2.1 some Laplace properties

Laplace transform properties

$\mathcal{L}$$f\left(t\right)=F\left(s\right)={\int }_0^{\mathrm{\infty }}f\left(t\right)e^{-st}dt$

$\mathcal{L}$$f(t-a)=e^{-as}F\left(s\right)$	$\mathcal{L}$$\delta \left(t\right)=1$impulse	$\mathcal{L}$$u\left(t\right)=\frac{1}{s}$	$\mathcal{L}$$u(t-a)=e^{-as}\frac{1}{s}$	$\mathcal{L}$$e^{-at}f\left(t\right)=F(s+a)$
$\mathcal{L}$$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}$