From 1a8e52ded09496a745756882253c1d3fd1e76996 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Wed, 31 Aug 2022 20:13:46 +0200
Subject: kugel: Feedback and minor changes, add reference

---
 buch/papers/kugel/spherical-harmonics.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 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 b540531..fb5a144 100644
--- a/buch/papers/kugel/spherical-harmonics.tex
+++ b/buch/papers/kugel/spherical-harmonics.tex
@@ -871,8 +871,8 @@ computational cost lower by a factor of six \cite{davari_new_2013}.
 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:
+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[
@@ -899,7 +899,7 @@ mathematical operations/rearrangements. We can start by listing all of them:
 \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
+  algebraic steps, we will obtain the relation we are looking for.
 \end{proof}
 
 \begin{proof}[Proof of \eqref{kugel:eqn:rec-sph-harm-2}]
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