From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS
---
source/know/concept/holomorphic-function/index.md | 32 ++++++++---------------
1 file changed, 11 insertions(+), 21 deletions(-)
(limited to 'source/know/concept/holomorphic-function')
diff --git a/source/know/concept/holomorphic-function/index.md b/source/know/concept/holomorphic-function/index.md
index 5dde240..cf252c0 100644
--- a/source/know/concept/holomorphic-function/index.md
+++ b/source/know/concept/holomorphic-function/index.md
@@ -61,6 +61,7 @@ and imaginary parts satisfy these equations. This gives an idea of how
strict the criteria are to qualify as holomorphic.
+
## Integration formulas
Holomorphic functions satisfy **Cauchy's integral theorem**, which states
@@ -73,11 +74,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-int-theorem" -%}
Just like before, we decompose $$f(z)$$ into its real and imaginary parts:
$$\begin{aligned}
@@ -97,8 +95,8 @@ $$\begin{aligned}
Since $$f(z)$$ is holomorphic, $$u$$ and $$v$$ satisfy the Cauchy-Riemann
equations, such that the integrands disappear and the final result is zero.
-
-
+{% include proof/end.html id="proof-int-theorem" %}
+
An interesting consequence is **Cauchy's integral formula**, which
states that the value of $$f(z)$$ at an arbitrary point $$z_0$$ is
@@ -110,11 +108,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-int-formula" -%}
Thanks to the integral theorem, we know that the shape and size
of $$C$$ is irrelevant. Therefore we choose it to be a circle with radius $$r$$,
such that the integration variable becomes $$z = z_0 + r e^{i \theta}$$. Then
@@ -133,9 +128,8 @@ $$\begin{aligned}
&= \frac{f(z_0)}{2 \pi} \int_0^{2 \pi} \dd{\theta}
= f(z_0)
\end{aligned}$$
+{% include proof/end.html id="proof-int-formula" %}
-
-
Similarly, **Cauchy's differentiation formula**,
or **Cauchy's integral formula for derivatives**
@@ -149,11 +143,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-dv-formula" -%}
By definition, the first derivative $$f'(z)$$ of a
holomorphic function exists and is:
@@ -186,6 +177,5 @@ $$\begin{aligned}
Since the second-order derivative $$f''(z)$$ is simply the derivative of $$f'(z)$$,
this proof works inductively for all higher orders $$n$$.
-
-
+{% include proof/end.html id="proof-dv-formula" %}
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