Laplace transform properties
\(\mathcal{L}\)\(f\left ( t\right ) =F\left ( s\right ) =\int _{0}^{\infty }f\left ( t\right ) e^{-st}dt\)
| \(\mathcal{L}\)\(f\left ( t-a\right ) =e^{-as}F\left ( s\right ) \) | \(\mathcal{L}\)\(\delta \left ( t\right ) =1\ \)impulse | \(\mathcal{L}\)\(u\left ( t\right ) =\frac{1}{s}\) | \(\mathcal{L}\)\(u\left ( t-a\right ) =e^{-as}\frac{1}{s}\) | \(\mathcal{L}\)\(e^{-at}f\left ( t\right ) =F\left ( s+a\right ) \) |
| \(\mathcal{L}\)\(f\left ( \frac{t}{\alpha }\right ) =\alpha F\left ( \alpha \ s\right ) \) | \(\mathcal{L}\)\(t=\frac{1}{s^{2}}\ \)ramp | \(\mathcal{L}\)\(\cos \omega t=\frac{s}{s^{2}+\omega ^{2}}\) | \(\mathcal{L}\)\(\sin \omega t=\frac{\omega }{s^{2}+\omega ^{2}}\) | |