From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/residue-theorem/index.md | 27 ++++++++++++++-------------
 1 file changed, 14 insertions(+), 13 deletions(-)

(limited to 'source/know/concept/residue-theorem/index.md')

diff --git a/source/know/concept/residue-theorem/index.md b/source/know/concept/residue-theorem/index.md
index bcbd6bd..b58e3c2 100644
--- a/source/know/concept/residue-theorem/index.md
+++ b/source/know/concept/residue-theorem/index.md
@@ -8,21 +8,21 @@ categories:
 layout: "concept"
 ---
 
-A function $f(z)$ is **meromorphic** if it is
+A function $$f(z)$$ is **meromorphic** if it is
 [holomorphic](/know/concept/holomorphic-function/)
 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 $z_p$.
-In other words, $f(z)$ can be approximated close to $z_p$:
+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 $$z_p$$.
+In other words, $$f(z)$$ can be approximated close to $$z_p$$:
 
 $$\begin{aligned}
     f(z)
     \approx \frac{R_p}{z - z_p}
 \end{aligned}$$
 
-Where 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 $z_p$:
+Where 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 $$z_p$$:
 
 $$\begin{aligned}
     \boxed{
@@ -32,8 +32,8 @@ $$\begin{aligned}
 
 **Cauchy's residue theorem** for meromorphic functions
 is a generalization of Cauchy's integral theorem for holomorphic functions,
-and states that the integral on a contour $C$
-purely depends on the simple poles $z_p$ enclosed by $C$:
+and states that the integral on a contour $$C$$
+purely depends on the simple poles $$z_p$$ enclosed by $$C$$:
 
 $$\begin{aligned}
     \boxed{
@@ -47,14 +47,14 @@ $$\begin{aligned}
 <div class="hidden" markdown="1">
 <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 $z_p$ are all its poles:
+we know that we can decompose $$f(z)$$ like so,
+where $$h(z)$$ is holomorphic and $$z_p$$ are all its poles:
 
 $$\begin{aligned}
     f(z) = h(z) + \sum_{z_p} \frac{R_p}{z - z_p}
 \end{aligned}$$
 
-We integrate this over a contour $C$ which contains all poles, and apply
+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:
 
 $$\begin{aligned}
@@ -62,11 +62,12 @@ $$\begin{aligned}
     &= \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, by cleverly choosing the contour $C$,
+but in fact, by cleverly choosing the contour $$C$$,
 it lets us 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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