Systèmes et représentations/Fiche/Table des transformées de Laplace

Fiche mémoire sur les transformées de Laplace

$\begin{array}{cc}\text{Domaine~de~Laplace}& \text{Domaine~temporel}\\ F(s)& f(t)\\ \\ F(s+a)& e^{-at}f(t)\\ \\ 1& \delta (t)\\ \\ \frac{1}{s}& \Gamma (t)\\ \\ \frac{1}{s^2}& t\\ \\ \frac{1}{1+\tau s}& \frac{1}{\tau }{e}^{-\frac{t}{\tau }}\\ \\ \frac{1}{s(1+\tau s)}& 1-e^{-\frac{t}{\tau }}\\ \\ \frac{1}{s^2(1+\tau s)}& t-\tau +\tau e^{-\frac{t}{\tau }}\\ \end{array}$

$\begin{array}{cc}\text{Domaine~de~Laplace}& \text{Domaine~temporel}\\ \frac{1}{(1+{\tau }_{1}s)(1+{\tau }_{2}s)}& \frac{1}{{\tau }_1-{\tau }_2}(e^{-\frac{t}{{\tau }_1}}-e^{-\frac{t}{{\tau }_2}})\\ \\ \frac{1}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}& \frac{1}{{\omega }_p}\sin ({\omega }_{p}t)e^{-\zeta {\omega }_{n}t}\\ \\ \frac{1}{s(s^2+2\zeta {\omega }_{n}s+{\omega }_n^2)}& \frac{1}{{\omega }_n^2}-\frac{1}{{\omega }_p{\omega }_n}\sin ({\omega }_{p}t+\phi )e^{-\zeta {\omega }_{n}t}\\ \\ \frac{s}{s^2+{\omega }^2}& \cos (\omega t)\\ \\ \frac{\omega }{s^2+{\omega }^2}& \sin (\omega t)\\ \end{array}$ ${\omega }_p={\omega }_n\sqrt{1-{\zeta }^2}$ $\phi =\arccos (\zeta )$