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'

---
 .../know/concept/kramers-kronig-relations/index.md | 46 +++++++++++-----------
 1 file changed, 23 insertions(+), 23 deletions(-)

(limited to 'source/know/concept/kramers-kronig-relations')

diff --git a/source/know/concept/kramers-kronig-relations/index.md b/source/know/concept/kramers-kronig-relations/index.md
index bb9b08c..3880113 100644
--- a/source/know/concept/kramers-kronig-relations/index.md
+++ b/source/know/concept/kramers-kronig-relations/index.md
@@ -10,22 +10,22 @@ categories:
 layout: "concept"
 ---
 
-Let $\chi(t)$ be a complex function describing
-the response of a system to an impulse $f(t)$ starting at $t = 0$.
-The **Kramers-Kronig relations** connect the real and imaginary parts of $\chi(t)$,
+Let $$\chi(t)$$ be a complex function describing
+the response of a system to an impulse $$f(t)$$ starting at $$t = 0$$.
+The **Kramers-Kronig relations** connect the real and imaginary parts of $$\chi(t)$$,
 such that one can be reconstructed from the other.
-Suppose we can only measure $\chi_r(t)$ or $\chi_i(t)$:
+Suppose we can only measure $$\chi_r(t)$$ or $$\chi_i(t)$$:
 
 $$\begin{aligned}
     \chi(t) = \chi_r(t) + i \chi_i(t)
 \end{aligned}$$
 
-Assuming that the system was at rest until $t = 0$,
-the response $\chi(t)$ cannot depend on anything from $t < 0$,
-since the known impulse $f(t)$ had not started yet,
+Assuming that the system was at rest until $$t = 0$$,
+the response $$\chi(t)$$ cannot depend on anything from $$t < 0$$,
+since the known impulse $$f(t)$$ had not started yet,
 This principle is called **causality**, and to enforce it,
 we use the [Heaviside step function](/know/concept/heaviside-step-function/)
-$\Theta(t)$ to create a **causality test** for $\chi(t)$:
+$$\Theta(t)$$ to create a **causality test** for $$\chi(t)$$:
 
 $$\begin{aligned}
     \chi(t) = \chi(t) \: \Theta(t)
@@ -34,7 +34,7 @@ $$\begin{aligned}
 If we [Fourier transform](/know/concept/fourier-transform/) this equation,
 then it will become a convolution in the frequency domain
 thanks to the [convolution theorem](/know/concept/convolution-theorem/),
-where $A$, $B$ and $s$ are constants from the FT definition:
+where $$A$$, $$B$$ and $$s$$ are constants from the FT definition:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
@@ -42,10 +42,10 @@ $$\begin{aligned}
     = B \int_{-\infty}^\infty \tilde{\chi}(\omega') \: \tilde{\Theta}(\omega - \omega') \dd{\omega'}
 \end{aligned}$$
 
-We look up the FT of the step function $\tilde{\Theta}(\omega)$,
-which involves the signum function $\mathrm{sgn}(t)$,
-the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$,
-and the Cauchy principal value $\pv{}$.
+We look up the FT of the step function $$\tilde{\Theta}(\omega)$$,
+which involves the signum function $$\mathrm{sgn}(t)$$,
+the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta$$,
+and the Cauchy principal value $$\pv{}$$.
 We arrive at:
 
 $$\begin{aligned}
@@ -58,7 +58,7 @@ $$\begin{aligned}
     \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
 \end{aligned}$$
 
-From the definition of the Fourier transform we know that $2 \pi A B / |s| = 1$:
+From the definition of the Fourier transform we know that $$2 \pi A B / |s| = 1$$:
 
 $$\begin{aligned}
     \tilde{\chi}(\omega)
@@ -66,7 +66,7 @@ $$\begin{aligned}
     + \mathrm{sgn}(s) \frac{i}{2 \pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}(\omega')}{\omega - \omega'} \dd{\omega'}}
 \end{aligned}$$
 
-We isolate this equation for $\tilde{\chi}(\omega)$
+We isolate this equation for $$\tilde{\chi}(\omega)$$
 to get the final version of the causality test:
 
 $$\begin{aligned}
@@ -76,7 +76,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-By inserting $\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$
+By inserting $$\tilde{\chi}(\omega) = \tilde{\chi}_r(\omega) + i \tilde{\chi}_i(\omega)$$
 and splitting the equation into real and imaginary parts,
 we get the Kramers-Kronig relations:
 
@@ -92,13 +92,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If the time-domain response function $\chi(t)$ is real
+If the time-domain response function $$\chi(t)$$ is real
 (so far we have assumed it to be complex),
 then we can take advantage of the fact that
 the FT of a real function satisfies
-$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$, i.e. $\tilde{\chi}_r(\omega)$
-is even and $\tilde{\chi}_i(\omega)$ is odd. We multiply the fractions by
-$(\omega' + \omega)$ above and below:
+$$\tilde{\chi}(-\omega) = \tilde{\chi}^*(\omega)$$, i.e. $$\tilde{\chi}_r(\omega)$$
+is even and $$\tilde{\chi}_i(\omega)$$ is odd. We multiply the fractions by
+$$(\omega' + \omega)$$ above and below:
 
 $$\begin{aligned}
     \tilde{\chi}_r(\omega)
@@ -110,8 +110,8 @@ $$\begin{aligned}
     + \frac{\omega}{\pi} \pv{\int_{-\infty}^\infty \frac{\tilde{\chi}_r(\omega')}{ {\omega'}^2 - \omega^2} \dd{\omega'}} \bigg)
 \end{aligned}$$
 
-For $\tilde{\chi}_r(\omega)$, the second integrand is odd, so we can drop it.
-Similarly, for $\tilde{\chi}_i(\omega)$, the first integrand is odd.
+For $$\tilde{\chi}_r(\omega)$$, the second integrand is odd, so we can drop it.
+Similarly, for $$\tilde{\chi}_i(\omega)$$, the first integrand is odd.
 We therefore find the following variant of the Kramers-Kronig relations:
 
 $$\begin{aligned}
@@ -126,7 +126,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-To reiterate: this version is only valid if $\chi(t)$ is real in the time domain.
+To reiterate: this version is only valid if $$\chi(t)$$ is real in the time domain.
 
 
 
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