Transformée de Laplace/Fiche/Table des transformées de Laplace

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Fiche mémoire sur les transformées de Laplace usuelles
En raison de limitations techniques, la typographie souhaitable du titre, « Fiche : Table des transformées de Laplace
Transformée de Laplace/Fiche/Table des transformées de Laplace
», n'a pu être restituée correctement ci-dessus.

Transformées de Laplace directes

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

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