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-rw-r--r--pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md8
1 files changed, 4 insertions, 4 deletions
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