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'

---
 .../concept/sokhotski-plemelj-theorem/index.md     | 32 +++++++++++-----------
 1 file changed, 16 insertions(+), 16 deletions(-)

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

diff --git a/source/know/concept/sokhotski-plemelj-theorem/index.md b/source/know/concept/sokhotski-plemelj-theorem/index.md
index 984a558..66e89bc 100644
--- a/source/know/concept/sokhotski-plemelj-theorem/index.md
+++ b/source/know/concept/sokhotski-plemelj-theorem/index.md
@@ -9,8 +9,8 @@ categories:
 layout: "concept"
 ---
 
-The goal is to evaluate integrals of the following form, where $a < 0 < b$,
-and $f(x)$ is assumed to be continuous in the integration interval $[a, b]$:
+The goal is to evaluate integrals of the following form, where $$a < 0 < b$$,
+and $$f(x)$$ is assumed to be continuous in the integration interval $$[a, b]$$:
 
 $$\begin{aligned}
     \lim_{\eta \to 0^+} \int_a^b \frac{f(x)}{x + i \eta} \dd{x}
@@ -26,7 +26,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 To evaluate the real part,
-we notice that for $\eta \to 0^+$ the integrand diverges for $x \to 0$,
+we notice that for $$\eta \to 0^+$$ the integrand diverges for $$x \to 0$$,
 and thus split the integral as follows:
 
 $$\begin{aligned}
@@ -35,7 +35,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This is simply the definition of the
-[Cauchy principal value](/know/concept/cauchy-principal-value/) $\mathcal{P}$,
+[Cauchy principal value](/know/concept/cauchy-principal-value/) $$\mathcal{P}$$,
 so the real part is given by:
 
 $$\begin{aligned}
@@ -45,7 +45,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Meanwhile, in the imaginary part,
-we substitute $\eta$ for $1 / m$, and introduce $\pi$:
+we substitute $$\eta$$ for $$1 / m$$, and introduce $$\pi$$:
 
 $$\begin{aligned}
     \lim_{\eta \to 0^+} \int_a^b \frac{\eta \: f(x)}{x^2 + \eta^2} \dd{x}
@@ -54,8 +54,8 @@ $$\begin{aligned}
     &= \lim_{m \to +\infty} \frac{\pi}{\pi} \int_a^b \frac{m}{1 + m^2 x^2} f(x) \dd{x}
 \end{aligned}$$
 
-The expression $m / \pi (1 + m^2 x^2)$ is a so-called *nascent delta function*,
-meaning that in the limit $m \to +\infty$ it converges to
+The expression $$m / \pi (1 + m^2 x^2)$$ is a so-called *nascent delta function*,
+meaning that in the limit $$m \to +\infty$$ it converges to
 the [Dirac delta function](/know/concept/dirac-delta-function/):
 
 $$\begin{aligned}
@@ -76,9 +76,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 However, this theorem is often written in the following sloppy way,
-where $\eta$ is defined up front to be small,
-the integral is hidden, and $f(x)$ is set to $1$.
-This awkwardly leaves $\mathcal{P}$ behind:
+where $$\eta$$ is defined up front to be small,
+the integral is hidden, and $$f(x)$$ is set to $$1$$.
+This awkwardly leaves $$\mathcal{P}$$ behind:
 
 $$\begin{aligned}
     \frac{1}{x + i \eta}
@@ -87,13 +87,13 @@ $$\begin{aligned}
 
 The full, complex version of the Sokhotski-Plemelj theorem
 evaluates integrals of the following form
-over a contour $C$ in the complex plane:
+over a contour $$C$$ in the complex plane:
 
 $$\begin{aligned}
     \phi(z) = \frac{1}{2 \pi i} \oint_C \frac{f(\zeta)}{\zeta - z} \dd{\zeta}
 \end{aligned}$$
 
-Where $f(z)$ must be [holomorphic](/know/concept/holomorphic-function/).
+Where $$f(z)$$ must be [holomorphic](/know/concept/holomorphic-function/).
 The Sokhotski-Plemelj theorem then states:
 
 $$\begin{aligned}
@@ -103,8 +103,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where the sign is positive if $z$ is inside $C$, and negative if it is outside.
-The real version follows by letting $C$ follow the whole real axis,
-making $C$ an infinitely large semicircle,
+Where the sign is positive if $$z$$ is inside $$C$$, and negative if it is outside.
+The real version follows by letting $$C$$ follow the whole real axis,
+making $$C$$ an infinitely large semicircle,
 so that the integrand vanishes away from the real axis,
-because $1 / (\zeta \!-\! z) \to 0$ for $|\zeta| \to \infty$.
+because $$1 / (\zeta \!-\! z) \to 0$$ for $$|\zeta| \to \infty$$.
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