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Diffstat (limited to 'content/know/concept/dirac-delta-function/index.pdc')
| -rw-r--r-- | content/know/concept/dirac-delta-function/index.pdc | 41 |
1 files changed, 24 insertions, 17 deletions
diff --git a/content/know/concept/dirac-delta-function/index.pdc b/content/know/concept/dirac-delta-function/index.pdc index 76b6e97..9eecefd 100644 --- a/content/know/concept/dirac-delta-function/index.pdc +++ b/content/know/concept/dirac-delta-function/index.pdc @@ -21,7 +21,7 @@ defined to be 1: $$\begin{aligned} \boxed{ - \delta(x) = + \delta(x) \equiv \begin{cases} +\infty & \mathrm{if}\: x = 0 \\ 0 & \mathrm{if}\: x \neq 0 @@ -56,12 +56,10 @@ following integral, which appears very often in the context of [Fourier transforms](/know/concept/fourier-transform/): $$\begin{aligned} - \boxed{ - \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\} - } + \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\} \end{aligned}$$ When the argument of $\delta(x)$ is scaled, the delta function is itself scaled: @@ -72,18 +70,22 @@ $$\begin{aligned} } \end{aligned}$$ -*__Proof.__ Because it is symmetric, $\delta(s x) = \delta(|s| x)$. Then by -substituting $\sigma = |s| x$:* +<div class="accordion"> +<input type="checkbox" id="proof-scale"/> +<label for="proof-scale">Proof</label> +<div class="hidden"> +<label for="proof-scale">Proof.</label> +Because it is symmetric, $\delta(s x) = \delta(|s| x)$. +Then by substituting $\sigma = |s| x$: $$\begin{aligned} \int \delta(|s| x) \dd{x} &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|} \end{aligned}$$ +</div> +</div> -*__Q.E.D.__* - -An even more impressive property is the behaviour of the derivative of -$\delta(x)$: +An even more impressive property is the behaviour of the derivative of $\delta(x)$: $$\begin{aligned} \boxed{ @@ -91,16 +93,21 @@ $$\begin{aligned} } \end{aligned}$$ -*__Proof.__ Note which variable is used for the -differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:* +<div class="accordion"> +<input type="checkbox" id="proof-dv1"/> +<label for="proof-dv1">Proof</label> +<div class="hidden"> +<label for="proof-dv1">Proof.</label> +Note which variable is used for the +differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$: $$\begin{aligned} \int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi} &= \dv{x} \int f(\xi) \: \delta(x - \xi) \dd{x} = f'(x) \end{aligned}$$ - -*__Q.E.D.__* +</div> +</div> This property also generalizes nicely for the higher-order derivatives: |