From cc295b5da8e3db4417523a507caf106d5839d989 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 2 Jun 2021 13:28:53 +0200
Subject: Introduce collapsible proofs to some articles

---
 .../know/concept/heaviside-step-function/index.pdc | 27 +++++++++++++++-------
 1 file changed, 19 insertions(+), 8 deletions(-)

(limited to 'content/know/concept/heaviside-step-function/index.pdc')

diff --git a/content/know/concept/heaviside-step-function/index.pdc b/content/know/concept/heaviside-step-function/index.pdc
index 0471acf..dbbca6f 100644
--- a/content/know/concept/heaviside-step-function/index.pdc
+++ b/content/know/concept/heaviside-step-function/index.pdc
@@ -50,7 +50,23 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The [Fourier transform](/know/concept/fourier-transform/)
-of $\Theta(t)$ is noteworthy.
+of $\Theta(t)$ is as follows,
+where $\pv{}$ is the Cauchy principal value,
+$A$ and $s$ are constants from the FT's definition,
+and $\mathrm{sgn}$ is the signum function:
+
+$$\begin{aligned}
+    \boxed{
+        \tilde{\Theta}(\omega)
+        = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big)
+    }
+\end{aligned}$$
+
+<div class="accordion">
+<input type="checkbox" id="proof-fourier"/>
+<label for="proof-fourier">Proof</label>
+<div class="hidden">
+<label for="proof-fourier">Proof.</label>
 In this case, it is easiest to use $\Theta(0) = 1/2$,
 such that the Heaviside step function can be expressed
 using the signum function $\mathrm{sgn}(t)$:
@@ -77,15 +93,10 @@ $$\begin{aligned}
     &= A \pi \delta(s \omega) + \frac{A}{2} \pv{\int_{-\infty}^\infty \mathrm{sgn}(t) \exp(i s \omega t) \dd{t}}
     = \frac{A}{|s|} \pi \delta(\omega) + i \frac{A}{s} \pv{\frac{1}{\omega}}
 \end{aligned}$$
+</div>
+</div>
 
 The use of $\pv{}$ without an integral is an abuse of notation,
 and means that this result only makes sense when wrapped in an integral.
 Formally, $\pv{\{1 / \omega\}}$ is a [Schwartz distribution](/know/concept/schwartz-distribution/).
-We thus have:
 
-$$\begin{aligned}
-    \boxed{
-        \tilde{\Theta}(\omega)
-        = \frac{A}{|s|} \Big( \pi \delta(\omega) + i \: \mathrm{sgn}(s) \pv{\frac{1}{\omega}} \Big)
-    }
-\end{aligned}$$
-- 
cgit v1.2.3