From d7b25fe85f3c05e03f59e919752dbfd8d76c45d7 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Fri, 2 Sep 2022 02:39:31 +0200
Subject: kugel: Add references to other chapters

---
 buch/papers/kugel/spherical-harmonics.tex | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

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

diff --git a/buch/papers/kugel/spherical-harmonics.tex b/buch/papers/kugel/spherical-harmonics.tex
index fb5a144..3100e36 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -311,10 +311,11 @@ such that they also become solutions of the associated Legendre equation
 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
-\kugeltodo{Reference book theory on extended power series method.}. Now, since
-we have a solution in our domain, namely $P_n(z)$, we can insert it in the lemma 
-obtain the \emph{associated Legendre functions}.
+Legendre equation, which is not possible only using power series (see sections
+\ref{buch:differentialgleichungen:section:potenzreihenmethode} and
+\ref{buch:differentialgleichungen:subsection:verallgemeinrt} for a discussion).
+Now, since we have a solution in our domain, namely $P_n(z)$, we can insert it
+in the lemma obtain the \emph{associated Legendre functions}.
 
 \begin{definition}[Ferrers or associated Legendre functions]
   \label{kugel:def:ferrers-functions}
@@ -595,9 +596,9 @@ 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}.
+\ref{buch:integrale:subsection:sturm-liouville-problem} and
+\ref{chapter:sturmliouville} 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}
-- 
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