From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 15 Feb 2024 05:11:07 +0100
Subject: Update

---
 .../one-complex-variable/cauchys-integral-formula.md              | 8 +++++---
 pages/complex-analysis/one-complex-variable/cauchys-theorem.md    | 2 +-
 pages/complex-analysis/one-complex-variable/power-series.md       | 5 +++--
 3 files changed, 9 insertions(+), 6 deletions(-)

(limited to 'pages/complex-analysis/one-complex-variable')

diff --git a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
index 3cf81f7..6ac0803 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -66,19 +66,21 @@ $$
 Note that the supremum is finite (and is attained),
 because $f$ is continuous and the circle is compact.
 Clearly, the integral evaluates to $2 \pi r / r^{n+1}$
-and the right hand side of the inequality reduces to the desired expression.
+and the right-hand side of the inequality reduces to the desired expression.
 {% endproof %}
 
 ---
 
-Recall that an *entire* function is a holomorphic function that is defined everywhere in the complex plane.
+Recall that an *entire* function is a holomorphic function
+that is defined everywhere in the complex plane.
 
 {% theorem * Liouville's Theorem %}
 Every bounded entire function is constant.
 {% endtheorem %}
 
 {% proof %}
-Consider an entire function $f$ and assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$.
+Consider an entire function $f$ and
+assume that $\norm{f(z)} \le M$ for all $z \in \CC$ and some $M > 0$.
 Since $f$ is holomorphic on the whole plane, we may make
 [Cauchy's Estimate](#cauchy-s-estimate)
 for all disks centered at any point $a \in \CC$ and with any radius $r>0$.
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
index 2445b8b..6d78e89 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -14,7 +14,7 @@ If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
 
 $$
 \int_{\gamma_0} \! f(z) \, dz =
-\int_{\gamma_1} \! f(z) \, dz 
+\int_{\gamma_1} \! f(z) \, dz
 $$
 
 If $\gamma$ is a null-homotopic closed curve in $G$, then
diff --git a/pages/complex-analysis/one-complex-variable/power-series.md b/pages/complex-analysis/one-complex-variable/power-series.md
index 31793ab..4876d30 100644
--- a/pages/complex-analysis/one-complex-variable/power-series.md
+++ b/pages/complex-analysis/one-complex-variable/power-series.md
@@ -21,7 +21,8 @@ $a$ is the *center* of the series.
 {% enddefinition %}
 
 {% lemma %}
-Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$ converges for $z = a + w$.
+Suppose the Banach space valued power series $\sum_{n=0}^{\infty} x_n (z - a)^n$
+converges for $z = a + w$.
 Then it converges absolutely for all $z$ with $\abs{z-a} < \abs{w}$.
 {% endlemma %}
 
@@ -35,7 +36,7 @@ Then either
 
 - the series converges only for $z=a$ (formally $R=0$), or
 - there exists a number $0<R<\infty$ such that
-  the series converges absolutely whenever $\abs{z-a} < R$ 
+  the series converges absolutely whenever $\abs{z-a} < R$
   and diverges whenever $\abs{z-a} > R$, or
 - the series converges absolutely for all $z \in \CC$ (formally $R=\infty$).
 
-- 
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