summaryrefslogtreecommitdiff
path: root/source/know/concept/dirac-delta-function
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/dirac-delta-function')
-rw-r--r--source/know/concept/dirac-delta-function/index.md18
1 files changed, 10 insertions, 8 deletions
diff --git a/source/know/concept/dirac-delta-function/index.md b/source/know/concept/dirac-delta-function/index.md
index 88a08cb..518eba1 100644
--- a/source/know/concept/dirac-delta-function/index.md
+++ b/source/know/concept/dirac-delta-function/index.md
@@ -8,10 +8,10 @@ categories:
layout: "concept"
---
-The **Dirac delta function** $\delta(x)$, often just the **delta function**,
+The **Dirac delta function** $$\delta(x)$$, often just the **delta function**,
is a function (or, more accurately, a [Schwartz distribution](/know/concept/schwartz-distribution/))
that is commonly used in physics.
-It is an infinitely narrow discontinuous "spike" at $x = 0$ whose area is
+It is an infinitely narrow discontinuous "spike" at $$x = 0$$ whose area is
defined to be 1:
$$\begin{aligned}
@@ -35,7 +35,7 @@ $$\begin{aligned}
}
\end{aligned}$$
-$\delta(x)$ is thus quite an effective weapon against integrals. This may not seem very
+$$\delta(x)$$ is thus quite an effective weapon against integrals. This may not seem very
useful due to its "unnatural" definition, but in fact it appears as the
limit of several reasonable functions:
@@ -57,7 +57,7 @@ $$\begin{aligned}
\:\:\propto\:\: \hat{\mathcal{F}}\{1\}
\end{aligned}$$
-When the argument of $\delta(x)$ is scaled, the delta function is itself scaled:
+When the argument of $$\delta(x)$$ is scaled, the delta function is itself scaled:
$$\begin{aligned}
\boxed{
@@ -70,17 +70,18 @@ $$\begin{aligned}
<label for="proof-scale">Proof</label>
<div class="hidden" markdown="1">
<label for="proof-scale">Proof.</label>
-Because it is symmetric, $\delta(s x) = \delta(|s| x)$.
-Then by substituting $\sigma = |s| x$:
+Because it is symmetric, $$\delta(s x) = \delta(|s| x)$$.
+Then by substituting $$\sigma = |s| x$$:
$$\begin{aligned}
\int \delta(|s| x) \dd{x}
&= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|}
\end{aligned}$$
+
</div>
</div>
-An even more impressive property is the behaviour of the derivative of $\delta(x)$:
+An even more impressive property is the behaviour of the derivative of $$\delta(x)$$:
$$\begin{aligned}
\boxed{
@@ -94,13 +95,14 @@ $$\begin{aligned}
<div class="hidden" markdown="1">
<label for="proof-dv1">Proof.</label>
Note which variable is used for the
-differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:
+differentiation, and that $$\delta'(x - \xi) = - \delta'(\xi - x)$$:
$$\begin{aligned}
\int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi}
&= \dv{}{x}\int f(\xi) \: \delta(x - \xi) \dd{x}
= f'(x)
\end{aligned}$$
+
</div>
</div>