Merge pull request #77 from NaoPross/master

Feedback and 1st application (EEG)

11 files changed, 830 insertions, 265 deletions

diff --git a/buch/papers/kugel/applications.tex b/buch/papers/kugel/applications.tex
index b2f227e..481a3a5 100644
--- a/buch/papers/kugel/applications.tex
+++ b/buch/papers/kugel/applications.tex
@@ -1,9 +1,206 @@
-% vim:ts=2 sw=2 et spell:
+% vim:ts=2 sw=2 et spell tw=80:
 
 \section{Applications}
 
-\subsection{Electroencephalography (EEG)}
+As suggested in the previous section, the fact that it is possible to write a
+Fourier style expansion of any function on the surface of the sphere is very
+useful in many fields of physics and engineering. Here we will present a few of
+the most interesting applications we came across during our research.
+
+\subsection{Electroencephalography}
+
+\begin{figure}
+  \centering
+  \subfigure[EEG Electrodes \label{kugel:fig:eeg-electrodes}]%
+    % {\kugelplaceholderfig{.4\linewidth}{5cm}}
+    {\includegraphics[width=.45\linewidth, frame]{papers/kugel/figures/electrodes}}
+  \qquad
+  \subfigure[Gauss' Law \label{kugel:fig:eeg-flux}]%
+    {\includegraphics[width=.4\linewidth]{papers/kugel/figures/flux}}
+  \caption{
+    Electroencephalography.
+    \label{kugel:fig:eeg}
+  }
+\end{figure}
+
+To start, we will look at an application that is from the field of medicine:
+electroencephalography. The \emph{electroencephalogram} (EEG) is a measurement
+of the electrical field on the scalp, which shows the brain's activity, and is
+used in many fields of research such as neurology and cognitive psychology.  The
+measurement is done by wearing a cap that contains a number of evenly
+distributed electrodes, each of which measures the electric potential (voltage)
+at their location (figure \ref{kugel:fig:eeg-electrodes}).  To see how this will
+relate to the spherical harmonics, we will first quickly recap a bit of physics,
+electrodynamics to be precise.
+
+\subsubsection{Electrodynamics}
+
+In section \ref{kugel:sec:construction:eigenvalue} we have shown that the
+spherical harmonics arise from the surface spherical Laplacian operator, whose
+origin we did not consider too much, which is how mathematicians do their work.
+On the contrary, physicists usually do the opposite and start by discussing what
+is happening in the real world, since variables represent physical quantities.
+So, we will quickly remind that the Laplacian operator does the following to an
+electric potential $\phi(x, y, z)$:
+\begin{equation*}
+  \nabla^2 \phi
+  = \nabla \cdot \nabla \phi
+  = \nabla \cdot \mathbf{E}
+  = \rho / \varepsilon,
+  \quad \text{or} \quad
+  \iiint_\Omega \nabla \cdot \mathbf{E} \, dv
+  = \iint_{\partial \Omega} \mathbf{E} \cdot d\mathbf{s}
+  = \Phi / \varepsilon.
+\end{equation*}
+Put into words: on the left we have the differential form, where we recall that
+the Laplacian (which is a second derivative) is the divergence of the gradient.
+Unpacking the notation we first see that we have the gradient of the potential,
+which is just the electric field $\mathbf{E}$, and then the divergence of said
+electric field is proportional to the charge density $\rho$. So, the Laplacian
+of the electric potential is the charge density! For those that are more
+familiar with the integral form of Maxwell's equation, we have also included an
+additional step using the divergence theorem, which brings us to the electric
+Flux, which by Gauss' law (shown in the iconic\footnote{Every electrical
+engineer has seen this picture so many times that is probably burnt in their
+eyes.} figure \ref{kugel:fig:eeg-flux}) equals the net electric charge.
+
+Now, an important observation is that if we switch to spherical coordinates, the
+physics does not change. So, the spherical Laplacian $\sphlaplacian$ of the
+electric potential $\phi(r, \vartheta, \varphi)$ is still the charge density (in
+spherical coordinates). And what about the surface spherical Laplacian
+$\surflaplacian$? To that case the physics is also indifferent, the only change
+is that the units result is a \emph{surface} charge density $\rho_s$. Thus, we
+are done with physics and finally arrive at the engineers' perspective: how can
+we use this fact to build something that reads the current flows on the surface
+of the brain?
+
+\subsubsection{EEG as Interpolation Problem}
+
+The details of how EEG actually works gets very complicated very quickly, but we
+will try our best to give an broad overview of the mathematical machinery that
+makes it possible to measure brain waves. The problem neither the physicist nor
+the mathematician considered is that we cannot measure the electric field in its
+entirety. As show in figure \ref{kugel:fig:eeg-electrodes} the electrodes give
+measurements that are only available at discrete locations, and we are thus
+missing quite a lot of data. Or in other words, we have an interpolation
+problem, which (at this point not so surprisingly) we will show can be solved
+using the spherical harmonics.
+
+To solve this new interpolation problem, we will start with a blatantly
+engineering assumption: the human head is a sphere of radius $R$, with the value
+of $R$ being the average radius of a human head (which is around 11 cm). So, we
+will assume that the potential distribution on the head can be written as a
+finite linear combination of spherical harmonics:
+\begin{equation*}
+  V(\vartheta, \varphi)
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n} Y^m_n(\vartheta, \varphi),
+\end{equation*}
+where the values $a_{m,n}$ are the unknowns of our interpolation problem. Now to
+the measurements: we let $\phi_1, \phi_2, \ldots, p_M$ be the measured voltages
+at points in space $p_1, p_2, \ldots, p_M$ (position of the electrodes). To
+simplify, we will assume that the electrodes are reasonably evenly distributed,
+which means that we have no points that are on top of each other or at wildly
+different radii from the origin. With that out of the way, we can now write a
+minimization problem:
+\begin{subequations}
+  \begin{align}
+    a_{m,n}^* &= \arg \min_{a_{m,n}}
+      \int_{\partial S} | \surflaplacian V |^2 \, ds 
+      = \int_0^{2\pi} \int_{0}^\pi | \surflaplacian V |^2
+        \sin \vartheta \, d\vartheta d\varphi, 
+        \label{kugel:eqn:eeg-min} \\
+    &\text{under the constraints} \quad V(p_j) = \phi_j
+      \quad \text{ for } \quad 1 \leq j \leq M.
+      \label{kugel:eqn:eeg-min-constraints}
+  \end{align}
+\end{subequations}
+Essentially, with \eqref{kugel:eqn:eeg-min} we are are asking for the solution
+to be smooth by minimizing the square of the total curvature (recall that the
+surface spherical Laplacian $\surflaplacian$ is a measure of curvature), while
+at the same time with \eqref{kugel:eqn:eeg-min-constraints}, we force the
+solution to go through the measured points. The latter is the reason why we
+needed to assumed that the measurements are at reasonable locations, something
+that (as every engineer show know) is not necessarily the case in the real
+world! Thus, to solve this problem, we will use the suspiciously convenient fact
+that (hint: eigenvalues)
+\begin{equation*}
+  \surflaplacian V(\vartheta, \varphi)
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n}
+      \surflaplacian Y^m_n(\vartheta, \varphi)
+    = \sum_{n=1}^N \sum_{m=-n}^n a_{m,n}
+      n(n+1) Y^m_n(\vartheta, \varphi).
+\end{equation*}
+So that when substituted into \eqref{kugel:eqn:eeg-min} results in
+\begin{align*}
+  \int_{\partial S} \left|
+    \sum_{n=1}^N \sum_{m=-n}^n n(n+1) a_{m,n}
+    Y^m_n(\vartheta, \varphi)
+  \right|^2 ds
+  = \sum_{m, m'} \sum_{n, n'} a_{m',n'} \overline{a_{m,n}}
+    n'(n'+1) n(n+1)
+    \underbrace{\int_{\partial S} Y^{m'}_{n'} \overline{Y^m_n} \, ds}_{
+      \langle Y^{m'}_{n'}, Y^m_n \rangle
+    },
+\end{align*}
+where we used a ``sloppy'' double sum notation to indicate that we have a bunch
+of terms of that form. We did not bother to properly expand the product of
+double sums, because we can see that at the end we end up with an inner product
+$\langle Y^{m'}_{n'}, Y^m_n \rangle$, which as we know equals $\delta_{m'm}
+\delta_{n'n}$, so all of the terms where $n' \neq n$ or $m' \neq m$ can be
+dropped and \eqref{kugel:eqn:eeg-min} simplifies down to
+\nocite{pascual-marqui_current_1988}
+\begin{equation}
+  a^*_{m,n} = \arg \min_{a_{m,n}} 
+    \sum_{n=1}^N \sum_{m=-n}^n n^2 (n+1)^2 |a_{m,n}|^2.
+\end{equation}
+
+At this point, we could continue solving for an analytical solution to the
+minimization problem, for example by differentiating with respect to some
+$a_{j,k}$, setting that to zero and so forth, but the job of the spherical
+harmonics ends here. So, we will not pursue this further, and instead briefly
+discuss a few interesting implications and problems. 
+
+\subsubsection{Sampling, Smoothness and Problems}
+\nocite{wingeier_spherical_2001, ruffini_spherical_2002}
+
+The most interesting perhaps unforeseen fact is that with this method we are
+getting a free (!) spectral analysis, since the coefficients $a_{m,n}$ are the
+spectrum of the interpolated electric field $V(\vartheta, \varphi)$. However,
+like in the non spherical Fourier transformation, we only get a \emph{finite}
+resolution since our measurement are spatially discrete. In fact, if we know the
+mean angular inter-electrode distance $\gamma$ we can actually formulate a
+Nyquist frequency just like in the usual Fourier theory:
+\begin{equation}
+  f_N = \frac{\pi}{2T}
+  \iff
+  n_N = \left\lfloor \frac{\pi}{2\gamma} \right\rfloor.
+\end{equation}
+
+Before concluding this overview of EEG, we should point out that in practice
+there are about a million problems with this oversimplified approach. We do not
+intend to give an in depth explanation (since the authors themselves are not
+experts in any of these fields), but there are a few problems that are too big
+to ignore, so we will very briefly discuss them now. The first important
+real-world problem is that the electrodes are not necessarily at a reasonable
+location, so the constraint \eqref{kugel:eqn:eeg-min-constraints} is a bit too
+strong, and may end up fitting some noise or disturbances in the measurement. A
+simple solution may for example be to introduce a smoothness factor $\lambda >
+0$ as follows:
+\begin{equation}
+  V(\vartheta, \varphi) = \sum_{n=1}^N \sum_{m=-n}^n 
+    \frac{a_{m,n}}{1 + \lambda n^2(n+1)^2} Y^m_n(\vartheta, \varphi).
+\end{equation}
+To find proper smoothness factor $\lambda$, is another problem of its own, thus
+we will not discuss it here, since this is getting too long already. Another
+important issue is that in the real world, we cannot ``evenly distribute'' the
+electrodes on our head. As shown in the image, most of the electrodes are on a
+cap, and then there are just a few on the face, and almost none near the jawline
+and chin. This not something that can be ignored, and in fact, makes the
+analysis much more difficult. Finally, the most obvious problem is that human
+heads are not perfect spheres. Here too, it is possible to account for this fact
+and model the head with a more complex shape at the cost of making the math
+quite unwieldy.
 
 \subsection{Measuring Gravitational Fields}
 
-\subsection{Quantisation of Angular Momentum}
+% \subsection{Quantisation of Angular Momentum}
diff --git a/buch/papers/kugel/figures/electrodes.jpg b/buch/papers/kugel/figures/electrodes.jpg
new file mode 100644
index 0000000..6c15de4
--- /dev/null
+++ b/buch/papers/kugel/figures/electrodes.jpg
diff --git a/buch/papers/kugel/figures/tikz/Makefile b/buch/papers/kugel/figures/tikz/Makefile
index 4ec4e5a..3b2df59 100644
--- a/buch/papers/kugel/figures/tikz/Makefile
+++ b/buch/papers/kugel/figures/tikz/Makefile
@@ -1,4 +1,4 @@
-FIGURES := spherical-coordinates.pdf curvature-1d.pdf
+FIGURES := spherical-coordinates.pdf curvature-1d.pdf legendre-substitution.pdf
 
 all: $(FIGURES)
 
diff --git a/buch/papers/kugel/figures/tikz/legendre-substitution.pdf b/buch/papers/kugel/figures/tikz/legendre-substitution.pdf
new file mode 100644
index 0000000..f77b6cb
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/legendre-substitution.pdf
diff --git a/buch/papers/kugel/figures/tikz/legendre-substitution.tex b/buch/papers/kugel/figures/tikz/legendre-substitution.tex
new file mode 100644
index 0000000..3a699b8
--- /dev/null
+++ b/buch/papers/kugel/figures/tikz/legendre-substitution.tex
@@ -0,0 +1,69 @@
+
+% vim: ts=2 sw=2 et :
+\documentclass[tikz, border=2mm]{standalone}
+
+\usepackage{amsmath}
+\usepackage{bm}
+
+\usepackage{times}
+\usepackage{txfonts}
+
+\usepackage{tikz-3dplot}
+
+\tdplotsetmaincoords{60}{130}
+\pgfmathsetmacro{\l}{2}
+
+\begin{document}
+  \begin{tikzpicture}[
+      >=latex,
+      tdplot_main_coords,
+    ]
+
+    % origin and poles
+    \coordinate (O) at (0,0,0);
+    \coordinate (NP) at (0,0,\l);
+    \coordinate (SP) at (0,0,-\l);
+
+    % gray unit circle
+    \tdplotdrawarc[lightgray, dashed]{(O)}{\l}{0}{360}{}{};
+    \draw[lightgray, dashed] (-\l, 0, 0) to (\l, 0, 0);
+    \draw[lightgray, dashed] (0, -\l, 0) to (0, \l, 0);
+
+    % axis
+    \draw[->] (O) -- ++(0,0,1.25*\l) node[above] {\(\mathbf{\hat{z}}\)};
+
+    % meridians
+    \foreach \phi in {0, 30, 60, ..., 150}{
+      \tdplotsetrotatedcoords{\phi}{90}{0};
+      \tdplotdrawarc[lightgray, densely dotted, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+    }
+
+    % dot above and its projection
+    \pgfmathsetmacro{\phi}{120}
+    \pgfmathsetmacro{\theta}{40}
+
+    \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l}
+    \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l}
+    \pgfmathsetmacro{\pz}{cos(\theta)*\l})
+
+    % Special meridian
+    \tdplotsetrotatedcoords{\phi-90}{90}{0};
+    \tdplotdrawarc[gray, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{};
+    
+    % point A
+    \coordinate (A) at (\px,\py,\pz);
+    \coordinate (Ap) at (\px,\py, 0);
+    
+    % lines
+    \draw[red!80!black, thick, ->] (O) -- (A);
+    \draw[red!80!black]
+      (O) -- node[midway, below, font=\small, sloped, fill=white] {$\sqrt{1 - z^2}$}
+      (Ap) -- node[midway, right, font=\small, sloped, fill=white, anchor=north] {$z = \cos \vartheta$}
+      (A) node[above right, fill=white] {$r = 1$};
+    
+    % theta arc
+    \tdplotsetrotatedcoords{\phi-90}{-90}{0};
+    \tdplotdrawarc[blue!80!black, ->, tdplot_rotated_coords]{(O)}{.95\l}{0}{\theta}{}{};
+    \node[above right = 1mm, blue!80!black] at (0,0,.8\l) {\(\bm{\hat{\vartheta}}\)};
+  \end{tikzpicture}
+\end{document}
diff --git a/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf b/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf
index 1bff016..67c7ea8 100644
--- a/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf
+++ b/buch/papers/kugel/figures/tikz/spherical-coordinates.pdf
diff --git a/buch/papers/kugel/figures/tikz/spherical-coordinates.tex b/buch/papers/kugel/figures/tikz/spherical-coordinates.tex
index 3a45385..d4e5088 100644
--- a/buch/papers/kugel/figures/tikz/spherical-coordinates.tex
+++ b/buch/papers/kugel/figures/tikz/spherical-coordinates.tex
@@ -2,9 +2,12 @@
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{bm}
-\usepackage{lmodern}
 \usepackage{tikz-3dplot}
 
+% \usepackage{lmodern}
+\usepackage{times}
+\usepackage{txfonts}
+
 \usetikzlibrary{arrows}
 \usetikzlibrary{intersections}
 \usetikzlibrary{math}
diff --git a/buch/papers/kugel/packages.tex b/buch/papers/kugel/packages.tex
index c02589f..3694ba3 100644
--- a/buch/papers/kugel/packages.tex
+++ b/buch/papers/kugel/packages.tex
@@ -9,6 +9,7 @@
 % following example
 %\usepackage{packagename}
 \usepackage{cases}
+\usepackage[export]{adjustbox}
 
 \newcommand{\kugeltodo}[1]{\textcolor{red!70!black}{\texttt{[TODO: #1]}}}
 \newcommand{\kugelplaceholderfig}[2]{ \begin{tikzpicture}%
diff --git a/buch/papers/kugel/preliminaries.tex b/buch/papers/kugel/preliminaries.tex
index 1fa78d7..c4c5cae 100644
--- a/buch/papers/kugel/preliminaries.tex
+++ b/buch/papers/kugel/preliminaries.tex
@@ -288,7 +288,7 @@ way that from now on we will not have to worry about the details of
 convergence.
 
 \begin{lemma}
-  \label{kugel:lemma:exp-1d}
+  \label{kugel:thm:exp-1d}
   The set of functions \(E_n(x) = e^{i2\pi nx}\) on the interval
   \([0; 1)\) with \(n \in \mathbb{Z} \) are orthonormal.
 \end{lemma}
@@ -318,7 +318,7 @@ convergence.
 \end{definition}
 
 \begin{theorem}[Fourier Theorem]
-  \label{fourier-theorem-1D}
+  \label{kugel:thm:fourier-theorem}
   \begin{equation*}
     \lim_{N \to \infty} \left \|
       f(x) - \sum_{n = -N}^N \hat{f}(n) E_n(x) 
@@ -331,7 +331,7 @@ convergence.
   on the square \([0; 1)^2\) with \(m, n \in \mathbb{Z} \) are orthonormal.
 \end{lemma}
 \begin{proof}
-  The proof is almost identical to lemma \ref{kugel:lemma:exp-1d}, with the
+  The proof is almost identical to lemma \ref{kugel:thm:exp-1d}, with the
   only difference that the inner product is given by
   \[
     \langle E_{m,n}, E_{m', n'} \rangle
diff --git a/buch/papers/kugel/references.bib b/buch/papers/kugel/references.bib
index 984d555..07e4d5d 100644
--- a/buch/papers/kugel/references.bib
+++ b/buch/papers/kugel/references.bib
@@ -17,15 +17,6 @@
 	file = {Submitted Version:/Users/npross/Zotero/storage/SN4YUNQC/Carvalhaes and de Barros - 2015 - The surface Laplacian technique in EEG Theory and.pdf:application/pdf},
 }
 
-@article{usecase_recursion_paper,
-	title = {New Implementation of Legendre Polynomials for Solving Partial Differential Equations},
-	issn = {272767969},
-	url = {https://www.researchgate.net/publication/272767969_New_Implementation_of_Legendre_Polynomials_for_Solving_Partial_Differential_Equations},
-	shorttitle = {Implementation og Legendre Polynom},
-	date = {2013-12},
-	author = {Ali Davari, Abozar Ahmadi}
-}
-
 @video{minutephysics_better_2021,
 	title = {A Better Way To Picture Atoms},
 	url = {https://www.youtube.com/watch?v=W2Xb2GFK2yc},
@@ -212,4 +203,143 @@ Created by Henry Reich},
 	publisher = {Dover Publ},
 	author = {Bell, William Wallace},
 	date = {2004},
+}
+
+@article{winch_geomagnetism_2005,
+	title = {Geomagnetism and Schmidt quasi-normalization},
+	volume = {160},
+	issn = {0956-540X},
+	url = {https://doi.org/10.1111/j.1365-246X.2004.02472.x},
+	doi = {10.1111/j.1365-246X.2004.02472.x},
+	abstract = {Spherical harmonic analysis of the main magnetic field of the Earth and its daily variations is the numerical determination of coefficients of solid spherical harmonics in the mathematical expressions used for the magnetic scalar potential of fields of internal and external origin. The coefficients are determined from vector components of the field and their purpose is to represent the vector field, not to reconstruct the magnetic scalar potential. An alternative interpretation of the spherical harmonic analysis is presented: namely the determination of the coefficients of a series representation of the magnetic vector field on a spherical surface in orthonormal real vector spherical harmonics, which correspond to the internal and external fields, and an additional non-potential toroidal field. The numerical values of the coefficients of an orthonormal vector spherical harmonic series have a direct physical significance, which is not obscured by some arbitrary normalization of the vector spherical harmonics. Therefore, we propose a Schmidt vector normalization to be used in conjunction with the Schmidt quasi-normalization of associated Legendre functions. A property of orthonormalized functions is that the standard deviations of the coefficients determined by the method of least squares from ideal data, which are uniformly accurate and uniformly globally distributed, are constant for all coefficients. The real vector spherical harmonic analysis of the geomagnetic field is extended to a spherical shell and conditions that restrict the radial dependence of the vector spherical harmonic coefficients are examined. In particular, two hypotheses for the current systems deriving from the non-potential toroidal component of the magnetic field over the surface of a sphere are presented, namely, Earth-air currents and field-aligned currents.},
+	pages = {487--504},
+	number = {2},
+	journaltitle = {Geophysical Journal International},
+	shortjournal = {Geophysical Journal International},
+	author = {Winch, D. E. and Ivers, D. J. and Turner, J. P. R. and Stening, R. J.},
+	urldate = {2022-08-19},
+	date = {2005-02-01},
+	file = {Full Text PDF:/Users/npross/Zotero/storage/KIJMWLLU/Winch et al. - 2005 - Geomagnetism and Schmidt quasi-normalization.pdf:application/pdf;Snapshot:/Users/npross/Zotero/storage/7ZIZYET9/659348.html:text/html},
+}
+
+@article{wingeier_spherical_2001,
+	title = {Spherical harmonic decomposition applied to spatial-temporal analysis of human high-density electroencephalogram},
+	volume = {64},
+	issn = {1063-651X, 1095-3787},
+	url = {https://link.aps.org/doi/10.1103/PhysRevE.64.051916},
+	doi = {10.1103/PhysRevE.64.051916},
+	pages = {051916},
+	number = {5},
+	journaltitle = {Physical Review E},
+	shortjournal = {Phys. Rev. E},
+	author = {Wingeier, B. M. and Nunez, P. L. and Silberstein, R. B.},
+	urldate = {2022-08-27},
+	date = {2001-10-26},
+	langid = {english},
+	file = {Submitted Version:/Users/npross/Zotero/storage/M6C7G2TZ/Wingeier et al. - 2001 - Spherical harmonic decomposition applied to spatia.pdf:application/pdf},
+}
+
+@artwork{sheerman-chase_volunteer_2012,
+	title = {"Volunteer Duty" Psychology Testing},
+	rights = {Attribution License},
+	url = {https://www.flickr.com/photos/tim_uk/8135755109/},
+	abstract = {Photo by Chris Hope
+
+{ASTROPHYSICIST} We have a very advanced program, something very different, [...]
+{MICROBIOLOGIST} An opportunity to reduce your sentence considerably...
+[...]
+{ASTROPHYSICIST} For a man in your position...an opportunity.
+{BOTANIST} Not to volunteer could be a real mistake.
+{MICROBIOLOGIST} (tapping his pencil again) Definitely a mistake! 
+{\textless}a href="http://scifiscripts.com/scripts/twelvemonkeys.txt" rel="noreferrer nofollow"{\textgreater}scifiscripts.com/scripts/twelvemonkeys.txt{\textless}/a{\textgreater}
+
+This photo was taken during studies that resulted in the publication: Hope, C, Sterr, A, Elangovan, P, Geades, N, Windridge, D, Young, K, Wells, K, Abbey, {CK} and {MelloThoms}, {CR} (2013) High Throughput Screening for Mammography using a Human-Computer Interface with Rapid Serial Visual Presentation ({RSVP}) Medical Imaging 2013: Image Perception, Observer Performance, And Technology Assessment, 8673. ? - ?. {ISSN} 0277-786X 
+
+{\textless}a href="http://epubs.surrey.ac.uk/804796/" rel="noreferrer nofollow"{\textgreater}epubs.surrey.ac.uk/804796/{\textless}/a{\textgreater}},
+	author = {Sheerman-Chase, Tim},
+	urldate = {2022-08-27},
+	date = {2012-10-29},
+	keywords = {12, brain, eeg, electroencephalography, mad, monkeys, psychology, research, science, surrey, university},
+	file = {"Volunteer Duty" Psychology Testing:/Users/npross/Zotero/storage/UXSB3HYM/Sheerman-Chase - 2012 - Volunteer Duty Psychology Testing.jpg:image/jpg},
+}
+
+@software{markwieczorek_shtoolsshtools_2022,
+	title = {{SHTOOLS}/{SHTOOLS}: v4.10},
+	rights = {Open Access},
+	url = {https://zenodo.org/record/6497293},
+	shorttitle = {{SHTOOLS}/{SHTOOLS}},
+	abstract = {Version 4.10 {\textless}strong{\textgreater}Enhancements{\textless}/strong{\textgreater} Change the preferred backend from 'shtools' to 'ducc'. Improve handing of switching backends. Changing backends will dynamically link the correct routine to the top level modules found in {\textless}code{\textgreater}pyshtools.expand{\textless}/code{\textgreater} and {\textless}code{\textgreater}pyshools.rotate{\textless}/code{\textgreater}. Update python examples so that they don't call routines directly in the shtools backend. [See deprecation note below.] Add historical lunar topography dataset {GLTM}-2B. Add historical martian magnetic field models {FSU}50 and {FSU}90. Add new Mars gravity model {MRO}120F as well as several historical Mars gravity models. Add historical Venus topography datasets {SHTJV}360A01 and {SHTJV}360A02. Add Thebault2021 Earth magnetic field dataset. Add Mars topography dataset {MarsTopo}719, which is a truncated version of {MarsTopo}2600. Update urls for databases hosted at {GSFC}. Reorder optional arguments in docs for {\textless}code{\textgreater}makegravgradgriddh{\textless}/code{\textgreater} and {\textless}code{\textgreater}makemaggravgradgrid{\textless}/code{\textgreater} for consistency with code. Allow 'shtools' and 'dov' file formats to contain floats for degree and order. Minor changes and enhancements to the documentation. {\textless}strong{\textgreater}Bug fixes{\textless}/strong{\textgreater} Fix typo regarding {\textless}code{\textgreater}nthreads{\textless}/code{\textgreater} in {\textless}code{\textgreater}{SHMagCoeffs}.rotate(){\textless}/code{\textgreater} method. Fix bug with {\textless}code{\textgreater}{SHGravCoeffs}.admittance(){\textless}/code{\textgreater} when using {\textless}code{\textgreater}function='geoid'{\textless}/code{\textgreater}. Fix bug in python wrapper of the routine {\textless}code{\textgreater}{MakeGrid}2D{\textless}/code{\textgreater} concerning the mandatory variable {\textless}code{\textgreater}interval{\textless}/code{\textgreater}. Add workaround to use pygmt with shading for versions \>=0.4. Convert all grids to {\textless}code{\textgreater}float{\textless}/code{\textgreater} before using the {\textless}code{\textgreater}ducc{\textless}/code{\textgreater} backend. {\textless}code{\textgreater}{SHGeoid}.to\_netcdf(){\textless}/code{\textgreater} now outputs double precision by default (consistent with the other grid classes). Fix bug with {\textless}code{\textgreater}{SHWindow}.multitaper\_cross\_spectrum(){\textless}/code{\textgreater} when using arbitrary localization regions. Fix bug with the c-wrapper for {\textless}code{\textgreater}{cMakeGradientDH}{\textless}/code{\textgreater} regarding the optional {\textless}code{\textgreater}radius{\textless}/code{\textgreater} parameter. Minor changes to remove deprecation warnings. Fixed {\textless}code{\textgreater}setup.py{\textless}/code{\textgreater} to work with setuptools 62.0.0 and 62.1.0. {\textless}strong{\textgreater}Note:{\textless}/strong{\textgreater} The module {\textless}code{\textgreater}pyshtools.shtools{\textless}/code{\textgreater} will be deprecated in the v4.11 release. This module represents 1 of 2 possible backends for pyshtools, and will henceforth be located at {\textless}code{\textgreater}pyshtools.backends.shtools{\textless}/code{\textgreater}. Unless explicitly required, the user should avoid using the {\textless}code{\textgreater}backends{\textless}/code{\textgreater} modules directly, and should instead call the routines that are located in the top level modules such as {\textless}code{\textgreater}pyshtools.expand{\textless}/code{\textgreater}. Setting the backend by use of the routine {\textless}code{\textgreater}pyshtools.backends.selected\_preferred\_backend(){\textless}/code{\textgreater} determines which backed to use when calling the routines in the top level modules. M. A. Wieczorek, M. Meschede, T. Brugere, A. Corbin, A. Hattori, K. Leinweber, I. Oshchepkov, M. Reinecke, E. Sales de Andrade, E. Schnetter, S. Schröder, A. Vasishta, A. Walker, B. Xu, J. Sierra (2022). {SHTOOLS}: Version 4.10, Zenodo, doi:10.5281/zenodo.592762},
+	version = {v4.10},
+	publisher = {Zenodo},
+	author = {Markwieczorek and Meschede, Matthias and Mreineck and Oshchepkov, Ilya and De Andrade, Elliott Sales and Corbin, Armin and Xoviat and Benda Xu and Schröder, Stefan and Hattori, Akihisa and Aaryaman Vasishta and Walker, Andrew and Schnetter, Erik and Sierra, Juan and Leinweber, Katrin},
+	urldate = {2022-08-27},
+	date = {2022-04-27},
+	doi = {10.5281/ZENODO.6497293},
+}
+
+@article{davari_new_2013,
+	title = {New Implementation of Legendre Polynomials for Solving Partial Differential Equations},
+	volume = {04},
+	issn = {2152-7385, 2152-7393},
+	url = {http://www.scirp.org/journal/doi.aspx?DOI=10.4236/am.2013.412224},
+	doi = {10.4236/am.2013.412224},
+	pages = {1647--1650},
+	number = {12},
+	journaltitle = {Applied Mathematics},
+	shortjournal = {{AM}},
+	author = {Davari, Ali and Ahmadi, Abozar},
+	urldate = {2022-08-28},
+	date = {2013},
+	file = {Full Text:/Users/npross/Zotero/storage/A8XM56WK/Davari and Ahmadi - 2013 - New Implementation of Legendre Polynomials for Sol.pdf:application/pdf},
+}
+
+@article{srinivasan_estimating_1998,
+	title = {Estimating the spatial Nyquist of the human {EEG}},
+	volume = {30},
+	issn = {0743-3808, 1532-5970},
+	url = {http://link.springer.com/10.3758/BF03209412},
+	doi = {10.3758/BF03209412},
+	pages = {8--19},
+	number = {1},
+	journaltitle = {Behavior Research Methods, Instruments, \& Computers},
+	shortjournal = {Behavior Research Methods, Instruments, \& Computers},
+	author = {Srinivasan, Ramesh and Tucker, Don M. and Murias, Michael},
+	urldate = {2022-08-28},
+	date = {1998-03},
+	langid = {english},
+	file = {Full Text:/Users/npross/Zotero/storage/RCY73VUB/Srinivasan et al. - 1998 - Estimating the spatial Nyquist of the human EEG.pdf:application/pdf},
+}
+
+@misc{ruffini_spherical_2002,
+	title = {Spherical Harmonics Interpolation, Computation of Laplacians and Gauge Theory},
+	url = {http://arxiv.org/abs/physics/0206007},
+	doi = {10.48550/arXiv.physics/0206007},
+	abstract = {The aim in this note is to define an algorithm to carry out minimal curvature spherical harmonics interpolation, which is then used to calculate the Laplacian for multi-electrode {EEG} data analysis. The approach taken is to respect the data. That is, we implement a minimal curvature condition for the interpolating surface subject to the constraints determined from the multi-electrode data. We implement this approach using spherical harmonics interpolation. In this elegant example we show that minimization requirement and constraints complement each other to fix all degrees of freedom automatically, as occurs in gauge theories. That is, the constraints are respected, while only the orthogonal subspace minimization constraints are enforced. As an example, we discuss the application to interpolate control data and calculate the temporal sequence of laplacians from an {EEG} Mismatch Negativity ({MMN}) experiment (using an implementation of the algorithm in {IDL}).},
+	number = {{arXiv}:physics/0206007},
+	publisher = {{arXiv}},
+	author = {Ruffini, Giulio and Marco, Josep and Grau, Carles},
+	urldate = {2022-08-30},
+	date = {2002-06-03},
+	eprinttype = {arxiv},
+	eprint = {physics/0206007},
+	keywords = {Physics - Data Analysis, Statistics and Probability, Physics - Medical Physics, Quantitative Biology - Neurons and Cognition},
+	file = {arXiv Fulltext PDF:/Users/npross/Zotero/storage/R7Z5FP8D/Ruffini et al. - 2002 - Spherical Harmonics Interpolation, Computation of .pdf:application/pdf;arXiv.org Snapshot:/Users/npross/Zotero/storage/ESQCQXAJ/0206007.html:text/html},
+}
+
+@article{pascual-marqui_current_1988,
+	title = {Current Source Density Estimation and Interpolation Based on the Spherical Harmonic Fourier Expansion},
+	volume = {43},
+	issn = {0020-7454},
+	url = {https://doi.org/10.3109/00207458808986175},
+	doi = {10.3109/00207458808986175},
+	abstract = {A method for the spatial analysis of {EEG} and {EP} data, based on the spherical harmonic Fourier expansion ({SHE}) of scalp potential measurements, is described. This model provides efficient and accurate formulas for: (1) the computation of the surface Laplacian and (2) the interpolation of electrical potentials, current source densities, test statistics and other derived variables. Physiologically based simulation experiments show that the {SHE} method gives better estimates of the surface Laplacian than the commonly used finite difference method. Cross-validation studies for the objective comparison of different interpolation methods demonstrate the superiority of the {SHE} over the commonly used methods based on the weighted (inverse distance) average of the nearest three and four neighbor values.},
+	pages = {237--249},
+	number = {3},
+	journaltitle = {International Journal of Neuroscience},
+	author = {Pascual-marqui, Roberto D. and Gonzalez-andino, Sara L. and Valdes-sosa, Pedro A.},
+	urldate = {2022-08-30},
+	date = {1988-01-01},
+	note = {Publisher: Taylor \& Francis
+\_eprint: https://doi.org/10.3109/00207458808986175},
+	keywords = {cross-validated current source density (csd) estimation, cross-validated functional brain mapping, cross-validated laplacian estimation, {ROLANDO} {BISCAY}-{LIRIO}, source derivations, spatial analysis, spherical harmonic fourier expansion},
 }
\ No newline at end of file
diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index 9349b61..b540531 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -33,16 +33,6 @@ that mathematics offers us.
 \subsection{Eigenvalue Problem}
 \label{kugel:sec:construction:eigenvalue}
 
-\begin{figure}
-  \centering
-  \includegraphics{papers/kugel/figures/tikz/spherical-coordinates}
-  \caption{
-    Spherical coordinate system. Space is described with the free variables $r
-    \in \mathbb{R}_0^+$, $\vartheta \in [0; \pi]$ and $\varphi \in [0; 2\pi)$.
-    \label{kugel:fig:spherical-coordinates}
-  }
-\end{figure}
-
 From Section \ref{buch:pde:section:kugel}, we know that the spherical Laplacian
 in the spherical coordinate system (shown in Figure
 \ref{kugel:fig:spherical-coordinates}) is is defined as
@@ -80,16 +70,17 @@ that deserves its own name.
   is called the surface spherical Laplacian.
 \end{definition}
 
-In the definition, the subscript ``$\partial S$'' was used to emphasize the
-fact that we are on the spherical surface, which can be understood as being the
-boundary of the sphere. But what does it actually do? To get an intuition,
-first of all, notice the fact that $\surflaplacian$ have second derivatives,
-which means that this a measure of \emph{curvature}; But curvature of what? To
-get an even stronger intuition we will go into geometry, were curvature can be
-grasped very well visually. Consider figure \ref{kugel:fig:curvature} where the
-curvature is shown using colors. First we have the curvature of a curve in 1D,
-then the curvature of a surface (2D), and finally the curvature of a function on
-the surface of the unit sphere.
+In the definition, the subscript ``$\partial S$'' was used to emphasize the fact
+that we are on the spherical surface, which can be understood as being the
+boundary of the sphere. But what does it actually do? To get an intuition, first
+of all, notice the fact that $\surflaplacian$ have second derivatives, which
+means that this a measure of \emph{curvature}; But curvature of what? To get an
+even stronger intuition we will go into geometry, were curvature can be grasped
+very well visually. Consider figure \ref{kugel:fig:curvature} where the
+curvature is shown using colors: positive curvature in red, and negative
+curvature in blue. First we have the curvature of a curve in 1D, then the
+curvature of a surface (2D), and finally the curvature of a function on the
+surface of the unit sphere.
 
 \begin{figure}
   \centering
@@ -111,12 +102,12 @@ that satisfy the equation
     \surflaplacian f = -\lambda f.
 \end{equation}
 Perhaps it may not be obvious at first glance, but we are in fact dealing with a
-partial differential equation (PDE)\footnote{
-  Considering the fact that we are dealing with a PDE, 
-  you may be wondering what are the boundary conditions. Well, since this eigenvalue problem is been developed on 
-  the spherical surface (boundary of a sphere), the boundary in this case are empty, i.e no boundary condition has to be considered.}.
-unpack the notation of the operator $\nabla^2_{\partial S}$ according to
-definition
+partial differential equation (PDE)\footnote{Considering the fact that we are
+dealing with a PDE, you may be wondering what are the boundary conditions. Well,
+since this eigenvalue problem is been developed on the spherical surface
+(boundary of a sphere), the boundary in this case are empty, i.e no boundary
+condition has to be considered.}. If we unpack the notation of the operator
+$\nabla^2_{\partial S}$ according to definition
 \ref{kugel:def:surface-laplacian}, we get:
 \begin{equation} \label{kugel:eqn:eigen-pde}
     \frac{1}{\sin\vartheta} \frac{\partial}{\partial \vartheta} \left(
@@ -189,23 +180,35 @@ require a dedicated section of its own.
 
 \begin{figure}
   \centering
-  \kugelplaceholderfig{.8\textwidth}{5cm}
+  \subfigure[Spherical coordinates. \label{kugel:fig:spherical-coordinates}]
+    {\includegraphics[width=.45\linewidth]{papers/kugel/figures/tikz/spherical-coordinates}}
+  \qquad
+  \subfigure[Substitution. \label{kugel:fig:legendre-substitution}]
+    {\includegraphics[
+      width=.45\linewidth
+      % scale = 1.2,
+      % trim = 0 40 0 0, clip,
+    ]{papers/kugel/figures/tikz/legendre-substitution}}
   \caption{
-    \kugeltodo{Why $z = \cos \vartheta$.}
+    (a) Spherical coordinate system. Space is described with the free variables
+    $r \in \mathbb{R}_0^+$, $\vartheta \in [0; \pi]$ and $\varphi \in [0;
+    2\pi)$. (b) Geometrical intuition for the substitution to obtain the
+    Legendre equation.
   }
 \end{figure}
 
 To solve \eqref{kugel:eqn:ode-theta} we start with the substitution $z = \cos
-\vartheta$ \kugeltodo{Explain geometric origin with picture}. The operator
-$\frac{d}{d \vartheta}$ becomes
+\vartheta$, which has a neat geometrical justification. To see it consider
+figure \ref{kugel:fig:legendre-substitution}, where we sketched the geometry of
+the problem. Algebraically, the operator $\frac{d}{d \vartheta}$ becomes
 \begin{equation*}
     \frac{d}{d \vartheta}
     = \frac{dz}{d \vartheta}\frac{d}{dz}
     = -\sin \vartheta \frac{d}{dz}
     = -\sqrt{1-z^2} \frac{d}{dz},
 \end{equation*} 
-since $\sin \vartheta = \sqrt{1 - \cos^2 \vartheta} = \sqrt{1 - z^2}$, and
-then \eqref{kugel:eqn:ode-theta} becomes 
+since $\sin \vartheta = \sqrt{1 - \cos^2 \vartheta} = \sqrt{1 - z^2}$, which
+agrees with our sketch. Thus, \eqref{kugel:eqn:ode-theta} becomes 
 \begin{align*}
   \frac{-\sqrt{1-z^2}}{\sqrt{1-z^2}} \frac{d}{dz} \left[
     \left(\sqrt{1-z^2}\right) \left(-\sqrt{1-z^2}\right) \frac{d \Theta}{dz}
@@ -236,8 +239,10 @@ can perform the substitution backwards and get back to our eigenvalue problem.
 However, the associated Legendre equation is not any easier, so to attack the
 problem we will look for the solutions in the easier special case when $m = 0$.
 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}:
+tricky to deal with\footnote{It however to notice that the differential equation
+is still singular because of the factor in front of the second derivative of
+$Z$}. 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^2}
   - 2z\frac{d Z}{dz}
@@ -316,11 +321,19 @@ obtain the \emph{associated Legendre functions}.
   The functions
   \begin{equation}
     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, \quad |m|<n
+      = \frac{1}{2^n n!}(1-z^2)^{\frac{m}{2}}
+        \frac{d^{m+n}}{dz^{m+n}}(1-z^2)^n, \quad |m|<n
   \end{equation}
   are known as Ferrers or associated Legendre functions.
 \end{definition}
-The constraint $|m|<n$, can be justified by considering eq.\eqref{kugel:eq:associated_leg_func}, where we differentiate $m+n$ times. We all know that a differentiation, to be well defined, must have an order that is greater than zero \kugeltodo{is that always true?}. Furthermore, it can be seen that this derivative is applied on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a polynomial of degree $2n$ more than $2n$ times, you get zero, that would be a trivial solution. This is the power of zero: It is almost always a (boring) solution.
+The constraint $|m|<n$, can be justified by considering equation
+\eqref{kugel:eq:associated_leg_func}, where we differentiate $m+n$ times. We all
+know that a differentiation, to be well defined, must have an order that is
+greater than zero. Furthermore, it can be seen that this derivative is applied
+on a polynomial of degree $2n$. As is known from Calculus 1, if you derive a
+polynomial of degree $2n$ more than $2n$ times, you get zero, that would be a
+trivial solution. This is the power of zero: It is almost always a (boring)
+solution.
 
 We can thus summarize these two conditions by writing:
 \begin{equation*}
@@ -337,8 +350,8 @@ section \ref{kugel:sec:construction:eigenvalue}. We had left off in the middle
 of the separation, were we had used the Ansatz $f(\vartheta, \varphi) =
 \Theta(\vartheta) \Phi(\varphi)$ to find that $\Phi(\varphi) = e^{im\varphi}$,
 and we were solving for $\Theta(\vartheta)$.  As you may recall, previously we
-performed the substitution $z = \cos \vartheta$. Now we can finally bring back the
-solution to the associated Legendre equation $P^m_n(z)$ into the $\vartheta$
+performed the substitution $z = \cos \vartheta$. Now we can finally bring back
+the solution to the associated Legendre equation $P^m_n(z)$ into the $\vartheta$
 domain and combine it with $\Phi(\varphi)$ to get the full result:
 \begin{equation*}
     f(\vartheta, \varphi)
@@ -534,6 +547,15 @@ product:
 \end{theorem}
 \begin{proof}
   We will begin by doing a bit of algebraic maipulaiton:
+  \footnote{
+    Essentially, what we just did was to turn
+    \eqref{kugel:eq:spherical-harmonics-inner-prod} in this form:
+    \(
+      \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S}
+      = \langle P^m_n, P^{m'}_{n'} \rangle_z
+      \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi
+    \).
+  }
   \begin{align*}
     \int_{0}^\pi \int_0^{2\pi}
       Y^m_n(\vartheta, \varphi) \overline{Y^{m'}_{n'}(\vartheta, \varphi)} 
@@ -561,17 +583,12 @@ product:
   \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}. 
-  We are allowed to use
-  the lemma because $m = m'$. After the just mentioned substitution we can write eq.\eqref{kugel:eq:spherical-harmonics-inner-prod} in this form
-  \begin{equation*}
-    \langle Y^m_n, Y^{m'}_{n'} \rangle_{\partial S} = \langle P^m_n, P^{m'}_{n'} \rangle_z \; \langle e^{im\varphi}, e^{-im'\varphi} \rangle_\varphi.
-  \end{equation*} 
-  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 exponential 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 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
+  exponential over the entire period, which always results in zero. Thus, we do
+  not need to do anything and the proof is complete.
 \end{proof}
 
 These proofs for the various orthogonality relations were quite long and
@@ -619,10 +636,12 @@ regrettably sometimes even ourselves, would write instead:
   reader.
 \end{proof}
 
-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 the separation Ansatz. So to
-prove their orthogonality using the Sturm-Liouville theory we argue that
+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 the separation Ansatz. 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)
@@ -639,8 +658,8 @@ 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
+Y^{m'}_{n'} \rangle = \delta_{nn'} \delta_{mm'}$. 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]
@@ -677,204 +696,314 @@ 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 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.
+numerical packages (such as \texttt{SHTOOLS}
+\cite{markwieczorek_shtoolsshtools_2022}) 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}.
 
-\subsection{Recurrence Relations}\kugeltodo{replace x with z}
-The idea of this subsection is to introduce first some recursive relations regarding the Associated Legendre Functions, defined in eq.\eqref{kugel:def:ferrers-functions}. Subsequently we will extend them, in order to derive recurrence formulas for the case of Spherical Harmonic functions as well. 
+\subsection{Recurrence Relations}
+
+The idea of this subsection is to introduce first some recursive relations
+regarding the associated Legendre functions, defined in equation
+\eqref{kugel:def:ferrers-functions}. Subsequently we will extend them, in order
+to derive recurrence formulas for the case of Spherical Harmonic functions as
+well.
+
 \subsubsection{Associated Legendre Functions}
-To start this journey, we can first write the following equations, which relate the Associated Legendre functions of different indeces $m$ and $n$ recursively:
+
+To start this journey, we can first write the following equations, which relate
+the associated Legendre functions of different indices $m$ and $n$ recursively:
 \begin{subequations}
   \begin{align}
-  P^m_n(z) &= \dfrac{1}{(2n+1)x} \left[ (m+n) P^m_{n-1}(z) + (n-m+1) P^m_{n+1}(z) \right] \label{kugel:eq:rec-leg-1} \\
-  P^m_n(z) &= \dfrac{\sqrt{1-z^2}}{2mz} \left[ P^{m+1}_n(z) + [n(n+1)-m(m-1)] P^{m-1}_n(z) \right] \label{kugel:eq:rec-leg-2} \\ 
-  P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[ P^{m+1}_{n+1}(z) - P^{m+1}_{n-1}(z) \right]  \label{kugel:eq:rec-leg-3} \\
-  P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[ (n+m)(n+m-1)P^{m-1}_{n-1}(z) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(z) \right]  \label{kugel:eq:rec-leg-4}
+    P^m_n(z) &= \dfrac{1}{(2n+1)x} \left[
+       (m+n) P^m_{n-1}(z) + (n-m+1) P^m_{n+1}(z)
+     \right] \label{kugel:eqn:rec-leg-1} \\
+    P^m_n(z) &= \dfrac{\sqrt{1-z^2}}{2mz} \left[
+        P^{m+1}_n(z) + [n(n+1)-m(m-1)] P^{m-1}_n(z)
+      \right] \label{kugel:eqn:rec-leg-2} \\ 
+    P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[
+        P^{m+1}_{n+1}(z) - P^{m+1}_{n-1}(z)
+      \right] \label{kugel:eqn:rec-leg-3} \\
+    P^m_n(z) &= \dfrac{1}{(2n+1)\sqrt{1-z^2}} \left[
+        (n+m)(n+m-1)P^{m-1}_{n-1}(z) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(z)
+      \right] \label{kugel:eqn:rec-leg-4}
   \end{align}
 \end{subequations}
-Much of the effort will be proving this bunch of equalities. Then, in the second part, where we will derive the recursion equations for $Y^m_n(\vartheta,\varphi)$, we will basically reuse the ones presented above.
-
-Maybe it is worth mentioning at least one use case for these relations: In some software implementations (that include lighting computations in computer graphics, antenna modelling softwares, 3-D modelling in medical applications, etc.) 
-they are widely used, as they lead to better numerical accuracy and computational cost lower by a factor of six\cite{usecase_recursion_paper}. 
-\begin{enumerate}[(i)]
-  \item 
-  \begin{proof}
-    This is the relation that links the associated Legendre functions with the same $m$ index but different $n$. Using \ref{} \kugeltodo{search the general equation of recursion for orthogonal polynomials (is somewhere in the book)}, we have 
-    \begin{equation*}
-      (n+1)P_{n+1}(z)-(2n+1)xP_n(z)+nP_{n-1}(z)=0,
-    \end{equation*}
-    that can be differentiated $m$ times, obtaining
-    \begin{equation}\label{kugel:eq:rec_1}
-      (n+1)\frac{d^mP_{n+1}}{dz^m}-(2n+1) \left[z \frac{d^m P_n}{dz^m}+ m\frac{d^{m-1}P_{n-1}}{dz^{m-1}} \right] + n\frac{d^m P_{n-1}}{dz^m}=0.
-    \end{equation}
-    To continue this derivation, we need the following relation:
-    \begin{equation}\label{kugel:eq:rec_2}
-      \frac{dP_{n+1}}{dz} - \frac{dP_{n-1}}{dz} = (2n+1)P_n.
-    \end{equation}
-    The latter will not be derived, because it suffices to use the definition of the Legendre Polynomials $P_n(x)$ to check it.
-    
-    We can now differentiate the just presented eq.\eqref{kugel:eq:rec_2} $m-1$ times, that will become
-    \begin{equation}\label{kugel:eq:rec_3}
-      \frac{d^mP_{n+1}}{dx^m} - \frac{d^mP_{n-1}}{dx^m} = (2n+1)\frac{d^{m-1}P_n}{dx^{m-1}}.
-    \end{equation}
-    Then, using eq.\eqref{kugel:eq:rec_3} in eq.\eqref{kugel:eq:rec_1}, we will have
-    \begin{equation}\label{kugel:eq:rec_4}
-      (n+1)\frac{d^mP_{n+1}}{dx^m}- (2n+1)\frac{d^mP_{n+1}}{dx^m} -m\left[\frac{d^m P_{n+1}}{dx^m}+ \frac{d^{m}P_{n-1}}{dx^m}\right] + n\frac{d^m P_{n-1}}{dx^m}=0.
-    \end{equation}
-    Finally, multiplying both sides by $(1-x^2)^{\frac{m}{2}}$ and simplifying the expression, we can rewrite eq.\eqref{kugel:eq:rec_4} in terms of $P^m_n(x)$, namely
-    \begin{equation*}
-      (n+1-m)P^m_{n+1}(x)-(2n+1)xP^m_n(x)+(m+n)P^m_{n-1}(x)=0,
-    \end{equation*}
-    that rearranged, will be
-    \begin{equation*}
-      (2n+1) x P^m_n(x)= (m+n) P^m_{n-1}(x) + (n-m+1) P^m_{n+1}(x).
-    \end{equation*}
-  \end{proof}
-  
-  \item
-  \begin{proof}
-    This relation, unlike the previous one, link three expression with the same $n$ index but different $m$. 
-    
-    In the proof of Lemma \ref{kugel:lemma:sol_associated_leg_eq}, at some point we ran into this expression.
-    \begin{equation*}
-      (1-x^2)\frac{d^{m+2}P_n}{dx^{m+2}} - 2(m+1)x \frac{d^{m+1}P_n}{dx^{m+1}} + [n(n+1)-m(m+1)]\frac{d^mP_n}{dx^m} = 0,
-    \end{equation*}
-    that, if multiplied by $(1-x^2)^{\frac{m}{2}}$, will be
-    \begin{equation*}
-      (1-x^2)^{\frac{m}{2}+1}\frac{d^{m+2}P_n}{dx^{m+2}} - 2(m+1)x (1-x^2)^{\frac{m}{2}}\frac{d^{m+1}P_n}{dx^{m+1}} + [n(n+1)-m(m+1)](1-x^2)^{\frac{m}{2}}\frac{d^mP_n}{dx^m} = 0.
-    \end{equation*}
-    Therefore, as before, expressing it in terms of $P^m_n(x)$:
-    \begin{equation*}
-      P^{m+2}_n(x) - \frac{2(m+1)x}{\sqrt{1-x^2}}P^{m+1}_n(x) + [n(n+1)-m(m+1)]P^m_n(x)=0.
-    \end{equation*}
-    Further, we can adjust the indeces and terms, obtaining
-    \begin{equation*}
-      \frac{2mx}{\sqrt{(1-x^2)}} P^m_n(x) = P^{m+1}_n(x) + [n(n+1)-m(m-1)] P^{m-1}_n(x).
-    \end{equation*}
-  
-  \end{proof}
-  
-  \item
-  \begin{proof}
-    To derive this expression, we can multiply eq.\eqref{kugel:eq:rec_3} by $(1-x^2)^{\frac{m}{2}}$ and, as always, we could express it in terms of $P^m_n(x)$:
-    \begin{equation*}
-      P^m_{n+1}(x) - P^m_{n-1}(x) = (2n+1)\sqrt{1-x^2}P^{m-1}_n(x).
-    \end{equation*}
-    Afer that we can divide by $2n+1$ resulting in
-    \begin{equation}\label{kugel:eq:helper}
-      \frac{1}{2n+1}[P^m_{n+1}(x) - P^m_{n-1}(x)] = \sqrt{1-x^2}P^{m-1}_n(x).
-    \end{equation}
-    To conclude, we arrange the indeces differently:
-    \begin{equation*}
-      \sqrt{1-x^2}P^{m}_n(x)=\frac{1}{2n+1}[P^{m+1}_{n+1}(x) - P^{m+1}_{n-1}(x)].
-    \end{equation*}
-  \end{proof}
-
-  \item
-  \begin{proof}
-    For this proof we can rely on eq.\eqref{kugel:eq:rec-leg-1}, and therefore rewrite eq.\eqref{kugel:eq:rec-leg-2} as 
-    \begin{equation*}
-      \frac{2m}{(2n+1)\sqrt{1-x^2}} \left[ (m+n)P^m_{n-1}(x) + (n-m+1)P^m_{n+1}(x) \right] = P^{m+1}_n(x) + [ n(n+1)-m(m-1) ]P^{m-1}_n(x).
-    \end{equation*}
-    Rewriting then $P^{m-1}_n(x)$ using eq.\eqref{kugel:eq:helper}, we will have
-    \begin{align*}
-      \frac{2m}{(2n+1)\sqrt{1-x^2}} &\left[ (m+n)P^m_{n-1}(x) + (n-m+1)P^m_{n+1}(x) \right] = P^{m+1}_n(x) \\
-      &+ \frac{n(n+1)-m(m-1)}{(2n+1)\sqrt{1-x^2}} \left[ P^m_{n+1}(x)-P^m_{n-1}(x) \right].
-    \end{align*}
-    The last equation, after some algebric rearrangements, it is easy to show that it is equivalent to
-    \begin{equation*}
-      \sqrt{1-x^2} P^m_n(x) = \dfrac{1}{2n+1} \left[ (n+m)(n+m-1)P^{m-1}_{n-1}(x) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(x) \right]
-    \end{equation*}
-  \end{proof}
-
-\end{enumerate}
+
+Much of the effort will be proving this bunch of equalities. Then, in the second
+part, where we will derive the recursion equations for
+$Y^m_n(\vartheta,\varphi)$, we will basically reuse the ones presented above.
+Maybe it is worth mentioning at least one use case for these relations: In some
+software implementations (that include lighting computations in computer
+graphics, antenna modelling software, 3D modelling in medical applications,
+etc.) they are widely used, as they lead to better numerical accuracy and
+computational cost lower by a factor of six \cite{davari_new_2013}.
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-leg-1}]
+  This is the relation that links the associated Legendre functions with the
+  same $m$ index but different $n$. Using theorem
+  \ref{buch:orthogonal:satz:drei-term-rekursion}, we have 
+  \begin{equation*}
+    (n+1)P_{n+1}(z)-(2n+1)zP_n(z)+nP_{n-1}(z)=0,
+  \end{equation*}
+  that can be differentiated $m$ times, obtaining
+  \begin{equation}
+    \label{kugel:eq:rec_1}
+    (n+1)\frac{d^mP_{n+1}}{dz^m}-(2n+1) \left[
+      z \frac{d^m P_n}{dz^m}+ m\frac{d^{m-1}P_{n-1}}{dz^{m-1}}
+    \right] + n\frac{d^m P_{n-1}}{dz^m} = 0.
+  \end{equation}
+  To continue this derivation, we need the following relation:
+  \begin{equation}
+    \label{kugel:eq:rec_2}
+    \frac{dP_{n+1}}{dz} - \frac{dP_{n-1}}{dz} = (2n+1)P_n.
+  \end{equation}
+  The latter will not be derived, because it suffices to use the definition of
+  the Legendre Polynomials $P_n(z)$ to check it.  We can now differentiate $m-1$
+  times the just presented equation \eqref{kugel:eq:rec_2}, that so that is
+  becomes
+  \begin{equation}
+    \label{kugel:eq:rec_3}
+    \frac{d^mP_{n+1}}{dz^m} - \frac{d^mP_{n-1}}{dz^m}
+    = (2n+1)\frac{d^{m-1}P_n}{dz^{m-1}}.
+  \end{equation}
+  Then, using eq. \eqref{kugel:eq:rec_3} in eq. \eqref{kugel:eq:rec_1}, we will
+  have
+  \begin{equation}
+    \label{kugel:eq:rec_4}
+    (n+1)\frac{d^mP_{n+1}}{dz^m}
+      - (2n+1)\frac{d^mP_{n+1}}{dz^m}
+      - m\left[\frac{d^m P_{n+1}}{dz^m}
+      + \frac{d^{m}P_{n-1}}{dz^m}\right]
+      + n\frac{d^m P_{n-1}}{dz^m}=0.
+  \end{equation}
+  Finally, multiplying both sides by $(1-z^2)^{\frac{m}{2}}$ and simplifying the
+  expression, we can rewrite eq. \eqref{kugel:eq:rec_4} in terms of $P^m_n(z)$,
+  namely
+  \begin{equation*}
+    (n+1-m)P^m_{n+1}(z)-(2n+1)zP^m_n(z)+(m+n)P^m_{n-1}(z)=0,
+  \end{equation*}
+  that rearranged, will be
+  \begin{equation*}
+    (2n+1) z P^m_n(z)= (m+n) P^m_{n-1}(z) + (n-m+1) P^m_{n+1}(z).
+    \qedhere
+  \end{equation*}
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-leg-2}]
+  This relation, unlike the previous one, link three expression with the same
+  $n$ index but different $m$. In the proof of Lemma
+  \ref{kugel:thm:extend-legendre}, at some point we ran into this expression.
+  \begin{equation*}
+    (1-z^2)\frac{d^{m+2}P_n}{dz^{m+2}}
+    - 2(m+1)z \frac{d^{m+1}P_n}{dz^{m+1}}
+    + [n(n+1)-m(m+1)]\frac{d^mP_n}{dz^m} = 0,
+  \end{equation*}
+  which, if multiplied by $(1-z^2)^{\frac{m}{2}}$, becomes
+  \begin{equation*}
+    (1-z^2)^{\frac{m}{2}+1}\frac{d^{m+2}P_n}{dz^{m+2}}
+    - 2(m+1)z (1-z^2)^{\frac{m}{2}}\frac{d^{m+1}P_n}{dz^{m+1}}
+    + [n(n+1)-m(m+1)](1-z^2)^{\frac{m}{2}}\frac{d^mP_n}{dz^m} = 0.
+  \end{equation*}
+  Therefore, as before, expressing it in terms of $P^m_n(z)$:
+  \begin{equation*}
+    P^{m+2}_n(z) - \frac{2(m+1)z}{\sqrt{1-z^2}}P^{m+1}_n(z)
+    + [n(n+1)-m(m+1)]P^m_n(z)=0.
+  \end{equation*}
+  Further, we can adjust the indices and terms, obtaining
+  \begin{equation*}
+    \frac{2mz}{\sqrt{(1-z^2)}} P^m_n(z)
+    = P^{m+1}_n(z) + [n(n+1)-m(m-1)] P^{m-1}_n(z).
+    \qedhere
+  \end{equation*}
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-leg-3}]
+  To derive this expression, we can multiply eq. \eqref{kugel:eq:rec_3} by
+  $(1-z^2)^{\frac{m}{2}}$ and, as always, we could express it in terms of
+  $P^m_n(z)$:
+  \begin{equation*}
+    P^m_{n+1}(z) - P^m_{n-1}(z) = (2n+1)\sqrt{1-z^2}P^{m-1}_n(z).
+  \end{equation*}
+  After that we can divide by $2n+1$ resulting in
+  \begin{equation}\label{kugel:eq:helper}
+    \frac{1}{2n+1}[P^m_{n+1}(z) - P^m_{n-1}(z)] = \sqrt{1-z^2}P^{m-1}_n(z).
+  \end{equation}
+  To conclude, we arrange the indices differently:
+  \begin{equation*}
+    \sqrt{1-z^2}P^{m}_n(z)=\frac{1}{2n+1}[P^{m+1}_{n+1}(z) - P^{m+1}_{n-1}(z)].
+    \qedhere
+  \end{equation*}
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-leg-4}]
+  For this proof we can rely on \eqref{kugel:eqn:rec-leg-1}, and therefore
+  rewrite \eqref{kugel:eqn:rec-leg-2} as 
+  \begin{equation*}
+    \frac{2m}{(2n+1)\sqrt{1-z^2}} \left[
+      (m+n)P^m_{n-1}(z) + (n-m+1)P^m_{n+1}(z)
+    \right] = P^{m+1}_n(z) + [ n(n+1)-m(m-1) ]P^{m-1}_n(z).
+  \end{equation*}
+  Rewriting then $P^{m-1}_n(z)$ using eq.\eqref{kugel:eq:helper}, we will have
+  \begin{align*}
+    \frac{2m}{(2n+1)\sqrt{1-z^2}}
+      &\left[ (m+n)P^m_{n-1}(z) + (n-m+1)P^m_{n+1}(z) \right] = P^{m+1}_n(z) \\
+      &+ \frac{n(n+1)-m(m-1)}{(2n+1)\sqrt{1-z^2}} \left[
+          P^m_{n+1}(z)-P^m_{n-1}(z)
+        \right].
+  \end{align*}
+  The last equation, after some algebraic rearrangements, it is easy to show
+  that it is equivalent to
+  \begin{equation*}
+    \sqrt{1-z^2} P^m_n(z) = \dfrac{1}{2n+1} \left[
+      (n+m)(n+m-1)P^{m-1}_{n-1}(z) - (n-m+1)(n-m+2)P^{m-1}_{n+1}(z)
+    \right].
+    \qedhere
+  \end{equation*}
+\end{proof}
 
 \subsubsection{Spherical Harmonics}
-The goal of this subsection's part is to apply the recurrence relations of the $P^m_n(z)$ functions to the Spherical Harmonics. 
-With some little adjustments we will be able to have recursion equations for them too. As previously written the most of the work is already done. Now it is only a matter of minor mathematical operations/rearrangements.
 
-We can start by listing all of them:
+The goal of this subsection's part is to apply the recurrence relations of the
+$P^m_n(z)$ functions to the Spherical Harmonics.  With some little adjustments
+we will be able to have recursion equations for them too. As previously written
+the most of the work is already done. Now it is only a matter of minor
+mathematical operations/rearrangements. We can start by listing all of them:
 \begin{subequations}
   \begin{align}
-  Y^m_n(\vartheta, \varphi) &= \dfrac{1}{(2n+1)\cos \vartheta} \left[ (m+n)Y^m_{n-1}(\vartheta, \varphi) + (m-n+1)Y^m_{n+1}(\vartheta, \varphi) \right] \label{kugel:eq:rec-sph_harm-1} \\
-  Y^m_n(\vartheta, \varphi) &= \dfrac{\tan \vartheta}{2m}\left[ Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi} \right] \label{kugel:eq:rec-sph_harm-2} \\
-  Y^m_n(\vartheta, \varphi) &= \dfrac{e^{-i\varphi}}{ (2n+1)\sin \vartheta } \left[ Y^{m+1}_{n+1}(\vartheta, \varphi) - Y^{m+1}_{n-1}(\vartheta, \varphi) \right] \label{kugel:eq:rec-sph_harm-3} \\
-  Y^m_n(\vartheta, \varphi) &= \dfrac{e^{i\varphi}}{(2n+1)\sin \vartheta} \left[ (n+m)(n+m-1)Y^{m-1}_{n-1}(\vartheta, \varphi) - (n-m+1)(n-m+2)Y^{m-1}_{n+1}(\vartheta, \varphi) \right]  \label{kugel:eq:rec-sph_harm-4} 
+  Y^m_n(\vartheta, \varphi) &= \dfrac{1}{(2n+1)\cos \vartheta} \left[
+      (m+n)Y^m_{n-1}(\vartheta, \varphi)
+      + (m-n+1)Y^m_{n+1}(\vartheta, \varphi)
+    \right] \label{kugel:eqn:rec-sph-harm-1} \\
+  Y^m_n(\vartheta, \varphi) &= \dfrac{\tan \vartheta}{2m}\left[
+      Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi}
+      + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi}
+    \right] \label{kugel:eqn:rec-sph-harm-2} \\
+  Y^m_n(\vartheta, \varphi) &= \dfrac{e^{-i\varphi}}{ (2n+1)\sin \vartheta}
+    \left[
+      Y^{m+1}_{n+1}(\vartheta, \varphi)
+      - Y^{m+1}_{n-1}(\vartheta, \varphi)
+    \right] \label{kugel:eqn:rec-sph-harm-3} \\
+  Y^m_n(\vartheta, \varphi) &= \dfrac{e^{i\varphi}}{(2n+1)\sin \vartheta}
+    \left[
+      (n+m)(n+m-1)Y^{m-1}_{n-1}(\vartheta, \varphi)
+      - (n-m+1)(n-m+2)Y^{m-1}_{n+1}(\vartheta, \varphi)
+    \right] \label{kugel:eqn:rec-sph-harm-4} 
   \end{align}
 \end{subequations}
 
-\begin{enumerate}[(i)]
-  \item
-  \begin{proof}
-    We can multiply both sides of equality in eq.\eqref{kugel:eq:rec-leg-1} by $e^{im \varphi}$ and perform the substitution $z=\cos \vartheta$. After a few simple algebraic steps, we will obtain the relation we are looking for
-  \end{proof}
-  \item 
-  \begin{proof}
-    In this proof, as before, we can perform the substitution $z=\cos \vartheta$, and notice that $\sqrt{1-z^2}=\sin \vartheta$, hence, the relation in eq.\eqref{kugel:eq:rec-leg-2} will be
-    \begin{equation*}
-      \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta) =  P^{m+1}_n(\cos \vartheta) + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta).
-    \end{equation*}
-    The latter, multiplied by $e^{im\varphi}$, becomes
-    \begin{align*}
-      \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta)e^{im\varphi} &=  P^{m+1}_n(\cos \vartheta)e^{im\varphi} + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta)e^{im\varphi} \\
-      &= P^{m+1}_n(\cos \vartheta)e^{i(m+1)\varphi}e^{-i\varphi} + [n(n+1)-m(m-1)]P^{m-1}_n (\cos \vartheta)e^{i(m-1)\varphi}e^{i\varphi} \\
-      &= Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi}.
-    \end{align*}
-    Finally, after some ``cleaning''
-    \begin{equation*}
-      Y^m_n(\vartheta, \varphi) = \frac{\tan \vartheta}{2m} \left[ Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi} + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi} \right]
-    \end{equation*}
-  \end{proof}
-  \item 
-  \begin{proof}
-    Now we can consider eq.\eqref{kugel:eq:rec-leg-3}, and multiply it by $e^{im\varphi}$. After the usual substitution $z=\cos \vartheta$, we have 
-    \begin{align*}
-      \sin \vartheta P^m_n(\cos \vartheta)e^{im\varphi} &= \dfrac{e^{im\varphi}}{2n+1}\left[ P^{m+1}_{n+1}(\cos \vartheta) - P^{m+1}_{n-1}(\cos \vartheta)\right] \\
-      &= \dfrac{e^{-i\varphi}}{2n+1}\left[ P^{m+1}_{n+1}(\cos \vartheta)e^{i(m+1)\varphi} - P^{m+1}_{n-1}(\cos \vartheta)e^{i(m+1)\varphi}\right].
-    \end{align*}
-    A few manipulations later, we will obtain
-    \begin{equation*}
-      Y^m_n(\vartheta, \varphi) = \frac{e^{-i\varphi}}{(2n+1)\sin \vartheta} \left[ Y^{m+1}_{n+1}(\vartheta, \varphi)-Y^{m+1}_{n-1}(\vartheta, \varphi) \right].
-    \end{equation*}
-  \end{proof}
-  \item 
-  \begin{proof}
-    This proof is very similar to the previous one. We just have to perform the substitution $z = \cos \vartheta$, as always. Secondly we can multiply the right side by $e^{im\varphi}$ and the left one too but in a different form, namely $e^{im\varphi}=e^{i(m-1)\varphi}e^{i\varphi}$. Then it is only a question of recalling the definition of $Y^m_n(\vartheta, \varphi)$.
-  \end{proof}
-\end{enumerate}
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-1}]
+  We can multiply both sides of equality in \eqref{kugel:eqn:rec-leg-1} by $e^{im
+  \varphi}$ and perform the substitution $z=\cos \vartheta$. After a few simple
+  algebraic steps, we will obtain the relation we are looking for
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-2}]
+  In this proof, as before, we can perform the substitution $z=\cos \vartheta$,
+  and notice that $\sqrt{1-z^2}=\sin \vartheta$, hence, the relation in
+  \eqref{kugel:eqn:rec-leg-2} will be
+  \begin{equation*}
+    \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta) 
+    = P^{m+1}_n(\cos \vartheta) + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta).
+  \end{equation*}
+  The latter, multiplied by $e^{im\varphi}$, becomes
+  \begin{align*}
+    \frac{2m \cos \vartheta}{\sin \vartheta} P^m_n(\cos \vartheta)e^{im\varphi}
+      &= P^{m+1}_n(\cos \vartheta)e^{im\varphi}
+       + [n(n+1)-m(m-1)]P^{m-1}_n P^m_n(\cos \vartheta)e^{im\varphi} \\
+      &= P^{m+1}_n(\cos \vartheta)e^{i(m+1)\varphi}e^{-i\varphi}
+       + [n(n+1)-m(m-1)]P^{m-1}_n (\cos \vartheta)e^{i(m-1)\varphi}e^{i\varphi} \\
+      &= Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi}
+       + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi}.
+  \end{align*}
+  Finally, after some ``cleaning''
+  \begin{equation*}
+    Y^m_n(\vartheta, \varphi) = \frac{\tan \vartheta}{2m} \left[
+      Y^{m+1}_n(\vartheta, \varphi)e^{-i\varphi}
+      + [n(n+1)-m(m-1)]Y^{m-1}_n(\vartheta, \varphi)e^{i\varphi}
+    \right]
+    \qedhere
+  \end{equation*}
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-3}]
+  Now we can consider \eqref{kugel:eqn:rec-leg-3}, and multiply it by
+  $e^{im\varphi}$. After the usual substitution $z=\cos \vartheta$, we have 
+  \begin{align*}
+    \sin \vartheta P^m_n(\cos \vartheta)e^{im\varphi}
+      &= \dfrac{e^{im\varphi}}{2n+1}\left[
+          P^{m+1}_{n+1}(\cos \vartheta)
+          - P^{m+1}_{n-1}(\cos \vartheta)
+        \right] \\
+      &= \dfrac{e^{-i\varphi}}{2n+1}\left[
+          P^{m+1}_{n+1}(\cos \vartheta)e^{i(m+1)\varphi}
+          - P^{m+1}_{n-1}(\cos \vartheta)e^{i(m+1)\varphi}
+        \right].
+  \end{align*}
+  A few manipulations later, we will obtain
+  \begin{equation*}
+    Y^m_n(\vartheta, \varphi) = \frac{e^{-i\varphi}}{(2n+1)\sin \vartheta}
+      \left[
+        Y^{m+1}_{n+1}(\vartheta, \varphi)-Y^{m+1}_{n-1}(\vartheta, \varphi)
+      \right].
+    \qedhere
+  \end{equation*}
+\end{proof}
+
+\begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-4}]
+  This proof is very similar to the previous one. We just have to perform the
+  substitution $z = \cos \vartheta$, as always. Secondly we can multiply the
+  right side by $e^{im\varphi}$ and the left one too but in a different form,
+  namely $e^{im\varphi}=e^{i(m-1)\varphi}e^{i\varphi}$. Then it is only a
+  question of recalling the definition of $Y^m_n(\vartheta, \varphi)$.
+\end{proof}
 
 \section{Series Expansions in $L^2(S^2)$}
-We have now reach a point where we have all the tools that are necessary to build something truly amazing: a general series expansion formula for
-function on the surface of the sphere.
-Before starting we want to recall the definition of the inner product on the spherical surface of definition \ref{kugel:def:inner-product-s2}
+
+We have now reach a point where we have all the tools that are necessary to
+build something truly amazing: a general series expansion formula for function
+on the surface of the sphere.  Before starting we want to recall the definition
+of the inner product on the spherical surface 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*}
-To be a bit technical we can say that the set of spherical harmonic functions, toghether with the inner product just showed, 
-form something that we call Hilbert Space\footnote{For more details about Hilber space you can take a look in section \ref{kugel:sec:preliminaries}}. 
-This function space is defined over the space of ``well-behaved'' \footnote{The definitions of ``well-behaved'' is pretty ambigous, even for mathematicians. 
-It depends basically on the context.
-You can sumarize it by saying: functions for which the theory we are considering (Fourier theorem) is always true. In our case we can say that well-behaved functions 
-are functions that follow some convergence contraints (pointwise, uniform, absolute, ...) that we don't want to consider further anyway.} functions. 
-We can say that the theory we are about to show can be applied on all twice differentiable complex valued functions, 
-to be more concise: complex valued $L^2$ functions $S^2 \to \mathbb{C}$.
-
-All these jargons are not really necessary for the practical applications of us mere mortals, namely physicists and engineers. 
-From now on we will therefore assume that the functions we will dealing with fulfill these ``minor'' conditions.
-
-The insiders could turn up their nose, but we don't want to dwell too much on the concept of Hilbert space, convergence, metric, well-behaved functions etc.
-We simply think that this rigorousness could be at the expense of the possibility to appreciate the beauty and elegance of this theory. 
-Furthermore, the risk of writing 300+ pages to prove that $1+1=2$\cite{principia-mathematica} is just around the corner (we apologize in advance to Mr. Whitehead and Mr. Russel for using their effort with a negative connotation). 
-
-Despite all, if you desire having definitions a bit more rigorous (as rigorous as two engineers can be), you could take a look at the chapter \ref{}.
+To be a bit technical we can say that the set of spherical harmonic functions,
+together with the inner product just showed, form something that we call Hilbert
+Space\footnote{For more details about Hilbert space you can take a look in
+section \ref{kugel:sec:preliminaries}}.  This function space is defined over the
+space of ``well-behaved'' \footnote{The definitions of ``well-behaved'' is
+pretty ambiguous, even for mathematicians.  It depends basically on the context.
+You can summarize it by saying: functions for which the theory we are considering
+(Fourier theorem) is always true. In our case we can say that well-behaved
+functions are functions that follow some convergence constraints (pointwise,
+uniform, absolute, ...) that we don't want to consider further anyway.}
+functions.  We can say that the theory we are about to show can be applied on
+all twice differentiable complex valued functions, to be more concise: complex
+valued $L^2$ functions $S^2 \to \mathbb{C}$.
+
+All this jargon is not really necessary for the practical applications of us
+mere mortals, namely physicists and engineers.  From now on we will therefore
+assume that the functions we will dealing with fulfill these ``minor''
+conditions.
+
+The insiders could turn up their nose, but we don't want to dwell too much on
+the concept of Hilbert space, convergence, metric, well-behaved functions etc.
+We simply think that this rigorousness could be at the expense of the
+possibility to appreciate the beauty and elegance of this theory.  Furthermore,
+the risk of writing 300+ pages to prove that $1+1=2$\cite{principia-mathematica}
+is just around the corner (we apologize in advance to Mr. Whitehead and Mr.
+Russel for using their effort with a negative connotation).
+
+Despite all, if you desire having definitions a bit more rigorous (as rigorous
+as two engineers can be), you could take a look at the chapter \ref{}.
 
 \subsection{Spherical Harmonics Series}
 
@@ -882,54 +1011,72 @@ 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]
-  \label{kugel:definition:spherical-harmonics-series}
+  \label{kugel:def:spherical-harmonics-series}
   \begin{equation}
     f(\vartheta, \varphi) 
     = \sum_{n=0}^\infty \sum_{m =-n}^n
-      c_{m,n} Y^m_n(\vartheta, \varphi). \label{kugel:definition:spherical-harmonics-series}
+      c_{m,n} Y^m_n(\vartheta, \varphi).
+      \label{kugel:eqn:spherical-harmonics-series}
   \end{equation}
 \end{definition}
 
-With this definition we are basically saying that any function defined on the spherical surface can be represented as a linear combination of spherical harmonics. 
-Does eq.\eqref{kugel:definition:spherical-harmonics-series} sound familiar? Well that is prefectly normal, since this is analog to the classical Fourier theory. 
-In the latter is stated that ``any'' $T$-periodic function $f(x)$, on any interval $[x_0-T/2,x_0+T/2]$, can be represented as a linear combination of complex exponentials. More compactly:
+With this definition we are basically saying that any function defined on the
+spherical surface can be represented as a linear combination of spherical
+harmonics.  Does equation \eqref{kugel:definition:spherical-harmonics-series}
+sound familiar? Well that is perfectly normal, since this is analog to the
+classical Fourier theory.  In the latter is stated that ``any'' $T$-periodic
+function $f(x)$, on any interval $[x_0-T/2,x_0+T/2]$, can be represented as a
+linear combination of complex exponentials. More compactly:
 \begin{equation*}
-  f(x) = \sum_{n \in \mathbb{Z}} c_n e^{i \omega_0 x}, \quad \omega_0=\frac{2\pi}{T}
+  f(x) = \sum_{n \in \mathbb{Z}} c_n e^{i \omega_0 x},
+    \quad \omega_0=\frac{2\pi}{T}
 \end{equation*}
-In the case of definition \ref{kugel:definition:spherical-harmonics-series} the kernels, instead of $e^{i\omega_0x}$, have become $Y^m_n$. In addition, the sum is now over the two indices $m$ and $n$.
+In the case of definition \ref{kugel:def:spherical-harmonics-series} the
+kernels, instead of $e^{i\omega_0x}$, have become $Y^m_n$. In addition, the sum
+is now over the two indices $m$ and $n$.
 
 \begin{lemma}[Spherical harmonic coefficients]
-  \label{kugel:lemma:spherical-harmonic-coefficient}
-  \begin{align*}
+  \label{kugel:thm:spherical-harmonic-coefficient}
+  \begin{equation*}
     c_{m,n} 
-    &= \langle f, Y^m_n \rangle_{\partial S} \\
-    &= \int_0^\pi \int_0^{2\pi} f(\vartheta,\varphi) \overline{Y^m_n(\vartheta,\varphi)} \sin\vartheta \,d\varphi\,d\vartheta
-  \end{align*}
+    = \langle f, Y^m_n \rangle_{\partial S}
+    = \int_0^\pi \int_0^{2\pi}
+      f(\vartheta,\varphi) \overline{Y^m_n(\vartheta,\varphi)}
+      \sin\vartheta \,d\varphi\,d\vartheta
+  \end{equation*}
+  \kugeltodo{Would be better as a definition? I don't get what is being proved.}
 \end{lemma}
+
 \begin{proof}
-  To develop this proof we will take advantage of the orthogonality property of the Spherical Harmonics. We can start and finish by applying the inner product on both sides of eq.\eqref{kugel:definition:spherical-harmonics-series}:
+  To develop this proof we will take advantage of the orthogonality property of
+  the Spherical Harmonics. We can start and finish by applying the inner product
+  on both sides of \eqref{kugel:eqn:spherical-harmonics-series}:
   \begin{align*}
     \langle f, Y^{m}_{n} \rangle_{\partial S}
     &= \left\langle \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
-      c_{m',n'} Y^{m'}_{n'}(\vartheta, \varphi) \right\rangle_{\partial S}  \\
-    &=  \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
+      c_{m',n'} Y^{m'}_{n'} \right\rangle_{\partial S}  \\
+    &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'}
       \langle c_{m',n'} Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} \\
-      &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'} c_{m',n'} \langle Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} = c_{m,n}
+    &= \sum_{n'=0}^\infty \sum_{m' =-n'}^{n'} c_{m',n'}
+      \langle Y^{m'}_{n'}, Y^{m}_{n} \rangle_{\partial S} = c_{m,n}
   \end{align*}
-  We omitted the $\vartheta, \varphi$ dependency to avoid overloading the notation.
 \end{proof}
-Thanks to Lemma \ref{kugel:lemma:spherical-harmonic-coefficient} we can now calculate the series expansion defined in \ref{kugel:definition:spherical-harmonics-series}.
 
-It can be shown that, for the famous ``well-behaved functions'' $f(\vartheta, \varphi)$ mentioned before, theorem \ref{fourier-theorem-spherical-surface} is true
+Thanks to Lemma \ref{kugel:thm:spherical-harmonic-coefficient} we can now
+calculate the series expansion defined in
+\ref{kugel:def:spherical-harmonics-series}. It can be shown that, for the famous
+``well-behaved functions'' $f(\vartheta, \varphi)$ mentioned before, the
+following theorem is true.  The connection to Theorem
+\ref{kugel:thm:fourier-theorem} is pretty obvious.
+
 \begin{theorem}[Fourier Theorem on $\partial S$]
   \label{fourier-theorem-spherical-surface}
   \begin{equation*}
-    \lim_{N \to \infty} 
+    \lim_{N \to \infty}
     \int_0^\pi \int_0^{2\pi} \left\| f(\vartheta,\varphi) - \sum_{n=0}^N\sum_{m=-n}^n c_{m,n} Y^m_n(\vartheta,\varphi)
     \right\|_2 \sin\vartheta \,d\varphi\,d\vartheta  = 0
   \end{equation*}
 \end{theorem}
-The connection to Theorem \ref{fourier-theorem-1D} is pretty obvious.
 
 \subsection{Spectrum}
 
@@ -974,4 +1121,22 @@ but with a spherical function $f(\vartheta, \varphi)$.
 \begin{proof}
 \end{proof}
 
-\subsection{Visualization}
\ No newline at end of file
+
+\if 0
+\begin{theorem}[Spherical harmonic series expansion]
+  A complex valued piecewise continuous function on the surface of the sphere
+  $f: S^2 \to \mathbb{C}$, $f \in L^2$ has a series expansion
+  \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*}
+  where $c_{m,n} = \langle f, Y^m_n \rangle$, that converges everywhere uniformly
+  on $f$, i.e. $\|f(\vartheta, \varphi) - \hat{f}(\vartheta, \varphi)\|_2 = 0$.
+\end{theorem}
+\begin{proof}
+  Sadly, this proof is beyond the scope of this text.
+\end{proof}
+\fi
+
+\subsection{Visualization}