From f83a8419ba9574fb68d64049abf039c38609f3ea Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 21 Feb 2021 16:13:31 +0100
Subject: Add "Fourier transform"

---
 latex/know/concept/dirac-delta-function/source.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

(limited to 'latex/know/concept/dirac-delta-function')

diff --git a/latex/know/concept/dirac-delta-function/source.md b/latex/know/concept/dirac-delta-function/source.md
index 478efb4..cb98c41 100644
--- a/latex/know/concept/dirac-delta-function/source.md
+++ b/latex/know/concept/dirac-delta-function/source.md
@@ -48,7 +48,7 @@ $$\begin{aligned}
         \delta(x)
         %= \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\}
         = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k}
-        \propto \hat{\mathcal{F}}\{1\}
+        \:\:\propto\:\: \hat{\mathcal{F}}\{1\}
     }
 \end{aligned}$$
 
@@ -79,12 +79,12 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-*__Proof.__ Be careful about which variable is used for the
-differentiation:*
+*__Proof.__ Note which variable is used for the
+differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:*
 
 $$\begin{aligned}
-    \int f(\xi) \: \frac{d\delta(x - \xi)}{dx} \dd{\xi}
-    &= \frac{d}{dx} \int f(\xi) \: \delta(x - \xi) \dd{x}
+    \int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi}
+    &= \dv{x} \int f(\xi) \: \delta(x - \xi) \dd{x}
     = f'(x)
 \end{aligned}$$
 
@@ -94,6 +94,6 @@ This property also generalizes nicely for the higher-order derivatives:
 
 $$\begin{aligned}
     \boxed{
-        \int f(\xi) \: \frac{d^n \delta(x - \xi)}{dx^n} \dd{\xi} = \dv[n]{f(x)}{x}
+        \int f(\xi) \: \dv[n]{\delta(x - \xi)}{x} \dd{\xi} = \dv[n]{f(x)}{x}
     }
 \end{aligned}$$
-- 
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