From a1b5de688d879069b5e1192057d71572c7bc5368 Mon Sep 17 00:00:00 2001
From: Justin Gassner <justin.gassner@mailbox.org>
Date: Thu, 29 Feb 2024 17:32:24 +0100
Subject: Update

---
 .../one-complex-variable/cauchys-integral-formula.md              | 8 ++++----
 pages/complex-analysis/one-complex-variable/cauchys-theorem.md    | 4 ++--
 2 files changed, 6 insertions(+), 6 deletions(-)

(limited to 'pages/complex-analysis')

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 6ac0803..f7414d5 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md
@@ -20,7 +20,7 @@ $$
 {% proof %}
 {% endproof %}
 
-{% theorem * Cauchy's Integral Formula (Generalization) %}
+{% theorem * Cauchy’s Integral Formula (Generalization) %}
 Let $f : G \to \CC$ be a function holomorphic in an open subset $G \subset \CC$.
 Then the $n$th derivative $f^{(n)}$ exists for every $n \in \NN$.
 If $\gamma$ is a contour in $G$ such that the interior of $\gamma$ is contained in $G$,
@@ -44,7 +44,7 @@ and is often used to compute the integral.
 
 ## Many Consequences
 
-{% theorem * Cauchy's Estimate %}
+{% theorem * Cauchy’s Estimate %}
 Let $f$ be holomorphic on an open set containing the disc with center $a$ and radius $r>0$.
 Then
 
@@ -74,7 +74,7 @@ and the right-hand side of the inequality reduces to the desired expression.
 Recall that an *entire* function is a holomorphic function
 that is defined everywhere in the complex plane.
 
-{% theorem * Liouville's Theorem %}
+{% theorem * Liouville’s Theorem %}
 Every bounded entire function is constant.
 {% endtheorem %}
 
@@ -82,7 +82,7 @@ Every bounded entire function is constant.
 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)
+[Cauchy’s Estimate](#cauchy-s-estimate)
 for all disks centered at any point $a \in \CC$ and with any radius $r>0$.
 For the first derivative, we have $\norm{f'(a)} \le M/r$, which tends to $0$ for $r \to \infty$.
 Hence $f' = 0$ in the whole plane. This
diff --git a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
index 6d78e89..eb040ca 100644
--- a/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
+++ b/pages/complex-analysis/one-complex-variable/cauchys-theorem.md
@@ -7,7 +7,7 @@ nav_order: 2
 
 # {{ page.title }}
 
-{% theorem Cauchy's Theorem (Homotopy Version) %}
+{% theorem Cauchy’s Theorem (Homotopy Version) %}
 Let $G$ be a connected open subset of the complex plane.
 Let $f : G \to \CC$ be a holomorphic function.
 If $\gamma_0$, $\gamma_1$ are homotopic closed curves in $G$, then
@@ -29,7 +29,7 @@ $$
 
 {{ page.title }} has a converse:
 
-{% theorem * Morera's Theorem %}
+{% theorem * Morera’s Theorem %}
 Let $G \subset \CC$ be open and let $f : G \to \CC$ be a continuous function.
 If $\int_{\gamma} f(z) \, dz = 0$ for every contour $\gamma$ contained in $G$,
 then $f$ is holomorphic in $G$.
-- 
cgit v1.2.3-54-g00ecf