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/impulse-response/index.md | 31 ++++++++++++++-------------
 1 file changed, 16 insertions(+), 15 deletions(-)

(limited to 'source/know/concept/impulse-response')

diff --git a/source/know/concept/impulse-response/index.md b/source/know/concept/impulse-response/index.md
index 533580d..397ac2d 100644
--- a/source/know/concept/impulse-response/index.md
+++ b/source/know/concept/impulse-response/index.md
@@ -8,9 +8,9 @@ categories:
 layout: "concept"
 ---
 
-The **impulse response** $u_p(t)$ of a system whose behaviour is described
-by a linear operator $\hat{L}$, is defined as the reponse of the system
-when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $\delta(t)$:
+The **impulse response** $$u_p(t)$$ of a system whose behaviour is described
+by a linear operator $$\hat{L}$$, is defined as the reponse of the system
+when forced by the [Dirac delta function](/know/concept/dirac-delta-function/) $$\delta(t)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -18,9 +18,9 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-This can be used to find the response $u(t)$ of $\hat{L}$ to
-*any* forcing function $f(t)$, i.e. not only $\delta(t)$,
-by simply taking the convolution with $u_p(t)$:
+This can be used to find the response $$u(t)$$ of $$\hat{L}$$ to
+*any* forcing function $$f(t)$$, i.e. not only $$\delta(t)$$,
+by simply taking the convolution with $$u_p(t)$$:
 
 $$\begin{aligned}
     \hat{L} \{ u(t) \} = f(t)
@@ -35,9 +35,9 @@ $$\begin{aligned}
 <label for="proof-main">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-main">Proof.</label>
-Starting from the definition of $u_p(t)$,
-we shift the argument by some constant $\tau$,
-and multiply both sides by the constant $f(\tau)$:
+Starting from the definition of $$u_p(t)$$,
+we shift the argument by some constant $$\tau$$,
+and multiply both sides by the constant $$f(\tau)$$:
 
 $$\begin{aligned}
     \hat{L} \{ u_p(t - \tau) \} &= \delta(t - \tau)
@@ -45,8 +45,8 @@ $$\begin{aligned}
     \hat{L} \{ f(\tau) \: u_p(t - \tau) \} &= f(\tau) \: \delta(t - \tau)
 \end{aligned}$$
 
-Where $f(\tau)$ can be moved inside using the
-linearity of $\hat{L}$. Integrating over $\tau$ then gives us:
+Where $$f(\tau)$$ can be moved inside using the
+linearity of $$\hat{L}$$. Integrating over $$\tau$$ then gives us:
 
 $$\begin{aligned}
     \int_0^\infty \hat{L} \{ f(\tau) \: u_p(t - \tau) \} \dd{\tau}
@@ -54,20 +54,21 @@ $$\begin{aligned}
     = f(t)
 \end{aligned}$$
 
-The integral and $\hat{L}$ are operators of different variables, so we reorder them:
+The integral and $$\hat{L}$$ are operators of different variables, so we reorder them:
 
 $$\begin{aligned}
     \hat{L} \int_0^\infty f(\tau) \: u_p(t - \tau) \dd{\tau}
     &= (f * u_p)(t) = \hat{L}\{ u(t) \} = f(t)
 \end{aligned}$$
+
 </div>
 </div>
 
 This is useful for solving initial value problems,
 because any initial condition can be satisfied
-due to the linearity of $\hat{L}$,
-by choosing the initial values of the homogeneous solution $\hat{L}\{ u_h(t) \} = 0$
-such that the total solution $(f * u_p)(t) + u_h(t)$
+due to the linearity of $$\hat{L}$$,
+by choosing the initial values of the homogeneous solution $$\hat{L}\{ u_h(t) \} = 0$$
+such that the total solution $$(f * u_p)(t) + u_h(t)$$
 has the desired values.
 
 Meanwhile, for boundary value problems,
-- 
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