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13.2: Перетворення Лапласа
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\[\begin{array} {rcl} {\mathcal{L} (e^{at} ; s)} & = & {\int_{0}^{\infty} e^{at} e^{-st}\ dt = \int_{0}^{\infty} e^{(a - s) t} \ dt = \dfrac{e^{(a - s) t}}{a - s} \vert_{0}^{\infty}} \\ {rcl} {} & = &... $\begin{array} {rcl} {\mathcal{L} (e^{at} ; s)} & = & {\int_{0}^{\infty} e^{at} e^{-st}\ dt = \int_{0}^{\infty} e^{(a - s) t} \ dt = \dfrac{e^{(a - s) t}}{a - s} \vert_{0}^{\infty}} \\ {rcl} {} & = & {= \begin{cases} \dfrac{1}{s - a} & \text{ if Re} (s) > \text{Re} (a) \\ \text{divergent} & \text{ otherwise} \end{cases}} \end{array}$