Transformées de Laplace usuelles

Bibliothèque wikiversitaire

Intitulé : Transformées de Laplace usuelles

Formulaires

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

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