0.
+  \end{equation*}
+  By dividing by the constant in front of the integral we have our first result.
+  Now we need to show that when $n = k$ the integral equals $2 / (2n + 1)$.
+  % \begin{equation*}
+  % \end{equation*}
+  \kugeltodo{Finish proof. Can we do it without the generating function of
+  $P_n$?}
+\end{proof}
+
+In a similarly algebraically tedious fashion, we can also continue to check for
+orthogonality for the Ferrers functions $P^m_n(z)$, since they are related to
+$P_n(z)$ by a $m$-th derivative, and obtain the following result.
+
+\begin{lemma} For the associated Legendre functions
+  \label{kugel:thm:associated-legendre-ortho}
+  \begin{equation*}
+    \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz
+    = \frac{2(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'}
+    = \begin{cases}
+      \frac{2(m + n)!}{(2n + 1)(n - m)!} & \text{if } n = n', \\
+      0 & \text{otherwise}.
+    \end{cases}
+  \end{equation*}
+\end{lemma}
+\begin{proof}
+  \kugeltodo{Is it worth showing? IMHO no, it is mostly the same as Lemma
+  \ref{kugel:thm:legendre-poly-ortho} with the difference that the $m$-th
+  derivative is a pain to deal with.}
+\end{proof}
+
+An interesting fact to observe in lemma
+\ref{kugel:thm:associated-legendre-ortho} is that the orthogonality is only
+affected in the lower index, while varying $m$ only changes the constant in
+front of the Kronecker delta. By having the orthogonality relations of the
+Legendre functions we can finally show that spherical harmonics are also
+orthogonal.
+
+\begin{lemma} For the spherical harmonics
+  \kugeltodo{Fix horizontal spacing, inner product definition is missing.}
+  \label{kugel:thm:spherical-harmonics-ortho}
+  \begin{equation*}
+    \langle Y^m_n, Y^{m'}_{n'} \rangle
+    = \int_{-\pi}^\pi \int_0^{2\pi}
+      Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
+      \sin \vartheta \, d\varphi \, d\vartheta
+    = \frac{-4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'}
+    = \begin{cases}
+      \frac{-4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} & \text{if } n = n', \\
+      0 & \text{otherwise}.
+    \end{cases}
+  \end{equation*}
+\end{lemma}
+\begin{proof}
+  We will begin by doing a bit of algebraic maipulaiton:
+  \begin{align*}
+    \int_{-\pi}^\pi \int_0^{2\pi}
+      Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
+      \sin \vartheta \, d\varphi \, d\vartheta
+    &= \int_{-\pi}^\pi \int_0^{2\pi}
+      e^{im\varphi} P^m_n(\cos \vartheta)
+      e^{-im'\varphi} P^{m'}_{n'}(\cos \vartheta)
+      \, d\varphi \sin \vartheta \, d\vartheta
+    \\
+    &= \int_{-\pi}^\pi
+      P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
+      \int_0^{2\pi} e^{i(m - m')\varphi}
+      \, d\varphi \sin \vartheta \, d\vartheta
+      .
+  \end{align*}
+  First, notice that the associated Legendre polynomials are assumed to be real,
+  and are thus unaffected by the complex conjugation. Then, we can see that when
+  $m = m'$ the inner integral simplifies to $\int_0^{2\pi} 1 \, d\varphi$ which
+  equals $2\pi$, so in this case the expression becomes
+  \begin{equation*}
+    2\pi \int_{-\pi}^\pi
+      P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
+    \sin \vartheta \, d\vartheta
+    = -2\pi \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz
+    = \frac{-4\pi(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'},
+  \end{equation*}
+  where in the second step we performed the substitution $z = \cos\vartheta$;
+  $d\vartheta = \frac{d\vartheta}{dz} dz= - dz / \sin \vartheta$, and then we
+  used lemma \ref{kugel:thm:associated-legendre-ortho}. Now we just need look at
+  the case when $m \neq m'$. Fortunately this is easy: the inner integral is
+  $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, or in other words we are
+  integrating a complex exponetial over the entire period, which always results
+  in zero. Thus, we do not need to do anything and the proof is complete.
+\end{proof}
+
 \subsection{Normalization}
 
 \kugeltodo{Discuss various normalizations.}
@@ -403,8 +590,6 @@ Ora, visto che la soluzione dell'eigenfunction problem è formata dalla moltipli
 
 \section{Series Expansions in $C(S^2)$}
 
-\subsection{Orthogonality of $P_n$, $P^m_n$ and $Y^m_n$}
-
-\subsection{Series Expansion}
+\subsection{Spherical Harmonics Series}
 
 \subsection{Fourier on $S^2$}
-- 
cgit v1.2.1


From c3261041f9bcf77a90ee0aa3e2dc73bf71edb923 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 18 Aug 2022 17:23:40 +0200
Subject: kugel: Corrections in orthogonality

---
 buch/papers/kugel/spherical-harmonics.tex | 65 +++++++++++++++++++++++--------
 1 file changed, 49 insertions(+), 16 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 2a00754..4f393d4 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -220,7 +220,7 @@ and $\lambda = n(n+1)$, we obtain what is known in the literature as the
 \emph{associated Legendre equation of order $m$}:
 \nocite{olver_introduction_2013}
 \begin{equation} \label{kugel:eqn:associated-legendre}
-  (1 - z^2)\frac{d^2 Z}{dz}
+  (1 - z^2)\frac{d^2 Z}{dz^2}
   - 2z\frac{d Z}{dz}
   + \left( n(n + 1) - \frac{m^2}{1 - z^2} \right) Z(z) = 0,
   \quad
@@ -236,7 +236,7 @@ This reduces the problem because it removes the double pole, which is always
 tricky to deal with. In fact, the reduced problem when $m = 0$ is known as the
 \emph{Legendre equation}:
 \begin{equation} \label{kugel:eqn:legendre}
-  (1 - z^2)\frac{d^2 Z}{dz}
+  (1 - z^2)\frac{d^2 Z}{dz^2}
   - 2z\frac{d Z}{dz}
   + n(n + 1) Z(z) = 0,
   \quad
@@ -275,7 +275,7 @@ Further, there are a few more interesting but not very relevant forms to write
 $P_n(z)$ such as \emph{Rodrigues' formula} and \emph{Laplace's integral
 representation} which are
 \begin{equation*}
-  P_n(z) = \frac{1}{2^n} \frac{d^n}{dz^n} (x^2 - 1)^n,
+  P_n(z) = \frac{1}{2^n n!} \frac{d^n}{dz^n} (z^2 - 1)^n,
   \qquad \text{and} \qquad
   P_n(z) = \frac{1}{\pi} \int_0^\pi \left(
     z + \cos\vartheta \sqrt{z^2 - 1}
@@ -312,8 +312,8 @@ obtain the \emph{associated Legendre functions}.
   \label{kugel:def:ferrers-functions}
   The functions
   \begin{equation}
-    P^m_n (z) = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z)
-      = \frac{1}{n!2^n}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n 
+    P^m_n (z) = (1-z^2)^{\frac{m}{2}}\frac{d^{m}}{dz^{m}} P_n(z)
+      = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}\frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n 
   \end{equation}
   are known as Ferrers or associated Legendre functions.
 \end{definition}
@@ -486,7 +486,8 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result.
     \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz
     = \frac{2(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'}
     = \begin{cases}
-      \frac{2(m + n)!}{(2n + 1)(n - m)!} & \text{if } n = n', \\
+      \frac{2(m + n)!}{(2n + 1)(n - m)!}
+        & \text{if } n = n' \text{ and } m = m', \\
       0 & \text{otherwise}.
     \end{cases}
   \end{equation*}
@@ -497,16 +498,26 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result.
   derivative is a pain to deal with.}
 \end{proof}
 
-An interesting fact to observe in lemma
-\ref{kugel:thm:associated-legendre-ortho} is that the orthogonality is only
-affected in the lower index, while varying $m$ only changes the constant in
-front of the Kronecker delta. By having the orthogonality relations of the
-Legendre functions we can finally show that spherical harmonics are also
-orthogonal.
+By having the orthogonality relations of the Legendre functions we can finally
+show that spherical harmonics are also orthogonal under the following inner
+product:
 
-\begin{lemma} For the spherical harmonics
-  \kugeltodo{Fix horizontal spacing, inner product definition is missing.}
+\begin{definition}[Inner product in $S^2$]
+  For 2 complex valued functions $f(\vartheta, \varphi)$ and $g(\vartheta,
+  \varphi)$ on the surface of the sphere the inner product is defined to be
+  \begin{equation*}
+    \langle f, g \rangle
+    = \int_{-\pi}^\pi \int_0^{2\pi}
+      f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)}
+      \sin \vartheta \, d\varphi \, d\vartheta.
+  \end{equation*}
+\end{definition}
+
+
+\begin{theorem} For the (unnormalized) spherical harmonics
   \label{kugel:thm:spherical-harmonics-ortho}
+  \kugeltodo{Why do I get a minus in front of $4\pi$??? It should not be there
+  right?}
   \begin{equation*}
     \langle Y^m_n, Y^{m'}_{n'} \rangle
     = \int_{-\pi}^\pi \int_0^{2\pi}
@@ -518,7 +529,7 @@ orthogonal.
       0 & \text{otherwise}.
     \end{cases}
   \end{equation*}
-\end{lemma}
+\end{theorem}
 \begin{proof}
   We will begin by doing a bit of algebraic maipulaiton:
   \begin{align*}
@@ -558,7 +569,29 @@ orthogonal.
 
 \subsection{Normalization}
 
-\kugeltodo{Discuss various normalizations.}
+At this point we have shown that the spherical harmonics form an orthogonal
+system, but in many applications we usually also want a normalization of some
+kind. For example the most obvious desirable property could be for the spherical
+harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n,
+Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add a
+normalization factor in front of the previous definition
+\ref{kugel:def:spherical-harmonics} as follows.
+
+\begin{definition}[Orthonormal spherical harmonics]
+  \label{kugel:def:spherical-harmonics-orthonormal}
+  The functions
+  \begin{equation*}
+    Y^m_n(\vartheta, \varphi)
+    = \sqrt{\frac{2n + 1}{4\pi} \frac{(n-m)!}{(m+n)!}}
+      P^m_n(\cos \vartheta) e^{im\varphi}
+  \end{equation*}
+  where $m, n \in \mathbb{Z}$ and $|m| < n$ are the orthonormal spherical
+  harmonics.
+\end{definition}
+
+Orthornomality is very useful indeed, but it is not the only common
+normalization that is found in the literature. In physics, quantum mechanics to
+be more specific, it is common to use the so called Schmidt semi-normalization.
 
 \if 0
 As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline 
-- 
cgit v1.2.1


From 6cc8d6c445305aa571f439d1945f53aac486ca72 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Fri, 19 Aug 2022 01:50:32 +0200
Subject: kugel: More corrections

---
 buch/papers/kugel/spherical-harmonics.tex | 45 ++++++++++++++++---------------
 1 file changed, 24 insertions(+), 21 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 4f393d4..9d055e0 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -394,7 +394,7 @@ will be our starting point.
   \end{equation*}
 \end{lemma}
 \begin{proof}
-  To start, consider the fact that that the Legendre equation
+  To start, consider the fact that the Legendre equation
   \eqref{kugel:eqn:legendre}, of which two distinct Legendre polynomials
   $P_n(z)$ and $P_k(z)$ are a solution ($n \neq k$), can be rewritten in the
   following form:
@@ -483,19 +483,19 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result.
 \begin{lemma} For the associated Legendre functions
   \label{kugel:thm:associated-legendre-ortho}
   \begin{equation*}
-    \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz
+    \int_{-1}^1 P^m_n(z) P^{m}_{n'}(z) \, dz
     = \frac{2(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'}
     = \begin{cases}
       \frac{2(m + n)!}{(2n + 1)(n - m)!}
-        & \text{if } n = n' \text{ and } m = m', \\
+        & \text{if } n = n', \\
       0 & \text{otherwise}.
     \end{cases}
   \end{equation*}
 \end{lemma}
 \begin{proof}
-  \kugeltodo{Is it worth showing? IMHO no, it is mostly the same as Lemma
-  \ref{kugel:thm:legendre-poly-ortho} with the difference that the $m$-th
-  derivative is a pain to deal with.}
+  \kugeltodo{Is this correct? And Is it worth showing? IMHO no, it is mostly the
+  same as Lemma \ref{kugel:thm:legendre-poly-ortho} with the difference that the
+  $m$-th derivative is a pain to deal with.}
 \end{proof}
 
 By having the orthogonality relations of the Legendre functions we can finally
@@ -507,7 +507,7 @@ product:
   \varphi)$ on the surface of the sphere the inner product is defined to be
   \begin{equation*}
     \langle f, g \rangle
-    = \int_{-\pi}^\pi \int_0^{2\pi}
+    = \int_{0}^\pi \int_0^{2\pi}
       f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)}
       \sin \vartheta \, d\varphi \, d\vartheta.
   \end{equation*}
@@ -520,12 +520,12 @@ product:
   right?}
   \begin{equation*}
     \langle Y^m_n, Y^{m'}_{n'} \rangle
-    = \int_{-\pi}^\pi \int_0^{2\pi}
+    = \int_{0}^\pi \int_0^{2\pi}
       Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
       \sin \vartheta \, d\varphi \, d\vartheta
-    = \frac{-4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'}
+    = \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'}
     = \begin{cases}
-      \frac{-4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} & \text{if } n = n', \\
+      \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} & \text{if } n = n', \\
       0 & \text{otherwise}.
     \end{cases}
   \end{equation*}
@@ -533,15 +533,15 @@ product:
 \begin{proof}
   We will begin by doing a bit of algebraic maipulaiton:
   \begin{align*}
-    \int_{-\pi}^\pi \int_0^{2\pi}
+    \int_{0}^\pi \int_0^{2\pi}
       Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
       \sin \vartheta \, d\varphi \, d\vartheta
-    &= \int_{-\pi}^\pi \int_0^{2\pi}
+    &= \int_{0}^\pi \int_0^{2\pi}
       e^{im\varphi} P^m_n(\cos \vartheta)
       e^{-im'\varphi} P^{m'}_{n'}(\cos \vartheta)
       \, d\varphi \sin \vartheta \, d\vartheta
     \\
-    &= \int_{-\pi}^\pi
+    &= \int_{0}^\pi
       P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
       \int_0^{2\pi} e^{i(m - m')\varphi}
       \, d\varphi \sin \vartheta \, d\vartheta
@@ -552,19 +552,22 @@ product:
   $m = m'$ the inner integral simplifies to $\int_0^{2\pi} 1 \, d\varphi$ which
   equals $2\pi$, so in this case the expression becomes
   \begin{equation*}
-    2\pi \int_{-\pi}^\pi
+    2\pi \int_{0}^\pi
       P^m_n(\cos \vartheta) P^{m'}_{n'}(\cos \vartheta)
     \sin \vartheta \, d\vartheta
-    = -2\pi \int_{-1}^1 P^m_n(z) P^{m'}_{n'}(z) \, dz
-    = \frac{-4\pi(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'},
+    = -2\pi \int_{1}^{-1} P^m_n(z) P^{m'}_{n'}(z) \, dz
+    = \frac{4\pi(m + n)!}{(2n + 1)(n - m)!} \delta_{nn'},
   \end{equation*}
   where in the second step we performed the substitution $z = \cos\vartheta$;
   $d\vartheta = \frac{d\vartheta}{dz} dz= - dz / \sin \vartheta$, and then we
-  used lemma \ref{kugel:thm:associated-legendre-ortho}. Now we just need look at
-  the case when $m \neq m'$. Fortunately this is easy: the inner integral is
-  $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, or in other words we are
-  integrating a complex exponetial over the entire period, which always results
-  in zero. Thus, we do not need to do anything and the proof is complete.
+  used lemma \ref{kugel:thm:associated-legendre-ortho}. We are allowed to use
+  the lemma because $m = m'$.
+
+  Now we just need look at the case when $m \neq m'$. Fortunately this is easy:
+  the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, or in
+  other words we are integrating a complex exponetial over the entire period,
+  which always results in zero. Thus, we do not need to do anything and the
+  proof is complete.
 \end{proof}
 
 \subsection{Normalization}
-- 
cgit v1.2.1


From 4e29e512c4f4f0f1244cbe38c804e46bafda225d Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Fri, 19 Aug 2022 21:57:24 +0200
Subject: kugel: Corrections and normalizations

---
 buch/papers/kugel/spherical-harmonics.tex | 104 ++++++++++++++++++------------
 1 file changed, 63 insertions(+), 41 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 9d055e0..72f7402 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -493,9 +493,12 @@ $P_n(z)$ by a $m$-th derivative, and obtain the following result.
   \end{equation*}
 \end{lemma}
 \begin{proof}
-  \kugeltodo{Is this correct? And Is it worth showing? IMHO no, it is mostly the
-  same as Lemma \ref{kugel:thm:legendre-poly-ortho} with the difference that the
-  $m$-th derivative is a pain to deal with.}
+  To show that the expression equals zero when $n \neq n'$ we can perform
+  exactly the same steps as in the proof of lemma
+  \ref{kugel:thm:legendre-poly-ortho}, so we will not repeat them here and prove
+  instead only the case when $n = n'$.
+  \kugeltodo{Finish proof, or not? I have to look and decide if it is
+  interesting enough.}
 \end{proof}
 
 By having the orthogonality relations of the Legendre functions we can finally
@@ -516,19 +519,19 @@ product:
 
 \begin{theorem} For the (unnormalized) spherical harmonics
   \label{kugel:thm:spherical-harmonics-ortho}
-  \kugeltodo{Why do I get a minus in front of $4\pi$??? It should not be there
-  right?}
-  \begin{equation*}
+  \begin{align*}
     \langle Y^m_n, Y^{m'}_{n'} \rangle
-    = \int_{0}^\pi \int_0^{2\pi}
+    &= \int_{0}^\pi \int_0^{2\pi}
       Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)}
       \sin \vartheta \, d\varphi \, d\vartheta
-    = \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'}
+    \\
+    &= \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} \delta_{nn'} \delta_{mm'}
     = \begin{cases}
-      \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!} & \text{if } n = n', \\
+      \frac{4\pi}{2n + 1} \frac{(m + n)!}{(n - m)!}
+        & \text{if } n = n' \text{ and } m = m', \\
       0 & \text{otherwise}.
     \end{cases}
-  \end{equation*}
+  \end{align*}
 \end{theorem}
 \begin{proof}
   We will begin by doing a bit of algebraic maipulaiton:
@@ -563,38 +566,15 @@ product:
   used lemma \ref{kugel:thm:associated-legendre-ortho}. We are allowed to use
   the lemma because $m = m'$.
 
-  Now we just need look at the case when $m \neq m'$. Fortunately this is easy:
-  the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$, or in
-  other words we are integrating a complex exponetial over the entire period,
-  which always results in zero. Thus, we do not need to do anything and the
-  proof is complete.
+  Now we just need look at the case when $m \neq m'$. Fortunately this is
+  easier: the inner integral is $\int_0^{2\pi} e^{i(m - m')\varphi} d\varphi$,
+  or in other words we are integrating a complex exponetial over the entire
+  period, which always results in zero. Thus, we do not need to do anything and
+  the proof is complete.
 \end{proof}
 
-\subsection{Normalization}
-
-At this point we have shown that the spherical harmonics form an orthogonal
-system, but in many applications we usually also want a normalization of some
-kind. For example the most obvious desirable property could be for the spherical
-harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n,
-Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add a
-normalization factor in front of the previous definition
-\ref{kugel:def:spherical-harmonics} as follows.
-
-\begin{definition}[Orthonormal spherical harmonics]
-  \label{kugel:def:spherical-harmonics-orthonormal}
-  The functions
-  \begin{equation*}
-    Y^m_n(\vartheta, \varphi)
-    = \sqrt{\frac{2n + 1}{4\pi} \frac{(n-m)!}{(m+n)!}}
-      P^m_n(\cos \vartheta) e^{im\varphi}
-  \end{equation*}
-  where $m, n \in \mathbb{Z}$ and $|m| < n$ are the orthonormal spherical
-  harmonics.
-\end{definition}
-
-Orthornomality is very useful indeed, but it is not the only common
-normalization that is found in the literature. In physics, quantum mechanics to
-be more specific, it is common to use the so called Schmidt semi-normalization.
+\kugeltodo{Briefly mention that we could have skipped the tedious proofs by
+showing that the (associated) Legendre equation is a Sturm Liouville problem.}
 
 \if 0
 As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline 
@@ -620,7 +600,49 @@ Inoltre, possiamo provare l'ortogonalità di $\Theta(\vartheta)$ utilizzando \eq
 Ora, visto che la soluzione dell'eigenfunction problem è formata dalla moltiplicazione di $\Phi_m(\varphi)$ e $P_{m,n}(x)$
 \fi
 
-\subsection{Properties}
+
+\subsection{Normalization and the Phase Factor}
+
+At this point we have shown that the spherical harmonics form an orthogonal
+system, but in many applications we usually also want a normalization of some
+kind. For example the most obvious desirable property could be for the spherical
+harmonics to be ortho\emph{normal}, by which we mean that $\langle Y^m_n,
+Y^{m'}_{n'} \rangle = \delta_{nn'}$. To obtain orthonormality, we simply add an
+ugly normalization factor in front of the previous definition
+\ref{kugel:def:spherical-harmonics} as follows.
+
+\begin{definition}[Orthonormal spherical harmonics]
+  \label{kugel:def:spherical-harmonics-orthonormal}
+  The functions
+  \begin{equation*}
+    Y^m_n(\vartheta, \varphi)
+    = \sqrt{\frac{2n + 1}{4\pi} \frac{(n-m)!}{(m+n)!}}
+      P^m_n(\cos \vartheta) e^{im\varphi}
+  \end{equation*}
+  where $m, n \in \mathbb{Z}$ and $|m| < n$ are the orthonormal spherical
+  harmonics.
+\end{definition}
+
+Orthornomality is very useful, but it is not the only common normalization that
+is found in the literature. In physics, geomagnetism to be more specific, it is
+common to use the so called Schmidt semi-normalization (or sometimes also called
+quasi-normalization).
+
+\begin{definition}[Schmidt semi-normalized spherical harmonics]
+  \label{kugel:def:spherical-harmonics-schmidt}
+  The Schmidt semi-normalized spherical harmonics are
+  \begin{equation*}
+    Y^m_n(\vartheta, \varphi)
+    = \sqrt{2 \frac{(n - m)!}{(n + m)!}}
+      P^m_n(\cos \vartheta) e^{im\varphi}
+  \end{equation*}
+  where $m, n \in \mathbb{Z}$ and $|m| < n$.
+\end{definition}
+
+However, for our purposes we will mostly only need the orthonormal spherical
+harmonics.  So from now on, unless specified otherwise, when we say spherical
+harmonics or write $Y^m_n$, we mean the orthonormal spherical harmonics of
+definition \ref{kugel:def:spherical-harmonics-orthonormal}.
 
 \subsection{Recurrence Relations}
 
-- 
cgit v1.2.1


From d2ae59bb9d2affc07bcb541d37a8f88fd009c167 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 19:49:13 +0200
Subject: kugel: mention Condon-Shortley phase factor

---
 buch/papers/kugel/spherical-harmonics.tex | 13 +++++++++++--
 1 file changed, 11 insertions(+), 2 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 72f7402..5d394a9 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -639,8 +639,17 @@ quasi-normalization).
   where $m, n \in \mathbb{Z}$ and $|m| < n$.
 \end{definition}
 
-However, for our purposes we will mostly only need the orthonormal spherical
-harmonics.  So from now on, unless specified otherwise, when we say spherical
+Additionally, there is another quirk in the literature that should be mentioned.
+In some other branches of physics such as seismology there is a so called
+Condon-Shortley phase factor $(-1)^m$ in front of the square root in the
+definition of the normalized spherical harmonics. It is yet another
+normalization that is added for reasons that are not very relevant to our
+discussion, but we are mentioning its existence since many numerical packages
+(such as \texttt{SHTOOLS} \kugeltodo{Reference}) offer an option to add or
+remove it from the computation.
+
+Though, for our purposes we will mostly only need the orthonormal spherical
+harmonics, so from now on, unless specified otherwise when we say spherical
 harmonics or write $Y^m_n$, we mean the orthonormal spherical harmonics of
 definition \ref{kugel:def:spherical-harmonics-orthonormal}.
 
-- 
cgit v1.2.1


From f05ad8165a516c7932a8137a51b247484c38403b Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 23:25:12 +0200
Subject: kugel: Orthogonality using Sturm-Liouville

---
 buch/papers/kugel/spherical-harmonics.tex | 92 +++++++++++++++++++++++++++----
 1 file changed, 82 insertions(+), 10 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 5d394a9..5a17b99 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -506,6 +506,7 @@ show that spherical harmonics are also orthogonal under the following inner
 product:
 
 \begin{definition}[Inner product in $S^2$]
+  \label{kugel:def:inner-product-s2}
   For 2 complex valued functions $f(\vartheta, \varphi)$ and $g(\vartheta,
   \varphi)$ on the surface of the sphere the inner product is defined to be
   \begin{equation*}
@@ -573,8 +574,51 @@ product:
   the proof is complete.
 \end{proof}
 
-\kugeltodo{Briefly mention that we could have skipped the tedious proofs by
-showing that the (associated) Legendre equation is a Sturm Liouville problem.}
+These proofs for the various orthogonality relations were quite long and
+algebraically tedious, mainly because they are ``low level'', by which we mean
+that they (arguably) do not rely on very abstract theory. However, if we allow
+ourselves to use the more abstract Sturm Liouville theory discussed in chapters
+\ref{buch:integrale:subsection:sturm-liouville-problem} and \kugeltodo{reference
+to chapter 17 of haddouche and Löffler} the proofs can become ridiculously
+short. Let's do for example lemma \ref{kugel:thm:associated-legendre-ortho}.
+
+\begin{proof}[
+    Shorter proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+  ]
+  The associated Legendre polynomials, of which we would like to prove an
+  orthogonality relation, are the solution to the associated Legendre equation,
+  which we can write as $LZ(z) = 0$, where
+  \begin{equation*}
+    L = \frac{d}{dz} (1 - z^2) \frac{d}{dz}
+      + n(n+1) - \frac{m^2}{1 - z^2}.
+  \end{equation*}
+  Notice that $L$ is in fact a Sturm-Liouville operator of the form
+  \begin{equation*}
+    L = \frac{1}{w(z)} \left[
+        \frac{d}{dz} p(z) \frac{d}{dz} - \lambda + q(z)
+      \right],
+  \end{equation*}
+  if we let $w(z) = 1$, $p(z) = (1 - z^2 )$, $q(z) = -m^2 / (1 - z^2)$, and
+  $\lambda = -n(n+1)$. By the theory of Sturm-Liouville operators, we know that
+  the each solution of the problem $LZ(z) = 0$, namely $P^m_n(z)$, is orthogonal
+  to every other solution that has a different $\lambda$. In our case $\lambda$
+  varies with $n$, so $P^m_n(z)$ with different $n$'s are orthogonal to each
+  other.
+\end{proof}
+
+But that was still rather informative and had a bit of explanation, which is
+terrible. Real snobs, such as Wikipedia contributors, some authors and sometimes
+regrettably even ourselves, would write instead:
+
+\begin{proof}[
+    Pretentiously short proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+  ]
+  The associated Legendre polynomials are solutions of the associated Legendre
+  equation which is a Sturm-Liouville problem and are thus orthogonal to each
+  other. The factor in front Kronecker delta is left as an exercise to the
+  reader.
+\end{proof}
+
 
 \if 0
 As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline 
@@ -640,13 +684,13 @@ quasi-normalization).
 \end{definition}
 
 Additionally, there is another quirk in the literature that should be mentioned.
-In some other branches of physics such as seismology there is a so called
-Condon-Shortley phase factor $(-1)^m$ in front of the square root in the
-definition of the normalized spherical harmonics. It is yet another
-normalization that is added for reasons that are not very relevant to our
-discussion, but we are mentioning its existence since many numerical packages
-(such as \texttt{SHTOOLS} \kugeltodo{Reference}) offer an option to add or
-remove it from the computation.
+In some other branches of physics such as seismology and quantum mechanics there
+is a so called Condon-Shortley phase factor $(-1)^m$ in front of the square root
+in the definition of the normalized spherical harmonics. It is yet another
+normalization that is added for physical reasons that are not very relevant to
+our discussion, but mention its existence this potential source of confusion
+since many numerical packages (such as \texttt{SHTOOLS} \kugeltodo{Reference})
+offer an option to add or remove it from the computation.
 
 Though, for our purposes we will mostly only need the orthonormal spherical
 harmonics, so from now on, unless specified otherwise when we say spherical
@@ -655,8 +699,36 @@ definition \ref{kugel:def:spherical-harmonics-orthonormal}.
 
 \subsection{Recurrence Relations}
 
-\section{Series Expansions in $C(S^2)$}
+\section{Series Expansions in $L^2(S^2)$}
+
+We have now reached a point were we have all of the tools that are necessary to
+build something truly amazing: a general series expansion formula for functions
+on the surface of the sphere. Using the jargon: we will now see that the
+spherical harmonics together with the inner product of definition
+\ref{kugel:def:inner-product-s2}
+\begin{equation*}
+  \langle f, g \rangle
+  = \int_{0}^\pi \int_0^{2\pi}
+    f(\vartheta, \varphi) \overline{g(\vartheta, \varphi)}
+    \sin \vartheta \, d\varphi \, d\vartheta
+\end{equation*}
+form a Hilbert space over the space of complex valued $L^2$ functions $S^2 \to
+\mathbb{C}$. We will see later that this fact is very consequential and is
+extremely useful for many types of applications. If the jargon was too much, no
+need to worry, we will now go back to normal words and explain it again in more
+detail.
 
 \subsection{Spherical Harmonics Series}
 
+To talk about a \emph{series expansion} we first need a series, so we shall
+build one using the spherical harmonics.
+
+\begin{definition}[Spherical harmonic series]
+  \begin{equation*}
+    \hat{f}(\vartheta, \varphi) 
+    = \sum_{n \in \mathbb{Z}} \sum_{m \in \mathbb{Z}}
+      c_{m,n} Y^m_n(\vartheta, \varphi)
+  \end{equation*}
+\end{definition}
+
 \subsection{Fourier on $S^2$}
-- 
cgit v1.2.1


From 63dee97e79f65a967f7d6b34bb8141ccaa226e20 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sat, 20 Aug 2022 23:40:29 +0200
Subject: kugel: Minor corrections

---
 buch/papers/kugel/spherical-harmonics.tex | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 5a17b99..54c8fa9 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -607,11 +607,11 @@ short. Let's do for example lemma \ref{kugel:thm:associated-legendre-ortho}.
 \end{proof}
 
 But that was still rather informative and had a bit of explanation, which is
-terrible. Real snobs, such as Wikipedia contributors, some authors and sometimes
-regrettably even ourselves, would write instead:
+terrible. Real snobs, such as Wikipedia contributors, some authors and
+regrettably sometimes even ourselves, would write instead:
 
 \begin{proof}[
-    Pretentiously short proof of lemma \ref{kugel:thm:associated-legendre-ortho}
+    Infuriatingly short proof of lemma \ref{kugel:thm:associated-legendre-ortho}
   ]
   The associated Legendre polynomials are solutions of the associated Legendre
   equation which is a Sturm-Liouville problem and are thus orthogonal to each
@@ -688,9 +688,9 @@ In some other branches of physics such as seismology and quantum mechanics there
 is a so called Condon-Shortley phase factor $(-1)^m$ in front of the square root
 in the definition of the normalized spherical harmonics. It is yet another
 normalization that is added for physical reasons that are not very relevant to
-our discussion, but mention its existence this potential source of confusion
-since many numerical packages (such as \texttt{SHTOOLS} \kugeltodo{Reference})
-offer an option to add or remove it from the computation.
+our discussion, but we mention this potential source of confusion since many
+numerical packages (such as \texttt{SHTOOLS} \kugeltodo{Reference}) offer an
+option to add or remove it from the computation.
 
 Though, for our purposes we will mostly only need the orthonormal spherical
 harmonics, so from now on, unless specified otherwise when we say spherical
-- 
cgit v1.2.1


From 288eb54f5089c48177434757b083309e05e30bf2 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Sun, 21 Aug 2022 11:48:48 +0200
Subject: kugel: More on Sturm-Liouville

---
 buch/papers/kugel/spherical-harmonics.tex | 42 ++++++++++++-------------------
 1 file changed, 16 insertions(+), 26 deletions(-)

(limited to 'buch/papers/kugel/spherical-harmonics.tex')

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 54c8fa9..bff91ef 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -107,7 +107,7 @@ the surface of the unit sphere.
 Now that we have defined an operator, we can go and study its eigenfunctions,
 which means that we would like to find the functions $f(\vartheta, \varphi)$
 that satisfy the equation
-\begin{equation} \label{kuvel:eqn:eigen}
+\begin{equation} \label{kugel:eqn:eigen}
     \surflaplacian f = -\lambda f.
 \end{equation}
 Perhaps it may not be obvious at first glance, but we are in fact dealing with a
@@ -619,31 +619,21 @@ regrettably sometimes even ourselves, would write instead:
   reader.
 \end{proof}
 
-
-\if 0
-As explained in the chapter \ref{}, the concept of orthogonality is very important and at the practical level it is very useful, because it allows us to develop very powerful techniques at the mathematical level.\newline 
-Throughout this book we have been confronted with the Sturm-Liouville theory (see chapter \ref{}). The latter, among other things, carries with it the concept of orthogonality. Indeed, if we consider the solutions of the Sturm-Liouville equation, which can be expressed in this form
-\begin{equation}\label{kugel:eq:sturm_liouville}
-    \mathcal{S}f := \frac{d}{dx}\left[p(x)\frac{df}{dx}\right]+q(x)f(x)
-\end{equation}
-possiamo dire che formano una base ortogonale.\newline
-Adesso possiamo dare un occhiata alle due equazioni che abbiamo ottenuto tramite la Separation Ansatz (Eqs.\eqref{kugel:eq:associated_leg_eq}\eqref{kugel:eq:ODE_1}), le quali possono essere riscritte come:
-\begin{align*}
-    \frac{d}{dx} \left[ (1-x^2) \cdot \frac{dP_{m,n}}{dx} \right] &+ \left(n(n+1)-\frac{m}{1-x^2} \right) \cdot P_{m,n}(x) = 0, \\
-    \frac{d}{d\varphi} \left[ 1 \cdot \frac{ d\Phi }{d\varphi} \right] &+ 1 \cdot \Phi(\varphi) = 0. 
-\end{align*}
-Si può concludere in modo diretto che sono due casi dell'equazione di Sturm-Liouville. Questo significa che le loro soluzioni sono ortogonali sotto l'inner product con weight function $w(x)=1$, dunque:
-\begin{align}
-\int_{0}^{2\pi} \Phi_m(\varphi)\Phi_m'(\varphi) d\varphi &= \delta_{m'm}, \nonumber \\
-\int_{-1}^1 P_{m,m'}(x)P_{n,n'}(x) dx &= \delta_{m'm}\delta_{n'n}. \label{kugel:eq:orthogonality_associated_func}
-\end{align}
-Inoltre, possiamo provare l'ortogonalità di $\Theta(\vartheta)$ utilizzando \eqref{kugel:eq:orthogonality_associated_func}:
-\begin{align}
-    x
-\end{align}
-Ora, visto che la soluzione dell'eigenfunction problem è formata dalla moltiplicazione di $\Phi_m(\varphi)$ e $P_{m,n}(x)$
-\fi
-
+Lemma \ref{kugel:thm:legendre-poly-ortho} has a very similar
+proof, while the theorem \ref{kugel:thm:spherical-harmonics-ortho} for the
+spherical harmonics is proved by the following argument. The spherical harmonics
+are the solutions to the eigenvalue problem $\surflaplacian f = -\lambda f$,
+which as discussed in the previous section is solved using separation. So to
+prove their orthogonality using the Sturm-Liouville theory we argue that
+\begin{equation*}
+  \surflaplacian = L_\vartheta L_\varphi \iff
+  \surflaplacian f(\vartheta, \varphi)
+    = L_\vartheta \Theta(\vartheta) L_\varphi \Phi(\varphi),
+\end{equation*}
+then we show that both $L_\vartheta$ and $L_\varphi$ are both Sturm-Liouville
+operators (we just did the former in the shorter proof above). Since both are
+Sturm-Liouville operators their combination, the surface spherical Laplacian, is
+also a Sturm-Liouville operator, which then implies orthogonality.
 
 \subsection{Normalization and the Phase Factor}
 
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