5. Some Rules

The \(s\)-differentiation rule: Multiplying \(f(t)\) by \(t\) applies \(-\dfrac{d}{ds}\) to the transform of \(f(t)\). \[\mathcal{L}[tf(t)] = -\frac{d}{ds}\left\{\mathcal{L}[f(t)] \right\} = -\frac{d}{ds}[F(s)]\]

Example:

\[\begin{aligned} \mathcal{L}[t\sin(\omega t)] &=-\frac{d}{ds}\left(\frac{\omega}{s^2+\omega^2} \right) \\ &= -\frac{(s^2+\omega^2)\times0-\omega\times(2s)}{(s^2+\omega^2)^2} \\ &= \frac{2\omega s}{(s^2+\omega^2)^2} \end{aligned}\]

First shifting rule: \[\mathcal{L}[e^{at}f(t)] = \mathcal{L}[f(t)]_{s\rightarrow(s-a)}\]

Example:

\[\begin{aligned} \mathcal{L}[e^{at}\sin(\omega t)] &= \mathcal{L}[\sin(\omega t)]_{s\rightarrow (s-a)} \\ &= \left.\frac{\omega}{s^2+\omega^2}\right|_{s\rightarrow (s-a)} \\ &= \frac{\omega}{(s-a)^2+\omega^2} \end{aligned}\]