From 598728629f690ddce63fd1e2ef8e02c09130cbe1 Mon Sep 17 00:00:00 2001
From: "Jiri (George) Lebl" <jiri.lebl@gmail.com>
Date: Mon, 6 May 2024 00:30:42 -0500
Subject: [PATCH] Minor nits

---
 ca.tex | 13 ++++++++-----
 1 file changed, 8 insertions(+), 5 deletions(-)

diff --git a/ca.tex b/ca.tex
index 1cde456..10d7946 100644
--- a/ca.tex
+++ b/ca.tex
@@ -14862,7 +14862,8 @@ \subsection{Isolated singularities}
 For harmonic functions, we get the following classification of removable
 singularities, which is sharp, that is, best possible.
 The harmonic function $\log \sabs{z}$
-has a nonremovable singularity at the origin, and any function
+has a nonremovable singularity at the origin.
+Any function
 that blows up any slower than that, doesn't actually blow up and, in fact,
 extends to be
 harmonic at the origin.
@@ -14901,13 +14902,15 @@ \subsection{Isolated singularities}
 \end{equation}
 The estimate \eqref{eq:removableestimateharmonic} holds also when
 $\sabs{z}=1$ as $g=0$ there.
-The function $-\log\sabs{z}$ is harmonic outside of the origin, so
+The functions $-\log\sabs{z}$ and $g$ are harmonic
+outside of the origin, so
 the maximum principle (the version in
 \exerciseref{exercise:maxprincsecondharmonic}) implies that
 \eqref{eq:removableestimateharmonic} holds also for $\delta < \sabs{z} < 1$,
 and thus for all $z \in \D \setminus \{ 0 \}$.
 As the estimate holds for all $\epsilon > 0$, we have $g(z) = 0$ for all
-$z \in \D \setminus \{0\}$.  So $u$ is the extension we are looking for.
+$z \in \D \setminus \{0\}$.  So $u$ is the extension near $0$ that
+we are looking for.
 \end{proof}
 
 An isolated singularity of a harmonic function $g$ could be very wild, for
@@ -14972,8 +14975,8 @@ \subsection{Isolated singularities}
 for some branch of the $\log$, where $\psi$ is a holomorphic function
 on $\D \setminus \{ 0 \}$.
 Taking the real part we get $f$, a well-defined function on
-$\D \setminus \{ 0 \}$.  That means that
-$c_{-1} \log z$ has real part well-defined in $\D \setminus \{ 0 \}$,
+$\D \setminus \{ 0 \}$.  Therefore,
+$c_{-1} \log z$ has real part that is well-defined in $\D \setminus \{ 0 \}$,
 meaning $c_{-1} \in \R$.
 So
 \begin{equation*}