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Diffstat (limited to 'source/know/concept/laplace-transform')
| -rw-r--r-- | source/know/concept/laplace-transform/index.md | 21 |
1 files changed, 7 insertions, 14 deletions
diff --git a/source/know/concept/laplace-transform/index.md b/source/know/concept/laplace-transform/index.md index c7f352a..94c3742 100644 --- a/source/know/concept/laplace-transform/index.md +++ b/source/know/concept/laplace-transform/index.md @@ -35,6 +35,7 @@ using [partial fraction decomposition](/know/concept/partial-fraction-decomposit and then looking up the individual terms. + ## Derivatives The derivative of a transformed function is the transform @@ -55,11 +56,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dv-s"/> -<label for="proof-dv-s">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dv-s">Proof.</label> + +{% include proof/start.html id="proof-dv-s" -%} The exponential $$\exp(- s t)$$ is the only thing that depends on $$s$$ here: $$\begin{aligned} @@ -69,9 +67,8 @@ $$\begin{aligned} &= \int_0^\infty (-t)^n f(t) \exp(- s t) \dd{t} = (-1)^n \hat{\mathcal{L}}\{t^n f(t)\} \end{aligned}$$ +{% include proof/end.html id="proof-dv-s" %} -</div> -</div> The Laplace transform of a derivative introduces the initial conditions into the result. Notice that $$f(0)$$ is the initial value in the original $$t$$-domain: @@ -98,11 +95,8 @@ and $$f^{(0)}(t) = f(t)$$. As an example, $$\hat{\mathcal{L}}\{f'''(t)\}$$ becomes $$- f''(0) - s f'(0) - s^2 f(0) + s^3 \tilde{f}(s)$$. -<div class="accordion"> -<input type="checkbox" id="proof-dv-t"/> -<label for="proof-dv-t">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dv-t">Proof.</label> + +{% include proof/start.html id="proof-dv-t" -%} We integrate by parts and use the fact that $$\lim_{x \to \infty} \exp(-x) = 0$$: $$\begin{aligned} @@ -116,8 +110,7 @@ $$\begin{aligned} And so on. By partially integrating $$n$$ times in total we arrive at the conclusion. -</div> -</div> +{% include proof/end.html id="proof-dv-t" %} |