Transformées inverses de Laplace usuelles

Transformée de Laplace de la fonction	Fonction
$1$	$\delta (t)$
$\frac{1}{p}$	$1$
$\frac{1}{p^2}$	$t$
$\frac{1}{p^n}$	$\frac{t^{n-1}}{(n-1)!}\mathrm{\forall }n\ge 1$
$\frac{1}{\sqrt{p}}$	$\frac{1}{\sqrt{\pi t}}\mathrm{\forall }t\in R_{+}^{*}$
$\frac{1}{\sqrt{p^3}}$	$2\sqrt{\frac{t}{\pi }}$
$\frac{1}{p+a}$	$e^{-at}$
$\frac{1}{p(p+a)}$	$\frac{1-e^{-at}}{a}\mathrm{\forall }a\ne 0$
$\frac{1}{p^2(p+a)}$	$\frac{e^{-at}-1+at}{a^2}\mathrm{\forall }a\ne 0$
$\frac{1}{p(p+a{)}^2}$	$\frac{1-e^{-at}(1+at)}{a^2}\mathrm{\forall }a\ne 0$
$\frac{1}{(p+a)(p+b)}$	$\frac{e^{-bt}-e^{-at}}{a-b}\mathrm{\forall }a\ne b$
$\frac{p}{(p+a)(p+b)}$	$\frac{a{e}^{-at}-b{e}^{-bt}}{a-b}\mathrm{\forall }a\ne b$
$\frac{1}{(p+a)(p+b)(p+c)}$	$\frac{(b-c)e^{-at}+(c-a)e^{-bt}+(a-b)e^{-ct}}{(a-b)(a-c)(b-c)}\mathrm{\forall }a\ne b\vee a\ne c\vee b\ne c$
$\frac{1}{(p+a{)}^2}$	$t{e}^{-at}$
$\frac{p}{(p+a{)}^2}$	$e^{-at}(1-at)$
$\frac{1}{(p+a)(p+b{)}^2}$	$\frac{e^{-at}+[(a-b)t-1]e^{-bt}}{(a-b{)}^2}\mathrm{\forall }a\ne b$
$\frac{1}{p(p+a)(p+b)}$	$\frac{b{e}^{-at}-a{e}^{-bt}+a-b}{a^{2}b-a{b}^2}\mathrm{\forall }a\ne b$
$\frac{p+c}{p(p+a)(p+b)}$	$\frac{b(c-a)e^{-at}+a(b-c)e^{-bt}+c(a-b)}{ab(a-b)}\mathrm{\forall }a\ne b$
$\frac{p^2+cp+d}{p(p+a)(p+b)}$	$\frac{b(a^2-ac+d)e^{-at}-a(b^2-bc+d)e^{-bt}+d(a-b)}{ab(a-b)}\mathrm{\forall }a\ne b$
$\frac{1}{(p+a{)}^3}$	$\frac{t^2{e}^{-at}}{2}$
$\ln \left(\frac{p+a}{p+b}\right)$	$\frac{e^{-bt}-e^{-at}}{t}\mathrm{\forall }t\in R_{+}^{*}$
$\frac{1}{p^2+a^2}$	$\frac{\sin (at)}{a}\mathrm{\forall }a\ne 0$
$\frac{1}{p(p^2+a^2)}$	$\frac{1-\cos (at)}{a^2}\mathrm{\forall }a\ne 0$
$\frac{p}{p^2+a^2}$	$\cos (at)$
$\frac{p+a}{p(p^2+b^2)}$	$\frac{a(1-\cos (bt))+b\sin (bt)}{b^2}\mathrm{\forall }b\ne 0$
$\frac{p^2+cp+d}{p(p^2+b^2)}$	$\frac{(b^2-d)\cos (bt)+bc\sin (bt)+d}{b^2}\mathrm{\forall }b\ne 0$
$\frac{1}{p^2-a^2}$	$\frac{\sinh (at)}{a}\mathrm{\forall }a\ne 0$
$\frac{p}{p^2-a^2}$	$\cosh (at)$
$\frac{1}{(p+b{)}^2+a^2}$	$\frac{e^{-bt}\sin (at)}{a}\mathrm{\forall }a\ne 0$
$\frac{p+b}{(p+b{)}^2+a^2}$	$e^{-bt}\cos (at)$
$\frac{1}{(p^2+a^2{)}^2}$	$\frac{\sin (at)-at\cos (at)}{2a^3}\mathrm{\forall }a\ne 0$
$\frac{p}{(p^2+a^2{)}^2}$	$\frac{t\sin (at)}{2a}\mathrm{\forall }a\ne 0$
$\frac{p^2}{(p^2+a^2{)}^2}$	$\frac{\sin (at)+at\cos (at)}{2a}\mathrm{\forall }a\ne 0$
$\frac{1}{p^3+a^3}$	$\frac{e^{-at}-e^{\frac{at}{2}}(\cos \left(\frac{\sqrt{3}}{2}at\right)-\sqrt{3}\sin \left(\frac{\sqrt{3}}{2}at\right))}{3a^2}\mathrm{\forall }a\ne 0$
$\frac{p}{p^3+a^3}$	$\frac{e^{\frac{at}{2}}(\sqrt{3}\sin \left(\frac{\sqrt{3}at}{2}\right)+\cos \left(\frac{\sqrt{3}at}{2}\right))-e^{-at}}{3a}\mathrm{\forall }a\ne 0$
$\frac{p^2}{p^3+a^3}$	$\frac{e^{-at}+2e^{\frac{at}{2}}\cos \left(\frac{\sqrt{3}at}{2}\right)}{3}$
$\frac{1}{p^2(1+{\tau }_{1}p)(1+{\tau }_{2}p)}$	$\frac{{\tau }_1^2{e}^{-t/{\tau }_1}-{\tau }_2^2{e}^{-t/{\tau }_2}+({\tau }_1-{\tau }_2)(t-{\tau }_1-{\tau }_2)}{{\tau }_1-{\tau }_2}\mathrm{\forall }{\tau }_1\ne {\tau }_2$