Théorie physique des distributions/Fiche/Table des transformées de Laplace

Fiche mémoire sur les transformées de Laplace usuelles

Transformées de Laplace directes () Fonction Transformée de Laplace de la fonction $\delta (t)$ $1$ $1$ ${\frac {1}{p}}$ $t$ ${\frac {1}{p^{2}}}$ $t^{n}\qquad \forall n\geq 0$ ${\frac {n!}{p^{n+1}}}$ ${\sqrt {t}}\qquad \forall t\in R_{+}$ ${\frac {1}{2}}{\sqrt {\frac {\pi }{p^{3}}}}$ ${\frac {1}{\sqrt {t}}}\qquad \forall t\in R_{+}^{*}$ ${\sqrt {\frac {\pi }{p}}}$ $e^{-ct}$ avec ($c\geq 0$) ${\frac {1}{p+c}}$ $te^{-ct}$ ${\frac {1}{(p+c)^{2}}}$ $t^{2}e^{-ct}$ ${\frac {2}{(p+c)^{3}}}$ $t^{n}e^{-ct}\qquad \forall n\geq 0$ ${\frac {n!}{(p+c)^{n+1}}}$ $a^{t}\qquad \forall a>0$ ${\frac {1}{p-\ln(a)}}$ $\sin(at)$ ${\frac {a}{p^{2}+a^{2}}}$ $t\sin(at)$ ${\frac {2ap}{(p^{2}+a^{2})^{2}}}$ $t^{2}\sin(at)$ ${\frac {2a(3p^{2}-a^{2})}{(p^{2}+a^{2})^{3}}}$ $\cos(at)$ ${\frac {p}{p^{2}+a^{2}}}$ $t\cos(at)$ ${\frac {p^{2}-a^{2}}{(p^{2}+a^{2})^{2}}}$ $t^{2}\cos(at)$ ${\frac {2p(p^{2}-3a^{2})}{(p^{2}+a^{2})^{3}}}$ $\sin(at+b)$ ${\frac {a\cos(b)+p\sin(b)}{p^{2}+a^{2}}}$ $\cos(at+b)$ ${\frac {p\cos(b)-a\sin(b)}{p^{2}+a^{2}}}$ $\sinh(at)$ ${\frac {a}{p^{2}-a^{2}}}$ $t\sinh(at)$ ${\frac {2ap}{(p^{2}-a^{2})^{2}}}$ $\cosh(at)$ ${\frac {p}{p^{2}-a^{2}}}$ $t\cosh(at)$ ${\frac {p^{2}+a^{2}}{(p^{2}-a^{2})^{2}}}$ $\sin(at)e^{-ct}$ ${\frac {a}{(p+c)^{2}+a^{2}}}$ $\cos(at)e^{-ct}$ ${\frac {p+c}{(p+c)^{2}+a^{2}}}$ $\sin(at+b)e^{-ct}$ ${\frac {a\cos(b)+(p+c)\sin(b)}{(p+c)^{2}+a^{2}}}$ $\cos(at+b)e^{-ct}$ ${\frac {(p+c)\cos(b)-a\sin(b)}{(p+c)^{2}+a^{2}}}$ $\sinh(at)e^{-ct}$ ${\frac {a}{(p+c)^{2}-a^{2}}}$ $\cosh(at)e^{-ct}$ ${\frac {p+c}{(p+c)^{2}-a^{2}}}$ $\sin ^{2}(at)$ ${\frac {2a^{2}}{p(p^{2}+4a^{2})}}$ $\sin ^{3}(at)$ ${\frac {6a^{3}}{(p^{2}+a^{2})(p^{2}+9a^{2})}}$ $\cos ^{2}(at)$ ${\frac {p^{2}+2a^{2}}{p(p^{2}+4a^{2})}}$ $\cos ^{3}(at)$ ${\frac {p(p^{2}+7a^{2})}{(p^{2}+a^{2})(p^{2}+9a^{2})}}$ $\sinh ^{2}(t)$ ${\frac {2}{p(p^{2}-4)}}\qquad \forall p\neq 2$ $\cosh ^{2}(t)$ ${\frac {p^{2}-2}{p(p^{2}-4)}}\qquad \forall p\neq 2$ $\sin(at)\sin(bt)$ ${\frac {2abp}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}$ $\cos(at)\cos(bt)$ ${\frac {p(p^{2}+a^{2}+b^{2})}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}$ $\sin(at)\cos(bt)$ ${\frac {a(p^{2}+a^{2}-b^{2})}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}$

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