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fix(appB): correct Laplace derivative formula

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2026-06-04 19:10:58 -05:00
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appendices/appB_transform_tables.tex
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@@ -63,7 +63,7 @@ The following algebraic and operational properties make the Laplace transform a
Linearity & $\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)$ \\[6pt] Linearity & $\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)$ \\[6pt]
First shift & $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ \\[6pt] First shift & $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ \\[6pt]
Second shift & $\mathcal{L}\{u_{c}(t)\,f(t-c)\} = e^{-cs}F(s)$ \\[6pt] Second shift & $\mathcal{L}\{u_{c}(t)\,f(t-c)\} = e^{-cs}F(s)$ \\[6pt]
Derivative & $\mathcal{L}\{f'(t)\} = s F(s) - s f(0)$ \\[6pt] Derivative & $\mathcal{L}\{f'(t)\} = s F(s) - f(0)$ \\[6pt]
Second derivative & $\mathcal{L}\{f''(t)\} = s^{2}F(s) - s f(0) - f'(0)$ \\[6pt] Second derivative & $\mathcal{L}\{f''(t)\} = s^{2}F(s) - s f(0) - f'(0)$ \\[6pt]
$n$th derivative & $\mathcal{L}\{f^{(n)}(t)\} = s^{n}F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$ \\[6pt] $n$th derivative & $\mathcal{L}\{f^{(n)}(t)\} = s^{n}F(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)$ \\[6pt]
Integral & $\displaystyle \mathcal{L}\!\left\{\int_{0}^{t} f(\tau)\,\diff\tau\right\} = \frac{F(s)}{s}$ \\[8pt] Integral & $\displaystyle \mathcal{L}\!\left\{\int_{0}^{t} f(\tau)\,\diff\tau\right\} = \frac{F(s)}{s}$ \\[8pt]