From a1b5de688d879069b5e1192057d71572c7bc5368 Mon Sep 17 00:00:00 2001 From: Justin Gassner Date: Thu, 29 Feb 2024 17:32:24 +0100 Subject: Update --- .../one-complex-variable/cauchys-integral-formula.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 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 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 -- cgit v1.2.3-54-g00ecf