Démontrer que
1
\(\mathcal{L}(1) (s)= \frac 1 s\)
2
\(\mathcal{L}(t^n)(s) = \frac {n!} {s^{n+1}}\)
3
\(\mathcal{L}(e^{at})(s) = \frac 1 {s-a}\)
4
\(\mathcal{L}(\sin at)(s) = \frac a {s^2+a^2}\)
5
\(\mathcal{L}(\cos at)(s) = \frac s {s^2+a^2}\)
6
\[\mathcal{L}(U_a)(s) = \frac {e^{-as}} s, \ \text{où}\ U_a(t) = \begin{cases} 1,&t\geq a,\\ 0,&t < a,\\ \end{cases} a \in \Rr, a \geq 0.\]