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')

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 cb815146f12661f320a771464c5083e5196bb783 Mon Sep 17 00:00:00 2001
From: tschwall <55748566+tschwall@users.noreply.github.com>
Date: Thu, 18 Aug 2022 16:51:08 +0200
Subject: Added graphic

---
 buch/papers/parzyl/img/Plane_2D.png | Bin 0 -> 99330 bytes
 buch/papers/parzyl/teil2.tex        |  20 +++++++++++++++-----
 2 files changed, 15 insertions(+), 5 deletions(-)
 create mode 100644 buch/papers/parzyl/img/Plane_2D.png

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/img/Plane_2D.png b/buch/papers/parzyl/img/Plane_2D.png
new file mode 100644
index 0000000..7c32877
Binary files /dev/null and b/buch/papers/parzyl/img/Plane_2D.png differ
diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index 573432a..d37c650 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -9,12 +9,22 @@
 
 Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld einer semi-infiniten Platte, wie in Abbildung \ref{parzyl:fig:leiterplatte} gezeigt, finden will.
 \begin{figure}
-	 \centering
-	\includegraphics[width=0.9\textwidth]{papers/parzyl/img/plane.pdf}
-	\caption{Semi-infinite Leiterplatte}
-	\label{parzyl:fig:leiterplatte}
+	\centering
+	\begin{minipage}{.7\textwidth}
+		 \centering
+		\includegraphics[width=\textwidth]{papers/parzyl/img/plane.pdf}
+		\caption{Semi-infinite Leiterplatte}
+		\label{parzyl:fig:leiterplatte}
+	\end{minipage}%
+	\begin{minipage}{.25\textwidth}
+		\centering
+	\includegraphics[width=\textwidth]{papers/parzyl/img/Plane_2D.png}
+	\caption{Semi-infinite Leiterplatte dargestellt in 2D}
+	\label{parzyl:fig:leiterplatte_2d}
+	\end{minipage}
 \end{figure}
-Das dies so ist kann im zwei Dimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Linie, was man in Abbildung TODO sieht.
+Das dies so ist kann im zwei Dimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Linie, was man in Abbildung \ref{parzyl:fig:leiterplatte_2d} sieht.
+
 Jede komplexe Funktion $F(z)$ kann geschrieben werden als
 \begin{equation}
 	F(s) = U(x,y) + iV(x,y) \qquad s \in \mathbb{C}; x,y \in \mathbb{R}.
-- 
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')

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 aea9cc922545bd617166b89edc353c2c2f180106 Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Thu, 18 Aug 2022 22:14:37 +0200
Subject: verbesserungen

---
 buch/papers/parzyl/teil1.tex | 12 ++++++------
 buch/papers/parzyl/teil3.tex | 10 +++++-----
 2 files changed, 11 insertions(+), 11 deletions(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil1.tex b/buch/papers/parzyl/teil1.tex
index 13d8109..2caabde 100644
--- a/buch/papers/parzyl/teil1.tex
+++ b/buch/papers/parzyl/teil1.tex
@@ -13,13 +13,13 @@ Die Lösung ist somit
 	i(z) 
 	= 
 	A\cos{ 
-		\left ( 
-		\sqrt{\lambda + \mu}z
+		\left ( z
+		\sqrt{\lambda + \mu}
 		\right )}
 	+
 	B\sin{ 
-		\left ( 
-		\sqrt{\lambda + \mu}z
+		\left ( z
+		\sqrt{\lambda + \mu}
 		\right )}.
 \end{equation}
 Die Differentialgleichungen \eqref{parzyl:sep_dgl_1} und \eqref{parzyl:sep_dgl_2} werden in \cite{parzyl:whittaker}
@@ -51,7 +51,7 @@ mit Hilfe der Whittaker Gleichung gelöst.
         M_{k, -m} \left(x\right)
     \end{equation*}
     gehören zu den Whittaker Funktionen und sind Lösungen
-    von der Whittaker Differentialgleichung
+    der Whittaker Differentialgleichung
     \begin{equation}
         \frac{d^2W}{d x^2} +
         \biggl( -\frac{1}{4}  + \frac{k}{x} + \frac{\frac{1}{4} - m^2}{x^2} \biggr) W = 0.
@@ -95,7 +95,7 @@ $w$ als Lösung haben.
 %              - k, {\textstyle \frac{3}{2}} ; {\textstyle \frac{1}{2}}z^2).
 %\end{align}
 
-In der Literatur gibt es verschiedene Standartlösungen für 
+In der Literatur gibt es verschiedene Standardlösungen für 
 \eqref{parzyl:eq:weberDiffEq}, wobei die Differentialgleichung jeweils 
 unterschiedlich geschrieben wird.
 Whittaker und Watson zeigen in \cite{parzyl:whittaker} die Lösung
diff --git a/buch/papers/parzyl/teil3.tex b/buch/papers/parzyl/teil3.tex
index 166eebf..6432905 100644
--- a/buch/papers/parzyl/teil3.tex
+++ b/buch/papers/parzyl/teil3.tex
@@ -12,9 +12,9 @@
 %Die parabolischen Zylinderfunktionen, welche in Gleichung \ref{parzyl:eq:solution_dgl} gegeben sind, 
 %können auch als Potenzreihen geschrieben werden
 Die parabolischen Zylinderfunktionen können auch als Potenzreihen geschrieben werden.
-Im folgenden Abschnitt werden die Terme welche nur von $n$ oder $a$ abhängig sind vernachlässigt.
+In diesem Abschnitt werden die Terme welche nur von $n$ oder $a$ abhängig sind vernachlässigt.
 Die parabolischen Zylinderfunktionen sind Linearkombinationen aus einem geraden Teil $w_1(\alpha, x)$ 
-und einem ungeraden Teil $w_2(\alpha, x)$, welche als Potenzreihe
+und einem ungeraden Teil $w_2(\alpha, x)$, welche als Potenzreihen
 \begin{align}
 	w_1(\alpha,x)
 	&=  
@@ -67,9 +67,9 @@ und
 \end{align}
 sind.
 Die Potenzreihen sind in der regel unendliche Reihen. 
-Es gibt allerdings die Möglichkeit für bestimmte $\alpha$ das die Terme in der Klammer gleich null werden 
+Es gibt allerdings die Möglichkeit, dass für bestimmte $\alpha$ die Terme in der Klammer gleich null werden 
 und die Reihe somit eine endliche Anzahl $n$ Summanden hat.
-Dies geschieht bei $w_1(\alpha,x)$ falls
+Dies geschieht bei $w_1(\alpha,x)$, falls
 \begin{equation}
 	\alpha =  -n \qquad n \in \mathbb{N}_0
 \end{equation}
@@ -77,7 +77,7 @@ und bei $w_2(\alpha,x)$ falls
 \begin{equation}
 	\alpha = -\frac{1}{2} - n \qquad n \in \mathbb{N}_0.
 \end{equation}
-Der Wert des von $\alpha$ ist abhängig, ob man $D_n(x)$ oder $U(a,x)$ / $V(a,x)$ verwendet.
+Der Wert von $\alpha$ ist abhängig, ob man $D_n(x)$ oder $U(a,x)$ / $V(a,x)$ verwendet.
 Bei $D_n(x)$ gilt $\alpha = -{\textstyle \frac{1}{2}} n$ und bei $U(a,z)$ oder $V(a,x)$ gilt 
 $\alpha = {\textstyle \frac{1}{2}} a + {\textstyle \frac{1}{4}}$.
 \subsection{Ableitung}
-- 
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')

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 aa2fec29136fb8eebab30b6c7bdd96917c58a298 Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Fri, 19 Aug 2022 08:47:28 +0200
Subject: verbesserungen

---
 buch/papers/parzyl/teil0.tex | 4 ++--
 buch/papers/parzyl/teil1.tex | 2 +-
 buch/papers/parzyl/teil3.tex | 8 ++++----
 3 files changed, 7 insertions(+), 7 deletions(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex
index 8be936d..70caa05 100644
--- a/buch/papers/parzyl/teil0.tex
+++ b/buch/papers/parzyl/teil0.tex
@@ -19,8 +19,8 @@ Die partielle Differentialgleichung
 \begin{equation}
 	\Delta f = \lambda f
 \end{equation}
-ist als Helmholtz-Gleichung bekannt und beschreibt das Eigenwert Problem für den Laplace-Operator. 
-Sie ist eine der Gleichungen welche auftritt wenn die Wellengleichung
+ist als Helmholtz-Gleichung bekannt und beschreibt das Eigenwertproblem für den Laplace-Operator. 
+Sie ist eine der Gleichungen, welche auftritt, wenn die Wellengleichung
 \begin{equation}
 	\left ( \nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}  \right ) u(\textbf{r},t)
 	=
diff --git a/buch/papers/parzyl/teil1.tex b/buch/papers/parzyl/teil1.tex
index 2caabde..0e1ad1b 100644
--- a/buch/papers/parzyl/teil1.tex
+++ b/buch/papers/parzyl/teil1.tex
@@ -94,7 +94,7 @@ $w$ als Lösung haben.
 %         ({\textstyle \frac{3}{4}} 
 %              - k, {\textstyle \frac{3}{2}} ; {\textstyle \frac{1}{2}}z^2).
 %\end{align}
-
+\subsection{Standardlösungen}
 In der Literatur gibt es verschiedene Standardlösungen für 
 \eqref{parzyl:eq:weberDiffEq}, wobei die Differentialgleichung jeweils 
 unterschiedlich geschrieben wird.
diff --git a/buch/papers/parzyl/teil3.tex b/buch/papers/parzyl/teil3.tex
index 6432905..1b59ed9 100644
--- a/buch/papers/parzyl/teil3.tex
+++ b/buch/papers/parzyl/teil3.tex
@@ -12,8 +12,8 @@
 %Die parabolischen Zylinderfunktionen, welche in Gleichung \ref{parzyl:eq:solution_dgl} gegeben sind, 
 %können auch als Potenzreihen geschrieben werden
 Die parabolischen Zylinderfunktionen können auch als Potenzreihen geschrieben werden.
-In diesem Abschnitt werden die Terme welche nur von $n$ oder $a$ abhängig sind vernachlässigt.
-Die parabolischen Zylinderfunktionen sind Linearkombinationen aus einem geraden Teil $w_1(\alpha, x)$ 
+Parabolische Zylinderfunktionen sind Linearkombinationen 
+$A(\alpha)w_1(\alpha, x) + B(\alpha)w_2(\alpha, x)$ aus einem geraden Teil $w_1(\alpha, x)$ 
 und einem ungeraden Teil $w_2(\alpha, x)$, welche als Potenzreihen
 \begin{align}
 	w_1(\alpha,x)
@@ -51,7 +51,7 @@ und
 	= 
 	xe^{-\frac{x^2}{4}}
 	\sum^{\infty}_{n=0}
-	\frac{\left ( \frac{3}{4} - k \right )_{n}}{\left ( \frac{3}{2}\right )_{n}}
+	\frac{\left ( \frac{1}{2} + \alpha \right )_{n}}{\left ( \frac{3}{2}\right )_{n}}
 	\frac{\left ( \frac{1}{2} x^2\right )^n}{n!} \\
 	&=
 	e^{-\frac{x^2}{4}}
@@ -77,7 +77,7 @@ und bei $w_2(\alpha,x)$ falls
 \begin{equation}
 	\alpha = -\frac{1}{2} - n \qquad n \in \mathbb{N}_0.
 \end{equation}
-Der Wert von $\alpha$ ist abhängig, ob man $D_n(x)$ oder $U(a,x)$ / $V(a,x)$ verwendet.
+Der Wert von $\alpha$ ist abhängig, ob man $D_n(x)$, $U(a,x)$ oder $V(a,x)$ verwendet.
 Bei $D_n(x)$ gilt $\alpha = -{\textstyle \frac{1}{2}} n$ und bei $U(a,z)$ oder $V(a,x)$ gilt 
 $\alpha = {\textstyle \frac{1}{2}} a + {\textstyle \frac{1}{4}}$.
 \subsection{Ableitung}
-- 
cgit v1.2.1


From 3dd42149bf496fe5cca749e69f839c2f6ab888a7 Mon Sep 17 00:00:00 2001
From: tschwall <55748566+tschwall@users.noreply.github.com>
Date: Fri, 19 Aug 2022 11:10:07 +0200
Subject: Corrected errors

---
 buch/papers/parzyl/img/Plane_2D.png | Bin 99330 -> 209118 bytes
 buch/papers/parzyl/teil2.tex        |  27 ++++++++++++++++++++-------
 2 files changed, 20 insertions(+), 7 deletions(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/img/Plane_2D.png b/buch/papers/parzyl/img/Plane_2D.png
index 7c32877..f55e3cf 100644
Binary files a/buch/papers/parzyl/img/Plane_2D.png and b/buch/papers/parzyl/img/Plane_2D.png differ
diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index d37c650..f0b5c34 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -12,7 +12,7 @@ Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld
 	\centering
 	\begin{minipage}{.7\textwidth}
 		 \centering
-		\includegraphics[width=\textwidth]{papers/parzyl/img/plane.pdf}
+		\includegraphics[width=\textwidth]{papers/parzyl/images/halfplane.pdf}
 		\caption{Semi-infinite Leiterplatte}
 		\label{parzyl:fig:leiterplatte}
 	\end{minipage}%
@@ -23,11 +23,12 @@ Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld
 	\label{parzyl:fig:leiterplatte_2d}
 	\end{minipage}
 \end{figure}
-Das dies so ist kann im zwei Dimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Linie, was man in Abbildung \ref{parzyl:fig:leiterplatte_2d} sieht.
+Das dies so ist kann im Zweidimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Halbgerade, was man in Abbildung \ref{parzyl:fig:leiterplatte_2d} sieht.
+
 
 Jede komplexe Funktion $F(z)$ kann geschrieben werden als
 \begin{equation}
-	F(s) = U(x,y) + iV(x,y) \qquad s \in \mathbb{C}; x,y \in \mathbb{R}.
+	F(s) = U(x,y) + iV(x,y) \quad s = x + iy \qquad s \in \mathbb{C}; x,y \in \mathbb{R}.
 \end{equation}  
 Dabei müssen, falls die Funktion differenzierbar ist, die Cauchy-Riemann Differentialgleichungen 
 \begin{equation}
@@ -59,23 +60,31 @@ Aus dieser Bedingung folgt
 	0
 	}_{\displaystyle{\nabla^2V(x,y) = 0}}.
 \end{equation}
-Zusätzlich kann auch gezeigt werden, dass die Funktion $F(z)$ eine winkeltreue Abbildung ist. 
+Zusätzlich kann auch gezeigt werden, dass die Funktion $F(z)$ eine winkeltreue Abbildung ist.
+
+ 
 Der Zusammenhang zum elektrischen Feld ist jetzt, dass das Potential an einem quellenfreien Punkt gegeben ist als 
 \begin{equation}
 	\nabla^2\phi(x,y) = 0.
 \end{equation}
-Dies ist eine Bedingung welche differenzierbare Funktionen, wie in Gleichung \eqref{parzyl_e_feld_zweite_ab} gezeigt wird, bereits besitzen. 
+Dies ist eine Bedingung, welche differenzierbare Funktionen, wie in Gleichung \eqref{parzyl_e_feld_zweite_ab} gezeigt wird, bereits besitzen. 
+
+
 Nun kann zum Beispiel $U(x,y)$ als das Potential angeschaut werden
 \begin{equation}
 	\phi(x,y) = U(x,y).
 \end{equation}
-Orthogonal zum Potential ist das elektrische Feld
+Orthogonal zu den Äquipotenzialfläche sind die Feldlinien des elektrische Feld
 \begin{equation}
 	E(x,y) = V(x,y).
 \end{equation}
+
+
 Um nun zu den parabolische Zylinderkoordinaten zu gelangen muss nur noch eine geeignete 
 komplexe Funktion $F(s)$ gefunden werden, 
 welche eine semi-infinite Platte beschreiben kann.
+
+
 Die gesuchte Funktion in diesem Fall ist
 \begin{equation}
 	F(s) 
@@ -93,6 +102,8 @@ Dies kann umgeformt werden zu
 	i\underbrace{\sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}}_{V(x,y)}
 	.
 \end{equation}
+
+
 Die Äquipotentialflächen können nun betrachtet werden, 
 indem man die Funktion, welche das Potential beschreibt, gleich eine Konstante setzt,
 \begin{equation}
@@ -103,7 +114,9 @@ Die Flächen mit der gleichen elektrischen Feldstärke können als
 	\tau = V(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}
 \end{equation}
 beschrieben werden. Diese zwei Gleichungen zeigen nun, wie man vom 
-kartesischen Koordinatensystem ins parabolische Zylinderkoordinatensystem kommt. 
+kartesischen Koordinatensystem ins parabolische Zylinderkoordinatensystem kommt.
+
+ 
 Werden diese Formeln nun nach $x$ und $y$ aufgelöst 
 \begin{equation}
 	x = \sigma \tau,
-- 
cgit v1.2.1


From ecd2ba28a3d26f232c45df6502d6e89d6b7bd05b Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Fri, 19 Aug 2022 14:22:52 +0200
Subject: verbesserungen

---
 buch/papers/parzyl/references.bib | 9 +++++++++
 buch/papers/parzyl/teil0.tex      | 4 ++--
 2 files changed, 11 insertions(+), 2 deletions(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/references.bib b/buch/papers/parzyl/references.bib
index 390d5ed..9639d0b 100644
--- a/buch/papers/parzyl/references.bib
+++ b/buch/papers/parzyl/references.bib
@@ -65,4 +65,13 @@
 	year = {2022},
 	month = {8},
 	day = {17}
+}
+
+@online{parzyl:scalefac,
+	title = {An introduction to curvlinear orthogonal coordinates},
+	url = {http://dslavsk.sites.luc.edu/courses/phys301/classnotes/scalefactorscomplete.pdf},
+	date = {2022-08-18},
+	year = {2022},
+	month = {08},
+	day = {18}
 }
\ No newline at end of file
diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex
index 70caa05..065c077 100644
--- a/buch/papers/parzyl/teil0.tex
+++ b/buch/papers/parzyl/teil0.tex
@@ -95,12 +95,12 @@ und
     konstantes $\sigma$ und die grünen ein konstantes $\tau$.}
     \label{parzyl:fig:cordinates}
 \end{figure}
-Abbildung \ref{parzyl:fig:cordinates} zeigt das Parabolische Koordinatensystem.
+Abbildung \ref{parzyl:fig:cordinates} zeigt das parabolische Koordinatensystem.
 Das parabolische Zylinderkoordinatensystem entsteht wenn die Parabeln aus der
 Ebene gezogen werden. 
 
 Um in diesem Koordinatensystem integrieren und differenzieren zu 
-können braucht es die Skalierungsfaktoren $h_{\tau}$, $h_{\sigma}$ und $h_{z}$.
+können braucht es die Skalierungsfaktoren $h_{\tau}$, $h_{\sigma}$ und $h_{z}$ \cite{parzyl:scalefac}.
 
 Eine infinitessimal kleine Distanz $ds$ zwischen zwei Punkten
 kann im kartesischen Koordinatensystem mit
-- 
cgit v1.2.1


From 029cf2a8c2ca284e92229a3e1a1b16ed39e7dbec Mon Sep 17 00:00:00 2001
From: tschwall <55748566+tschwall@users.noreply.github.com>
Date: Fri, 19 Aug 2022 14:28:37 +0200
Subject: Updated Paper

---
 buch/papers/parzyl/teil2.tex | 1 +
 1 file changed, 1 insertion(+)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index f0b5c34..e6a7e28 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -23,6 +23,7 @@ Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld
 	\label{parzyl:fig:leiterplatte_2d}
 	\end{minipage}
 \end{figure}
+Die Äquipotentiallinien sind dabei rot dargestellt und die des elektrischen Feldes sind grün. Die semi-infinite Platte ist in blau dargestellt.
 Das dies so ist kann im Zweidimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Halbgerade, was man in Abbildung \ref{parzyl:fig:leiterplatte_2d} sieht.
 
 
-- 
cgit v1.2.1


From 8b79b30b55c44b0bbf7cc484c7eb2c5f7e273088 Mon Sep 17 00:00:00 2001
From: tschwall <55748566+tschwall@users.noreply.github.com>
Date: Fri, 19 Aug 2022 14:36:18 +0200
Subject: Updated Paper

---
 buch/papers/parzyl/teil2.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index e6a7e28..5ba9de8 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -23,7 +23,7 @@ Die parabolischen Zylinderkoordinaten tauchen auf, wenn man das elektrische Feld
 	\label{parzyl:fig:leiterplatte_2d}
 	\end{minipage}
 \end{figure}
-Die Äquipotentiallinien sind dabei rot dargestellt und die des elektrischen Feldes sind grün. Die semi-infinite Platte ist in blau dargestellt.
+Die Äquipotentiallinien sind dabei in rot ,die des elektrischen Feldes in grün und semi-infinite Platte ist in blau dargestellt.
 Das dies so ist kann im Zweidimensionalen mit Hilfe von komplexen Funktionen gezeigt werden. Die Platte ist dann nur eine Halbgerade, was man in Abbildung \ref{parzyl:fig:leiterplatte_2d} sieht.
 
 
-- 
cgit v1.2.1


From cc6b7320b8de4f36bc4a6516af87c66a108bc81c Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Fri, 19 Aug 2022 15:01:01 +0200
Subject: 1 satz

---
 buch/papers/parzyl/teil0.tex | 1 +
 1 file changed, 1 insertion(+)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex
index 065c077..f9e34d5 100644
--- a/buch/papers/parzyl/teil0.tex
+++ b/buch/papers/parzyl/teil0.tex
@@ -98,6 +98,7 @@ und
 Abbildung \ref{parzyl:fig:cordinates} zeigt das parabolische Koordinatensystem.
 Das parabolische Zylinderkoordinatensystem entsteht wenn die Parabeln aus der
 Ebene gezogen werden. 
+Die Flächen mit $\tau = 0$ oder $\sigma = 0$ stellen somit Halbebenen entlang der $z$-Achse dar.
 
 Um in diesem Koordinatensystem integrieren und differenzieren zu 
 können braucht es die Skalierungsfaktoren $h_{\tau}$, $h_{\sigma}$ und $h_{z}$ \cite{parzyl:scalefac}.
-- 
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')

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 1d82e0588b95188264168223fd0337529e88acf0 Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Sat, 20 Aug 2022 12:10:36 +0200
Subject: =?UTF-8?q?koordinatensystem=20=C3=A4nderungen?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 buch/papers/parzyl/img/coordinates.png | Bin 0 -> 1215422 bytes
 buch/papers/parzyl/teil0.tex           |  20 ++++++++++----------
 buch/papers/parzyl/teil2.tex           |  24 +++++++++++++++++-------
 3 files changed, 27 insertions(+), 17 deletions(-)
 create mode 100644 buch/papers/parzyl/img/coordinates.png

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/img/coordinates.png b/buch/papers/parzyl/img/coordinates.png
new file mode 100644
index 0000000..0ea3701
Binary files /dev/null and b/buch/papers/parzyl/img/coordinates.png differ
diff --git a/buch/papers/parzyl/teil0.tex b/buch/papers/parzyl/teil0.tex
index f9e34d5..3bf9257 100644
--- a/buch/papers/parzyl/teil0.tex
+++ b/buch/papers/parzyl/teil0.tex
@@ -73,26 +73,26 @@ Das parabolischen Zylinderkoordinatensystem \cite{parzyl:coordinates} ist ein kr
 bei dem parabolische Zylinder die Koordinatenflächen bilden.
 Die Koordinate $(\sigma, \tau, z)$ sind in kartesischen Koordinaten ausgedrückt mit
 \begin{align}
-    x & = \sigma \tau \\
+    x & = \frac{1}{2}\left(\tau^2 - \sigma^2\right) \\
     \label{parzyl:coordRelationsa}
-    y & = \frac{1}{2}\left(\tau^2 - \sigma^2\right) \\
+    y & =  \sigma \tau\\
     z & = z.
     \label{parzyl:coordRelationse}
 \end{align}
-Wird $\tau$ oder $\sigma$ konstant gesetzt, resultieren die Parabeln
+Wird $\sigma$ oder $\tau$ konstant gesetzt, resultieren die Parabeln
 \begin{equation}
-    y = \frac{1}{2} \left( \frac{x^2}{\sigma^2} - \sigma^2 \right)
+    x = \frac{1}{2} \left( \frac{y^2}{\sigma^2} - \sigma^2 \right)
 \end{equation}
 und 
 \begin{equation}
-    y = \frac{1}{2} \left( -\frac{x^2}{\tau^2} + \tau^2 \right).
+    x = \frac{1}{2} \left( -\frac{y^2}{\tau^2} + \tau^2 \right).
 \end{equation}
 
 \begin{figure}
     \centering
-    \includegraphics[scale=0.4]{papers/parzyl/img/koordinaten.png}
-    \caption{Das parabolische Koordinatensystem. Die roten Parabeln haben ein 
-    konstantes $\sigma$ und die grünen ein konstantes $\tau$.}
+    \includegraphics[scale=0.32]{papers/parzyl/img/coordinates.png}
+    \caption{Das parabolische Koordinatensystem. Die grünen Parabeln haben ein 
+    konstantes $\sigma$ und die roten ein konstantes $\tau$.}
     \label{parzyl:fig:cordinates}
 \end{figure}
 Abbildung \ref{parzyl:fig:cordinates} zeigt das parabolische Koordinatensystem.
@@ -124,11 +124,11 @@ von \eqref{parzyl:coordRelationsa} - \eqref{parzyl:coordRelationse} als
     dx  &= \frac{\partial x }{\partial \sigma} d\sigma + 
         \frac{\partial x }{\partial \tau} d\tau + 
         \frac{\partial x }{\partial \tilde{z}} d \tilde{z} 
-        = \tau d\sigma + \sigma d \tau \\
+        = \tau d\tau - \sigma d \sigma \\
     dy &= \frac{\partial y }{\partial \sigma} d\sigma + 
         \frac{\partial y }{\partial \tau} d\tau +
         \frac{\partial y }{\partial \tilde{z}} d \tilde{z} 
-        = \tau d\tau - \sigma d \sigma \\
+        = \tau d\sigma + \sigma d \tau  \\
     dz &= \frac{\partial \tilde{z} }{\partial \sigma} d\sigma + 
         \frac{\partial \tilde{z} }{\partial \tau} d\tau +
         \frac{\partial \tilde{z} }{\partial \tilde{z}} d \tilde{z} 
diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index 5ba9de8..fbe5711 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -108,21 +108,31 @@ Dies kann umgeformt werden zu
 Die Äquipotentialflächen können nun betrachtet werden, 
 indem man die Funktion, welche das Potential beschreibt, gleich eine Konstante setzt,
 \begin{equation}
-	\sigma = U(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} + x}{2}}.
+%	\sigma = U(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} + x}{2}}.
+	c_1 = U(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} + x}{2}}.
 \end{equation}
 Die Flächen mit der gleichen elektrischen Feldstärke können als
 \begin{equation}
-	\tau = V(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}
+%	\tau = V(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}
+	c_2 = V(x,y) = \sqrt{\frac{\sqrt{x^2+y^2} - x}{2}}
 \end{equation}
 beschrieben werden. Diese zwei Gleichungen zeigen nun, wie man vom 
 kartesischen Koordinatensystem ins parabolische Zylinderkoordinatensystem kommt.
-
- 
+%Werden diese Formeln nun nach $x$ und $y$ aufgelöst 
+%\begin{equation}
+%	x = \sigma \tau,
+%\end{equation}
+%\begin{equation}
+%	y = \frac{1}{2}\left ( \tau^2 - \sigma^2 \right ),
+%\end{equation}
+%so beschreibe sie, wie man aus dem parabolischen Zylinderkoordinatensystem zurück ins kartesische rechnen kann.
 Werden diese Formeln nun nach $x$ und $y$ aufgelöst 
 \begin{equation}
-	x = \sigma \tau,
+	x =  c_1^2 - c_2^2 ,
 \end{equation}
 \begin{equation}
-	y = \frac{1}{2}\left ( \tau^2 - \sigma^2 \right ),
+	y = 2c_1 c_2,
 \end{equation}
-so beschreibe sie, wie man aus dem parabolischen Zylinderkoordinatensystem zurück ins kartesische rechnen kann.
\ No newline at end of file
+so beschreibe sie mit $c_1 = \tau \sqrt{2}$ und $c_2 = \sigma \sqrt{2}$ die Beziehung 
+zwischen dem parabolischen Zylinderkoordinatensystem und dem kartesischen Koordinatensystem.
+
-- 
cgit v1.2.1


From db498caa668ca8d0fa3e51de0f3668a47325d2e5 Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Sat, 20 Aug 2022 12:16:00 +0200
Subject: align

---
 buch/papers/parzyl/teil2.tex | 12 +++++-------
 1 file changed, 5 insertions(+), 7 deletions(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index fbe5711..ab0e971 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -127,12 +127,10 @@ kartesischen Koordinatensystem ins parabolische Zylinderkoordinatensystem kommt.
 %\end{equation}
 %so beschreibe sie, wie man aus dem parabolischen Zylinderkoordinatensystem zurück ins kartesische rechnen kann.
 Werden diese Formeln nun nach $x$ und $y$ aufgelöst 
-\begin{equation}
-	x =  c_1^2 - c_2^2 ,
-\end{equation}
-\begin{equation}
-	y = 2c_1 c_2,
-\end{equation}
-so beschreibe sie mit $c_1 = \tau \sqrt{2}$ und $c_2 = \sigma \sqrt{2}$ die Beziehung 
+\begin{align}
+	x &=  c_1^2 - c_2^2 ,\\
+	y &= 2c_1 c_2,
+\end{align}
+so beschreiben sie mit $c_1 = \tau \sqrt{2}$ und $c_2 = \sigma \sqrt{2}$ die Beziehung 
 zwischen dem parabolischen Zylinderkoordinatensystem und dem kartesischen Koordinatensystem.
 
-- 
cgit v1.2.1


From d2cb5860b83b16ef466a23473e8366a686ce20f7 Mon Sep 17 00:00:00 2001
From: Alain <mceagle117@gmail.com>
Date: Sat, 20 Aug 2022 12:23:30 +0200
Subject: jetz aber

---
 buch/papers/parzyl/teil2.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

(limited to 'buch/papers')

diff --git a/buch/papers/parzyl/teil2.tex b/buch/papers/parzyl/teil2.tex
index ab0e971..0cf4283 100644
--- a/buch/papers/parzyl/teil2.tex
+++ b/buch/papers/parzyl/teil2.tex
@@ -131,6 +131,6 @@ Werden diese Formeln nun nach $x$ und $y$ aufgelöst
 	x &=  c_1^2 - c_2^2 ,\\
 	y &= 2c_1 c_2,
 \end{align}
-so beschreiben sie mit $c_1 = \tau \sqrt{2}$ und $c_2 = \sigma \sqrt{2}$ die Beziehung 
+so beschreiben sie mit $\tau = c_1 \sqrt{2}$ und $\sigma = c_2 \sqrt{2}$ die Beziehung 
 zwischen dem parabolischen Zylinderkoordinatensystem und dem kartesischen Koordinatensystem.
 
-- 
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')

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')

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')

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')

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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