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authorJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
committerJustin Gassner <justin.gassner@mailbox.org>2024-02-15 05:11:07 +0100
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downloadsite-7c66b227a494748e2a546fb85317accd00aebe53.tar.zst
Update
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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 $r>0$.