From 7c66b227a494748e2a546fb85317accd00aebe53 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 15 Feb 2024 05:11:07 +0100 Subject: Update --- .../one-complex-variable/cauchys-integral-formula.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) (limited to 'pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md') 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 $r>0$. -- cgit v1.2.3-54-g00ecf