# Transformées inverses de Laplace usuelles

| Transformée de Laplace de la fonction | Fonction |
| $1$ | $\delta (t)$ |
| ${\frac {1}{p}}$ | $1$ |
| ${\frac {1}{p^{2}}}$ | $t$ |
| ${\frac {1}{p^{n}}}$ | ${\frac {t^{n-1}}{(n-1)!}}\qquad \forall n\geq 1$ |
| ${\frac {1}{\sqrt {p}}}$ | ${\frac {1}{\sqrt {\pi t}}}\qquad \forall t\in R_{+}^{*}$ |
| ${\frac {1}{\sqrt {p^{3}}}}$ | $2{\sqrt {\frac {t}{\pi }}}$ |
| ${\frac {1}{p+a}}$ | $e^{-at}$ |
| ${\frac {1}{p(p+a)}}$ | ${1-e^{-at} \over a}\qquad \forall a\neq 0$ |
| ${\frac {1}{p^{2}(p+a)}}$ | ${\frac {e^{-at}-1+at}{a^{2}}}\qquad \forall a\neq 0$ |
| ${\frac {1}{p(p+a)^{2}}}$ | ${1-e^{-at}(1+at) \over a^{2}}\qquad \forall a\neq 0$ |
| ${\frac {1}{(p+a)(p+b)}}$ | ${\frac {e^{-bt}-e^{-at}}{a-b}}\qquad \forall a\neq b$ |
| ${\frac {p}{(p+a)(p+b)}}$ | ${\frac {ae^{-at}-be^{-bt}}{a-b}}\qquad \forall a\neq b$ |
| ${\frac {1}{(p+a)(p+b)(p+c)}}$ | ${(b-c)e^{-at}+(c-a)e^{-bt}+(a-b)e^{-ct} \over (a-b)(a-c)(b-c)}\qquad \forall a\neq b\lor a\neq c\lor b\neq c$ |
| ${\frac {1}{(p+a)^{2}}}$ | $te^{-at}$ |
| ${\frac {p}{(p+a)^{2}}}$ | $e^{-at}(1-at)$ |
| ${\frac {1}{(p+a)(p+b)^{2}}}$ | ${\frac {e^{-at}+\left[(a-b)t-1\right]e^{-bt}}{(a-b)^{2}}}\qquad \forall a\neq b$ |
| ${\frac {1}{p(p+a)(p+b)}}$ | ${be^{-at}-ae^{-bt}+a-b \over \ a^{2}b-ab^{2}}\qquad \forall a\neq b$ |
| ${\frac {p+c}{p(p+a)(p+b)}}$ | ${b(c-a)e^{-at}+a(b-c)e^{-bt}+c(a-b) \over \ ab(a-b)}\qquad \forall a\neq b$ |
| ${\frac {p^{2}+cp+d}{p(p+a)(p+b)}}$ | ${b(a^{2}-ac+d)e^{-at}-a(b^{2}-bc+d)e^{-bt}+d(a-b) \over \ ab(a-b)}\qquad \forall a\neq b$ |
| ${\frac {1}{(p+a)^{3}}}$ | ${\frac {t^{2}e^{-at}}{2}}$ |
| $\ln \left({\frac {p+a}{p+b}}\right)$ | ${\frac {e^{-bt}-e^{-at}}{t}}\qquad \forall t\in R_{+}^{*}$ |
| ${\frac {1}{p^{2}+a^{2}}}$ | ${\sin(at) \over a}\qquad \forall a\neq 0$ |
| ${\frac {1}{p(p^{2}+a^{2})}}$ | ${1-\cos(at) \over a^{2}}\qquad \forall a\neq 0$ |
| ${\frac {p}{p^{2}+a^{2}}}$ | $\cos(at)$ |
| ${\frac {p+a}{p(p^{2}+b^{2})}}$ | ${a(1-\cos(bt))+b\sin(bt) \over b^{2}}\qquad \forall b\neq 0$ |
| ${\frac {p^{2}+cp+d}{p(p^{2}+b^{2})}}$ | ${(b^{2}-d)\cos(bt)+bc\sin(bt)+d \over b^{2}}\qquad \forall b\neq 0$ |
| ${\frac {1}{p^{2}-a^{2}}}$ | ${\sinh(at) \over a}\qquad \forall a\neq 0$ |
| ${\frac {p}{p^{2}-a^{2}}}$ | $\cosh(at)$ |
| ${\frac {1}{(p+b)^{2}+a^{2}}}$ | ${e^{-bt}\sin(at) \over a}\qquad \forall a\neq 0$ |
| ${\frac {p+b}{(p+b)^{2}+a^{2}}}$ | $e^{-bt}\cos(at)$ |
| ${\frac {1}{(p^{2}+a^{2})^{2}}}$ | ${\sin(at)-at\cos(at) \over 2a^{3}}\qquad \forall a\neq 0$ |
| ${\frac {p}{(p^{2}+a^{2})^{2}}}$ | ${t\sin(at) \over 2a}\qquad \forall a\neq 0$ |
| ${\frac {p^{2}}{(p^{2}+a^{2})^{2}}}$ | ${\sin(at)+at\cos(at) \over 2a}\qquad \forall a\neq 0$ |
| ${\frac {1}{p^{3}+a^{3}}}$ | ${e^{-at}-e^{\frac {at}{2}}\left(\cos \left({\frac {\sqrt {3}}{2}}at\right)-{\sqrt {3}}\sin \left({\frac {\sqrt {3}}{2}}at\right)\right) \over 3a^{2}}\qquad \forall a\neq 0$ |
| ${\frac {p}{p^{3}+a^{3}}}$ | ${e^{at \over 2}\left({\sqrt {3}}\sin \left({\frac {{\sqrt {3}}at}{2}}\right)+\cos \left({\frac {{\sqrt {3}}at}{2}}\right)\right)-e^{-at} \over 3a}\qquad \forall a\neq 0$ |
| ${\frac {p^{2}}{p^{3}+a^{3}}}$ | ${e^{-at}+2e^{\frac {at}{2}}\cos \left({\frac {{\sqrt {3}}at}{2}}\right) \over 3}$ |
| ${\frac {1}{p^{2}(1+\tau _{1}p)(1+\tau _{2}p)}}$ | ${\tau _{1}^{2}e^{-t/\tau _{1}}-\tau _{2}^{2}e^{-t/\tau _{2}}+(\tau _{1}-\tau _{2})(t-\tau _{1}-\tau _{2}) \over \tau _{1}-\tau _{2}}\qquad \forall \tau _{1}\neq \tau _{2}$ |