Transformées inverses de Laplace usuelles

Bibliothèque wikiversitaire

Intitulé : Transformées inverses de Laplace usuelles

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Cet élément de bibliothèque est rattaché au département Outils mathématiques et informatiques pour la physique.

Transformée de Laplace	Fonction
1	$\delta (t)$
${\frac {1}{p}}$	$1$
${\frac {1}{p^{2}}}$	$t$
${\frac {1}{p^{n}}}$	${\frac {t^{n-1}}{(n-1)!}}$
${\frac {1}{\sqrt {p}}}$	${\frac {1}{\sqrt {\pi t}}}$
${\frac {1}{\sqrt {p^{3}}}}$	$2{\sqrt {\frac {t}{\pi }}}$
${\frac {1}{p+a}}$	$e^{-a.t}$
${\frac {1}{p(p+a)}}$	${\frac {1}{a}}\left(1-e^{-a.t}\right)$
${\frac {1}{p^{2}(p+a)}}$	${\frac {e^{-a.t}}{a^{2}}}+{\frac {t}{a}}-{\frac {1}{a^{2}}}$
${\frac {1}{p(p+a)^{2}}}$	${\frac {1}{a^{2}}}\left(1-e^{-a.t}-a.t.e^{-a.t}\right)$
${\frac {1}{(p+a)(p+b)}}$	${\frac {e^{-b.t}-e^{-a.t}}{a-b}}$
${\frac {p}{(p+a)(p+b)}}$	${\frac {ae^{-a.t}-be^{-b.t}}{a-b}}$
${\frac {1}{(p+a)(p+b)(p+c)}}$	${\frac {e^{-a.t}}{(b-a)(c-a)}}+{\frac {e^{-b.t}}{(a-b)(c-b)}}+{\frac {e^{-c.t}}{(a-c)(b-c)}}$
${\frac {1}{(p+a)^{2}}}$	$t.e^{-a.t}$
${\frac {p}{(p+a)^{2}}}$	$e^{-a.t}(1-a.t)$
${\frac {1}{(p+a)(p+b)^{2}}}$	${\frac {e^{-a.t}-\left[1+(b-a)t\right]e^{-b.t}}{(b-a)^{2}}}$
${\frac {1}{p(p+a)(p+b)}}$	${\frac {1}{a.b}}\left(1+{\frac {b.e^{-a.t}-a.e^{-b.t}}{a-b}}\right)$
${\frac {p+c}{p(p+a)(p+b)}}$	${\frac {c}{a.b}}+{\frac {c-a}{a(a-b)}}.e^{-a.t}+{\frac {c-b}{b(b-a)}}.e^{-b.t}$
${\frac {p^{2}+c.p+d}{p(p+a)(p+b)}}$	${\frac {d}{a.b}}+{\frac {a^{2}-a.c+d}{a(a-b)}}.e^{-a.t}+{\frac {b^{2}-b.c+d}{b(b-a)}}.e^{-b.t}$
${\frac {1}{(p+a)^{3}}}$	${\frac {t^{2}.e^{-a.t}}{2}}$
$\ln \left({\frac {p+a}{p+b}}\right)$	${\frac {e^{-b.t}-e^{-a.t}}{t}}$
${\frac {1}{p^{2}+a^{2}}}$	${\frac {1}{a}}\sin {(a.t)}$
${\frac {1}{p(p^{2}+a^{2})}}$	${\frac {1}{a^{2}}}(1-\cos {(a.t)})$
${\frac {p}{p^{2}+a^{2}}}$	$\cos {(a.t)}$
${\frac {p+a}{p(p^{2}+b^{2})}}$	${\frac {a}{b^{2}}}-{\frac {\sqrt {a^{2}+b^{2}}}{b^{2}}}\cos \left(b.t+\arctan {\frac {b}{a}}\right)$
${\frac {p^{2}+c.p+d}{p(p^{2}+b^{2})}}$	${\frac {d}{b^{2}}}-{\frac {\sqrt {(d-b^{2})^{2}+c^{2}b^{2}}}{b^{2}}}\cos \left(b.t+\arctan {\frac {bc}{d-b^{2}}}\right)$
${\frac {1}{p^{2}-a^{2}}}$	${\frac {1}{a}}\sinh {(a.t)}$
${\frac {p}{p^{2}-a^{2}}}$	$\cosh {(a.t)}$
${\frac {1}{(p+b)^{2}+a^{2}}}$	${\frac {1}{a}}e^{-b.t}.\sin {(a.t)}$
${\frac {p+b}{(p+b)^{2}+a^{2}}}$	$e^{-bt}.\cos {(a.t)}$
${\frac {1}{(p^{2}+a^{2})^{2}}}$	${\frac {\sin(a.t)}{2a^{3}}}-{\frac {t.\cos(a.t)}{2a^{2}}}$
${\frac {p}{(p^{2}+a^{2})^{2}}}$	${\frac {t}{2a}}\sin(a.t)$
${\frac {p^{2}}{(p^{2}+a^{2})^{2}}}$	${\frac {1}{2a}}(\sin(a.t)+a.t.\cos(a.t))$
${\frac {1}{p^{3}+a^{3}}}$	${\frac {1}{3a^{2}}}\left[e^{-at}-e^{\frac {at}{2}}\left(\cos \left({\frac {\sqrt {3}}{2}}at\right)-{\sqrt {3}}\sin \left({\frac {\sqrt {3}}{2}}at\right)\right)\right]$
${\frac {p}{p^{3}+a^{3}}}$	${\frac {1}{3a}}\left[-e^{-at}-e^{\frac {at}{2}}\left(\cos \left({\frac {\sqrt {3}}{2}}at\right)+{\sqrt {3}}\sin \left({\frac {\sqrt {3}}{2}}at\right)\right)\right]$
${\frac {p^{2}}{p^{3}+a^{3}}}$	Échec de l’analyse (fonction inconnue « \fr »): {\displaystyle \fr\right]}
${\frac {1}{(\tau 1p+1)(\tau 2p+1)p^{2}}}$	$t-(\tau 1+\tau 2)+{\frac {1}{(\tau 1-\tau 2)}}.(\tau 1^{2}.e^{-t/\tau 1}-\tau 2^{2}.e^{-t/\tau 2})$