Transformée de Laplace/Fiche/Table des transformées de Laplace

Fiche mémoire sur les transformées de Laplace usuelles

Transformées de Laplace directes () Fonction Transformée de Laplace et inverse $\delta (t)$ 1 $1$ ${\frac {1}{p}}$ $t$ ${\frac {1}{p^{2}}}$ $t^{n}$ ${\frac {n!}{p^{n+1}}}$ ${\sqrt {t}}$ ${\frac {1}{2}}{\sqrt {\frac {\pi }{p^{3}}}}$ ${\frac {1}{\sqrt {t}}}$ ${\sqrt {\frac {\pi }{p}}}$ $e^{-c.t}$ ${\frac {1}{p+c}}$ $t.e^{-c.t}$ ${\frac {1}{(p+c)^{2}}}$ $t^{2}.e^{-c.t}$ ${\frac {2}{(p+c)^{3}}}$ $t^{n}.e^{-c.t}$ ${\frac {n!}{(p+c)^{n+1}}}$ $a^{t}$ ${\frac {1}{p-\ln a}}$ $\sin(a.t)$ ${\frac {a}{p^{2}+a^{2}}}$ $t.\sin(a.t)$ ${\frac {2a.p}{(p^{2}+a^{2})^{2}}}$ $t^{2}.\sin(a.t)$ ${\frac {2a(3p^{2}-a^{2})}{(p^{2}+a^{2})^{3}}}$ $\cos(a.t)$ ${\frac {p}{p^{2}+a^{2}}}$ $t.\cos(a.t)$ ${\frac {p^{2}-a^{2}}{(p^{2}+a^{2})^{2}}}$ $t^{2}.\cos(a.t)$ ${\frac {2p(p^{2}-3a^{2})}{(p^{2}+a^{2})^{3}}}$ $\sin(a.t+b)$ ${\frac {a\cos b+p\sin b}{p^{2}+a^{2}}}$ $\cos(a.t+b)$ ${\frac {p.\cos b-a\sin b}{p^{2}+a^{2}}}$ $\sinh(a.t)$ ${\frac {a}{p^{2}-a^{2}}}$ $t.\sinh(a.t)$ ${\frac {2a.p}{(p^{2}-a^{2})^{2}}}$ $\cosh(a.t)$ ${\frac {p}{p^{2}-a^{2}}}$ $t.\cosh(a.t)$ ${\frac {p^{2}+a^{2}}{(p^{2}-a^{2})^{2}}}$ $e^{-c.t}.\sin(a.t)$ ${\frac {a}{(p+c)^{2}+a^{2}}}$ $e^{-c.t}.\cos(a.t)$ ${\frac {p+c}{(p+c)^{2}+a^{2}}}$ $e^{-c.t}.\sin(a.t+b)$ ${\frac {a\cos b+(p+c)\sin b}{(p+c)^{2}+a^{2}}}$ $e^{-c.t}.\cos(a.t+b)$ ${\frac {(p+c)\cos b+a\sin b}{(p+c)^{2}+a^{2}}}$ $e^{-c.t}.\sinh(a.t)$ ${\frac {a}{(p+c)^{2}-a^{2}}}$ $e^{-c.t}.\cosh(a.t)$ ${\frac {p+c}{(p+c)^{2}-a^{2}}}$ $\sin ^{2}(a.t)$ ${\frac {2a^{2}}{p(p^{2}+4a^{2})}}$ $\sin ^{3}(a.t)$ ${\frac {6a^{3}}{(p^{2}+a^{2})(p^{2}+9a^{2})}}$ $\cos ^{2}(a.t)$ ${\frac {p^{2}+2a^{2}}{p(p^{2}+4a^{2})}}$ $\cos ^{3}(a.t)$ ${\frac {p(p^{2}+7a^{2})}{(p^{2}+a^{2})(p^{2}+9a^{2})}}$ $\sinh ^{2}t$ ${\frac {2}{p(p^{2}-4)}}$ $\cosh ^{2}t$ ${\frac {p^{2}-2}{p(p^{2}-4)}}$ $\sin(a.t).\sin(b.t)$ ${\frac {2a.b.p}{\left[(p^{2}+(a-b)^{2}\right].\left[(p^{2}+(a+b)^{2}\right]}}$ $\cos(a.t).\cos(b.t)$ ${\frac {p^{2}(p^{2}+a^{2}+b^{2})}{\left[(p^{2}+(a-b)^{2}\right].\left[(p^{2}+(a+b)^{2}\right]}}$ $\sin(a.t).\cos(b.t)$ ${\frac {a(p^{2}+a^{2}-b^{2})}{\left[(p^{2}+(a-b)^{2}\right].\left[(p^{2}+(a+b)^{2}\right]}}$

Transformées de Laplace inverses () Transformée de Laplace Fonction 1 $\delta (t)$ ${\frac {1}{p}}$ $1$ ${\frac {1}{p^{2}}}$ $t$ ${\frac {1}{p^{n}}}$ ${\frac {t^{n-1}}{(n-1)!}}$ ${\frac {1}{\sqrt {p}}}$ ${\frac {1}{\sqrt {\pi t}}}$ ${\frac {1}{\sqrt {p^{3}}}}$ $2{\sqrt {\frac {t}{\pi }}}$ ${\frac {1}{p+a}}$ $e^{-a.t}$ ${\frac {1}{p(p+a)}}$ ${\frac {1}{a}}\left(1-e^{-a.t}\right)$ ${\frac {1}{p^{2}(p+a)}}$ ${\frac {e^{-a.t}}{a^{2}}}+{\frac {t}{a}}-{\frac {1}{a^{2}}}$ ${\frac {1}{p(p+a)^{2}}}$ ${\frac {1}{a^{2}}}\left(1-e^{-a.t}-a.t.e^{-a.t}\right)$ ${\frac {1}{(p+a)(p+b)}}$ ${\frac {e^{-b.t}-e^{-a.t}}{a-b}}$ ${\frac {p}{(p+a)(p+b)}}$ ${\frac {ae^{-a.t}-be^{-b.t}}{a-b}}$ ${\frac {1}{(p+a)(p+b)(p+c)}}$ ${\frac {e^{-a.t}}{(b-a)(c-a)}}+{\frac {e^{-b.t}}{(a-b)(c-b)}}+{\frac {e^{-c.t}}{(a-c)(b-c)}}$ ${\frac {1}{(p+a)^{2}}}$ $t.e^{-a.t}$ ${\frac {p}{(p+a)^{2}}}$ $e^{-a.t}(1-a.t)$ ${\frac {1}{(p+a)(p+b)^{2}}}$ ${\frac {e^{-a.t}-\left[1+(b-a)t\right]e^{-b.t}}{(b-a)^{2}}}$ ${\frac {1}{p(p+a)(p+b)}}$ ${\frac {1}{a.b}}\left(1+{\frac {b.e^{-a.t}-a.e^{-b.t}}{a-b}}\right)$ ${\frac {p+c}{p(p+a)(p+b)}}$ ${\frac {c}{a.b}}+{\frac {c-a}{a(a-b)}}.e^{-a.t}+{\frac {c-b}{b(b-a)}}.e^{-b.t}$ ${\frac {p^{2}+c.p+d}{p(p+a)(p+b)}}$ ${\frac {d}{a.b}}+{\frac {a^{2}-a.c+d}{a(a-b)}}.e^{-a.t}+{\frac {b^{2}-b.c+d}{b(b-a)}}.e^{-b.t}$ ${\frac {1}{(p+a)^{3}}}$ ${\frac {t^{2}.e^{-a.t}}{2}}$ $\ln \left({\frac {p+a}{p+b}}\right)$ ${\frac {e^{-b.t}-e^{-a.t}}{t}}$ ${\frac {1}{p^{2}+a^{2}}}$ ${\frac {1}{a}}\sin {(a.t)}$ ${\frac {1}{p(p^{2}+a^{2})}}$ ${\frac {1}{a^{2}}}(1-\cos {(a.t)})$ ${\frac {p}{p^{2}+a^{2}}}$ $\cos {(a.t)}$ ${\frac {p+a}{p(p^{2}+b^{2})}}$ ${\frac {a}{b^{2}}}-{\frac {\sqrt {a^{2}+b^{2}}}{b^{2}}}\cos \left(b.t+\arctan {\frac {b}{a}}\right)$ ${\frac {p^{2}+c.p+d}{p(p^{2}+b^{2})}}$ ${\frac {d}{b^{2}}}-{\frac {\sqrt {(d-b^{2})^{2}+c^{2}b^{2}}}{b^{2}}}\cos \left(b.t+\arctan {\frac {bc}{d-b^{2}}}\right)$ ${\frac {1}{p^{2}-a^{2}}}$ ${\frac {1}{a}}\sinh {(a.t)}$ ${\frac {p}{p^{2}-a^{2}}}$ $\cosh {(a.t)}$ ${\frac {1}{(p+b)^{2}+a^{2}}}$ ${\frac {1}{a}}e^{-b.t}.\sin {(a.t)}$ ${\frac {p+b}{(p+b)^{2}+a^{2}}}$ $e^{-bt}.\cos {(a.t)}$ ${\frac {1}{(p^{2}+a^{2})^{2}}}$ ${\frac {\sin(a.t)}{2a^{3}}}-{\frac {t.\cos(a.t)}{2a^{2}}}$ ${\frac {p}{(p^{2}+a^{2})^{2}}}$ ${\frac {t}{2a}}\sin(a.t)$ ${\frac {p^{2}}{(p^{2}+a^{2})^{2}}}$ ${\frac {1}{2a}}(\sin(a.t)+a.t.\cos(a.t))$ ${\frac {1}{p^{3}+a^{3}}}$ ${\frac {1}{3a^{2}}}\left[e^{-at}-e^{\frac {at}{2}}\left(\cos \left({\frac {\sqrt {3}}{2}}at\right)-{\sqrt {3}}\sin \left({\frac {\sqrt {3}}{2}}at\right)\right)\right]$ ${\frac {p}{p^{3}+a^{3}}}$ ${\frac {1}{3a}}\left[-e^{-at}-e^{\frac {at}{2}}\left(\cos \left({\frac {\sqrt {3}}{2}}at\right)+{\sqrt {3}}\sin \left({\frac {\sqrt {3}}{2}}at\right)\right)\right]$ ${\frac {p^{2}}{p^{3}+a^{3}}}$ ${\frac {1}{3}}\left[e^{-at}+2e^{\frac {at}{2}}\cos \left({\frac {\sqrt {3}}{2}}at\right)\right]$ ${\frac {1}{(\tau 1p+1)(\tau 2p+1)p^{2}}}$ $t-(\tau 1+\tau 2)+{\frac {1}{(\tau 1-\tau 2)}}.(\tau 1^{2}.e^{-t/\tau 1}-\tau 2^{2}.e^{-t/\tau 2})$