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13.4: Властивості перетворення Лапласа
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$\mathcal{L} (f''; s) = \mathcal{L} ((f')'; s) = s \mathcal{L} (f'; s) - f'(0) = s(sF(s) - f(0)) - f'(0) = s^2 F(s) - sf(0) - f'(0). \text{ QED}$ \(\begin{array} {rcl} {\mathcal{L} (f(t - a); s)} & ... $\mathcal{L} (f''; s) = \mathcal{L} ((f')'; s) = s \mathcal{L} (f'; s) - f'(0) = s(sF(s) - f(0)) - f'(0) = s^2 F(s) - sf(0) - f'(0). \text{ QED}$ $\begin{array} {rcl} {\mathcal{L} (f(t - a); s)} & = & {\int_{0}^{\infty} f(t - a) e^{-st}\ dt = \int_{a}^{\infty} f(t - a) e^{-st}\ dt} \\ {} & = & {\int_{0}^{\infty} f(\tau) e^{-s(\tau + a)} \ d \tau = e^{-sa} \int_{0}^{\infty} f(\tau) e^{-s \tau} \ d \tau = e^{-sa} F(s).} \end{array}$