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Diffstat (limited to 'source/know/concept/dirac-delta-function/index.md')
| -rw-r--r-- | source/know/concept/dirac-delta-function/index.md | 18 |
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> |