From c0d352dd0f66b47ee91fb96eaf320f895fa78790 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 14 Nov 2021 17:54:04 +0100
Subject: Expand knowledge base

---
 .../know/concept/holomorphic-function/index.pdc    | 57 ----------------------
 1 file changed, 57 deletions(-)

(limited to 'content/know/concept/holomorphic-function')

diff --git a/content/know/concept/holomorphic-function/index.pdc b/content/know/concept/holomorphic-function/index.pdc
index 4b7221c..3e3984a 100644
--- a/content/know/concept/holomorphic-function/index.pdc
+++ b/content/know/concept/holomorphic-function/index.pdc
@@ -193,60 +193,3 @@ this proof works inductively for all higher orders $n$.
 </div>
 </div>
 
-
-## Residue theorem
-
-A function $f(z)$ is **meromorphic** if it is holomorphic except in
-a finite number of **simple poles**, which are points $z_p$ where
-$f(z_p)$ diverges, but where the product $(z - z_p) f(z)$ is non-zero and
-still holomorphic close to $z_p$.
-
-The **residue** $R_p$ of a simple pole $z_p$ is defined as follows, and
-represents the rate at which $f(z)$ diverges close to $z_p$:
-
-$$\begin{aligned}
-    \boxed{
-        R_p = \lim_{z \to z_p} (z - z_p) f(z)
-    }
-\end{aligned}$$
-
-**Cauchy's residue theorem** generalizes Cauchy's integral theorem
-to meromorphic functions, and states that the integral of a contour $C$
-depends on the simple poles $p$ it encloses:
-
-$$\begin{aligned}
-    \boxed{
-        \oint_C f(z) \dd{z} = i 2 \pi \sum_{p} R_p
-    }
-\end{aligned}$$
-
-<div class="accordion">
-<input type="checkbox" id="proof-res-theorem"/>
-<label for="proof-res-theorem">Proof</label>
-<div class="hidden">
-<label for="proof-res-theorem">Proof.</label>
-From the definition of a meromorphic function,
-we know that we can decompose $f(z)$ like so,
-where $h(z)$ is holomorphic and $p$ are all its poles:
-
-$$\begin{aligned}
-    f(z) = h(z) + \sum_{p} \frac{R_p}{z - z_p}
-\end{aligned}$$
-
-We integrate this over a contour $C$ which contains all poles, and apply
-both Cauchy's integral theorem and Cauchy's integral formula to get:
-
-$$\begin{aligned}
-    \oint_C f(z) \dd{z}
-    &= \oint_C h(z) \dd{z} + \sum_{p} R_p \oint_C \frac{1}{z - z_p} \dd{z}
-    = \sum_{p} R_p \: 2 \pi i
-\end{aligned}$$
-</div>
-</div>
-
-This theorem might not seem very useful,
-but in fact, thanks to some clever mathematical magic,
-it allows us to evaluate many integrals along the real axis,
-most notably [Fourier transforms](/know/concept/fourier-transform/).
-It can also be used to derive the [Kramers-Kronig relations](/know/concept/kramers-kronig-relations).
-
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