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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-29 17:32:24 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-29 17:32:24 +0100 |
commit | a1b5de688d879069b5e1192057d71572c7bc5368 (patch) | |
tree | a0f4801d14bfbcc75a6091bdc7d17aceab71f6d4 /pages/complex-analysis | |
parent | 8b9bb9346c217874670b0f1798ab6f1cb28fdb83 (diff) | |
download | site-a1b5de688d879069b5e1192057d71572c7bc5368.tar.zst |
Update
Diffstat (limited to 'pages/complex-analysis')
-rw-r--r-- | pages/complex-analysis/one-complex-variable/cauchys-integral-formula.md | 8 | ||||
-rw-r--r-- | pages/complex-analysis/one-complex-variable/cauchys-theorem.md | 4 |
2 files changed, 6 insertions, 6 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 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 $G$. |