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
(
Modifier le tableau ci-dessous
)
Fonction
Transformée de Laplace et inverse
δ
(
t
)
{\displaystyle \delta (t)}
1
1
{\displaystyle 1}
1
p
{\displaystyle {\frac {1}{p}}}
t
{\displaystyle t}
1
p
2
{\displaystyle {\frac {1}{p^{2}}}}
t
n
{\displaystyle t^{n}}
n
!
p
n
+
1
{\displaystyle {\frac {n!}{p^{n+1}}}}
t
{\displaystyle {\sqrt {t}}}
1
2
π
p
3
{\displaystyle {\frac {1}{2}}{\sqrt {\frac {\pi }{p^{3}}}}}
1
t
{\displaystyle {\frac {1}{\sqrt {t}}}}
π
p
{\displaystyle {\sqrt {\frac {\pi }{p}}}}
e
−
c
.
t
{\displaystyle e^{-c.t}}
1
p
+
c
{\displaystyle {\frac {1}{p+c}}}
t
.
e
−
c
.
t
{\displaystyle t.e^{-c.t}}
1
(
p
+
c
)
2
{\displaystyle {\frac {1}{(p+c)^{2}}}}
t
2
.
e
−
c
.
t
{\displaystyle t^{2}.e^{-c.t}}
2
(
p
+
c
)
3
{\displaystyle {\frac {2}{(p+c)^{3}}}}
t
n
.
e
−
c
.
t
{\displaystyle t^{n}.e^{-c.t}}
n
!
(
p
+
c
)
n
+
1
{\displaystyle {\frac {n!}{(p+c)^{n+1}}}}
a
t
{\displaystyle a^{t}}
1
p
−
ln
a
{\displaystyle {\frac {1}{p-\ln a}}}
sin
(
a
.
t
)
{\displaystyle \sin(a.t)}
a
p
2
+
a
2
{\displaystyle {\frac {a}{p^{2}+a^{2}}}}
t
.
sin
(
a
.
t
)
{\displaystyle t.\sin(a.t)}
2
a
.
p
(
p
2
+
a
2
)
2
{\displaystyle {\frac {2a.p}{(p^{2}+a^{2})^{2}}}}
t
2
.
sin
(
a
.
t
)
{\displaystyle t^{2}.\sin(a.t)}
2
a
(
3
p
2
−
a
2
)
(
p
2
+
a
2
)
3
{\displaystyle {\frac {2a(3p^{2}-a^{2})}{(p^{2}+a^{2})^{3}}}}
cos
(
a
.
t
)
{\displaystyle \cos(a.t)}
p
p
2
+
a
2
{\displaystyle {\frac {p}{p^{2}+a^{2}}}}
t
.
cos
(
a
.
t
)
{\displaystyle t.\cos(a.t)}
p
2
−
a
2
(
p
2
+
a
2
)
2
{\displaystyle {\frac {p^{2}-a^{2}}{(p^{2}+a^{2})^{2}}}}
t
2
.
cos
(
a
.
t
)
{\displaystyle t^{2}.\cos(a.t)}
2
p
(
p
2
−
3
a
2
)
(
p
2
+
a
2
)
3
{\displaystyle {\frac {2p(p^{2}-3a^{2})}{(p^{2}+a^{2})^{3}}}}
sin
(
a
.
t
+
b
)
{\displaystyle \sin(a.t+b)}
a
cos
b
+
p
sin
b
p
2
+
a
2
{\displaystyle {\frac {a\cos b+p\sin b}{p^{2}+a^{2}}}}
cos
(
a
.
t
+
b
)
{\displaystyle \cos(a.t+b)}
p
.
cos
b
−
a
sin
b
p
2
+
a
2
{\displaystyle {\frac {p.\cos b-a\sin b}{p^{2}+a^{2}}}}
sinh
(
a
.
t
)
{\displaystyle \sinh(a.t)}
a
p
2
−
a
2
{\displaystyle {\frac {a}{p^{2}-a^{2}}}}
t
.
sinh
(
a
.
t
)
{\displaystyle t.\sinh(a.t)}
2
a
.
p
(
p
2
−
a
2
)
2
{\displaystyle {\frac {2a.p}{(p^{2}-a^{2})^{2}}}}
cosh
(
a
.
t
)
{\displaystyle \cosh(a.t)}
p
p
2
−
a
2
{\displaystyle {\frac {p}{p^{2}-a^{2}}}}
t
.
cosh
(
a
.
t
)
{\displaystyle t.\cosh(a.t)}
p
2
+
a
2
(
p
2
−
a
2
)
2
{\displaystyle {\frac {p^{2}+a^{2}}{(p^{2}-a^{2})^{2}}}}
e
−
c
.
t
.
sin
(
a
.
t
)
{\displaystyle e^{-c.t}.\sin(a.t)}
a
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {a}{(p+c)^{2}+a^{2}}}}
e
−
c
.
t
.
cos
(
a
.
t
)
{\displaystyle e^{-c.t}.\cos(a.t)}
p
+
c
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {p+c}{(p+c)^{2}+a^{2}}}}
e
−
c
.
t
.
sin
(
a
.
t
+
b
)
{\displaystyle e^{-c.t}.\sin(a.t+b)}
a
cos
b
+
(
p
+
c
)
sin
b
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {a\cos b+(p+c)\sin b}{(p+c)^{2}+a^{2}}}}
e
−
c
.
t
.
cos
(
a
.
t
+
b
)
{\displaystyle e^{-c.t}.\cos(a.t+b)}
(
p
+
c
)
cos
b
+
a
sin
b
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {(p+c)\cos b+a\sin b}{(p+c)^{2}+a^{2}}}}
e
−
c
.
t
.
sinh
(
a
.
t
)
{\displaystyle e^{-c.t}.\sinh(a.t)}
a
(
p
+
c
)
2
−
a
2
{\displaystyle {\frac {a}{(p+c)^{2}-a^{2}}}}
e
−
c
.
t
.
cosh
(
a
.
t
)
{\displaystyle e^{-c.t}.\cosh(a.t)}
p
+
c
(
p
+
c
)
2
−
a
2
{\displaystyle {\frac {p+c}{(p+c)^{2}-a^{2}}}}
sin
2
(
a
.
t
)
{\displaystyle \sin ^{2}(a.t)}
2
a
2
p
(
p
2
+
4
a
2
)
{\displaystyle {\frac {2a^{2}}{p(p^{2}+4a^{2})}}}
sin
3
(
a
.
t
)
{\displaystyle \sin ^{3}(a.t)}
6
a
3
(
p
2
+
a
2
)
(
p
2
+
9
a
2
)
{\displaystyle {\frac {6a^{3}}{(p^{2}+a^{2})(p^{2}+9a^{2})}}}
cos
2
(
a
.
t
)
{\displaystyle \cos ^{2}(a.t)}
p
2
+
2
a
2
p
(
p
2
+
4
a
2
)
{\displaystyle {\frac {p^{2}+2a^{2}}{p(p^{2}+4a^{2})}}}
cos
3
(
a
.
t
)
{\displaystyle \cos ^{3}(a.t)}
p
(
p
2
+
7
a
2
)
(
p
2
+
a
2
)
(
p
2
+
9
a
2
)
{\displaystyle {\frac {p(p^{2}+7a^{2})}{(p^{2}+a^{2})(p^{2}+9a^{2})}}}
sinh
2
t
{\displaystyle \sinh ^{2}t}
2
p
(
p
2
−
4
)
{\displaystyle {\frac {2}{p(p^{2}-4)}}}
cosh
2
t
{\displaystyle \cosh ^{2}t}
p
2
−
2
p
(
p
2
−
4
)
{\displaystyle {\frac {p^{2}-2}{p(p^{2}-4)}}}
sin
(
a
.
t
)
.
sin
(
b
.
t
)
{\displaystyle \sin(a.t).\sin(b.t)}
2
a
.
b
.
p
[
(
p
2
+
(
a
−
b
)
2
]
.
[
(
p
2
+
(
a
+
b
)
2
]
{\displaystyle {\frac {2a.b.p}{\left[(p^{2}+(a-b)^{2}\right].\left[(p^{2}+(a+b)^{2}\right]}}}
cos
(
a
.
t
)
.
cos
(
b
.
t
)
{\displaystyle \cos(a.t).\cos(b.t)}
p
2
(
p
2
+
a
2
+
b
2
)
[
(
p
2
+
(
a
−
b
)
2
]
.
[
(
p
2
+
(
a
+
b
)
2
]
{\displaystyle {\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
)
{\displaystyle \sin(a.t).\cos(b.t)}
a
(
p
2
+
a
2
−
b
2
)
[
(
p
2
+
(
a
−
b
)
2
]
.
[
(
p
2
+
(
a
+
b
)
2
]
{\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
(
Modifier le tableau ci-dessous
)
Transformée de Laplace
Fonction
1
δ
(
t
)
{\displaystyle \delta (t)}
1
p
{\displaystyle {\frac {1}{p}}}
1
{\displaystyle 1}
1
p
2
{\displaystyle {\frac {1}{p^{2}}}}
t
{\displaystyle t}
1
p
n
{\displaystyle {\frac {1}{p^{n}}}}
t
n
−
1
(
n
−
1
)
!
{\displaystyle {\frac {t^{n-1}}{(n-1)!}}}
1
p
{\displaystyle {\frac {1}{\sqrt {p}}}}
1
π
t
{\displaystyle {\frac {1}{\sqrt {\pi t}}}}
1
p
3
{\displaystyle {\frac {1}{\sqrt {p^{3}}}}}
2
t
π
{\displaystyle 2{\sqrt {\frac {t}{\pi }}}}
1
p
+
a
{\displaystyle {\frac {1}{p+a}}}
e
−
a
.
t
{\displaystyle e^{-a.t}}
1
p
(
p
+
a
)
{\displaystyle {\frac {1}{p(p+a)}}}
1
a
(
1
−
e
−
a
.
t
)
{\displaystyle {\frac {1}{a}}\left(1-e^{-a.t}\right)}
1
p
2
(
p
+
a
)
{\displaystyle {\frac {1}{p^{2}(p+a)}}}
e
−
a
.
t
a
2
+
t
a
−
1
a
2
{\displaystyle {\frac {e^{-a.t}}{a^{2}}}+{\frac {t}{a}}-{\frac {1}{a^{2}}}}
1
p
(
p
+
a
)
2
{\displaystyle {\frac {1}{p(p+a)^{2}}}}
1
a
2
(
1
−
e
−
a
.
t
−
a
.
t
.
e
−
a
.
t
)
{\displaystyle {\frac {1}{a^{2}}}\left(1-e^{-a.t}-a.t.e^{-a.t}\right)}
1
(
p
+
a
)
(
p
+
b
)
{\displaystyle {\frac {1}{(p+a)(p+b)}}}
e
−
b
.
t
−
e
−
a
.
t
a
−
b
{\displaystyle {\frac {e^{-b.t}-e^{-a.t}}{a-b}}}
p
(
p
+
a
)
(
p
+
b
)
{\displaystyle {\frac {p}{(p+a)(p+b)}}}
a
e
−
a
.
t
−
b
e
−
b
.
t
a
−
b
{\displaystyle {\frac {ae^{-a.t}-be^{-b.t}}{a-b}}}
1
(
p
+
a
)
(
p
+
b
)
(
p
+
c
)
{\displaystyle {\frac {1}{(p+a)(p+b)(p+c)}}}
e
−
a
.
t
(
b
−
a
)
(
c
−
a
)
+
e
−
b
.
t
(
a
−
b
)
(
c
−
b
)
+
e
−
c
.
t
(
a
−
c
)
(
b
−
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)}}}
1
(
p
+
a
)
2
{\displaystyle {\frac {1}{(p+a)^{2}}}}
t
.
e
−
a
.
t
{\displaystyle t.e^{-a.t}}
p
(
p
+
a
)
2
{\displaystyle {\frac {p}{(p+a)^{2}}}}
e
−
a
.
t
(
1
−
a
.
t
)
{\displaystyle e^{-a.t}(1-a.t)}
1
(
p
+
a
)
(
p
+
b
)
2
{\displaystyle {\frac {1}{(p+a)(p+b)^{2}}}}
e
−
a
.
t
−
[
1
+
(
b
−
a
)
t
]
e
−
b
.
t
(
b
−
a
)
2
{\displaystyle {\frac {e^{-a.t}-\left[1+(b-a)t\right]e^{-b.t}}{(b-a)^{2}}}}
1
p
(
p
+
a
)
(
p
+
b
)
{\displaystyle {\frac {1}{p(p+a)(p+b)}}}
1
a
.
b
(
1
+
b
.
e
−
a
.
t
−
a
.
e
−
b
.
t
a
−
b
)
{\displaystyle {\frac {1}{a.b}}\left(1+{\frac {b.e^{-a.t}-a.e^{-b.t}}{a-b}}\right)}
p
+
c
p
(
p
+
a
)
(
p
+
b
)
{\displaystyle {\frac {p+c}{p(p+a)(p+b)}}}
c
a
.
b
+
c
−
a
a
(
a
−
b
)
.
e
−
a
.
t
+
c
−
b
b
(
b
−
a
)
.
e
−
b
.
t
{\displaystyle {\frac {c}{a.b}}+{\frac {c-a}{a(a-b)}}.e^{-a.t}+{\frac {c-b}{b(b-a)}}.e^{-b.t}}
p
2
+
c
.
p
+
d
p
(
p
+
a
)
(
p
+
b
)
{\displaystyle {\frac {p^{2}+c.p+d}{p(p+a)(p+b)}}}
d
a
.
b
+
a
2
−
a
.
c
+
d
a
(
a
−
b
)
.
e
−
a
.
t
+
b
2
−
b
.
c
+
d
b
(
b
−
a
)
.
e
−
b
.
t
{\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}}
1
(
p
+
a
)
3
{\displaystyle {\frac {1}{(p+a)^{3}}}}
t
2
.
e
−
a
.
t
2
{\displaystyle {\frac {t^{2}.e^{-a.t}}{2}}}
ln
(
p
+
a
p
+
b
)
{\displaystyle \ln \left({\frac {p+a}{p+b}}\right)}
e
−
b
.
t
−
e
−
a
.
t
t
{\displaystyle {\frac {e^{-b.t}-e^{-a.t}}{t}}}
1
p
2
+
a
2
{\displaystyle {\frac {1}{p^{2}+a^{2}}}}
1
a
sin
(
a
.
t
)
{\displaystyle {\frac {1}{a}}\sin {(a.t)}}
1
p
(
p
2
+
a
2
)
{\displaystyle {\frac {1}{p(p^{2}+a^{2})}}}
1
a
2
(
1
−
cos
(
a
.
t
)
)
{\displaystyle {\frac {1}{a^{2}}}(1-\cos {(a.t)})}
p
p
2
+
a
2
{\displaystyle {\frac {p}{p^{2}+a^{2}}}}
cos
(
a
.
t
)
{\displaystyle \cos {(a.t)}}
p
+
a
p
(
p
2
+
b
2
)
{\displaystyle {\frac {p+a}{p(p^{2}+b^{2})}}}
a
b
2
−
a
2
+
b
2
b
2
cos
(
b
.
t
+
arctan
b
a
)
{\displaystyle {\frac {a}{b^{2}}}-{\frac {\sqrt {a^{2}+b^{2}}}{b^{2}}}\cos \left(b.t+\arctan {\frac {b}{a}}\right)}
p
2
+
c
.
p
+
d
p
(
p
2
+
b
2
)
{\displaystyle {\frac {p^{2}+c.p+d}{p(p^{2}+b^{2})}}}
d
b
2
−
(
d
−
b
2
)
2
+
c
2
b
2
b
2
cos
(
b
.
t
+
arctan
b
c
d
−
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)}
1
p
2
−
a
2
{\displaystyle {\frac {1}{p^{2}-a^{2}}}}
1
a
sinh
(
a
.
t
)
{\displaystyle {\frac {1}{a}}\sinh {(a.t)}}
p
p
2
−
a
2
{\displaystyle {\frac {p}{p^{2}-a^{2}}}}
cosh
(
a
.
t
)
{\displaystyle \cosh {(a.t)}}
1
(
p
+
b
)
2
+
a
2
{\displaystyle {\frac {1}{(p+b)^{2}+a^{2}}}}
1
a
e
−
b
.
t
.
sin
(
a
.
t
)
{\displaystyle {\frac {1}{a}}e^{-b.t}.\sin {(a.t)}}
p
+
b
(
p
+
b
)
2
+
a
2
{\displaystyle {\frac {p+b}{(p+b)^{2}+a^{2}}}}
e
−
b
t
.
cos
(
a
.
t
)
{\displaystyle e^{-bt}.\cos {(a.t)}}
1
(
p
2
+
a
2
)
2
{\displaystyle {\frac {1}{(p^{2}+a^{2})^{2}}}}
sin
(
a
.
t
)
2
a
3
−
t
.
cos
(
a
.
t
)
2
a
2
{\displaystyle {\frac {\sin(a.t)}{2a^{3}}}-{\frac {t.\cos(a.t)}{2a^{2}}}}
p
(
p
2
+
a
2
)
2
{\displaystyle {\frac {p}{(p^{2}+a^{2})^{2}}}}
t
2
a
sin
(
a
.
t
)
{\displaystyle {\frac {t}{2a}}\sin(a.t)}
p
2
(
p
2
+
a
2
)
2
{\displaystyle {\frac {p^{2}}{(p^{2}+a^{2})^{2}}}}
1
2
a
(
sin
(
a
.
t
)
+
a
.
t
.
cos
(
a
.
t
)
)
{\displaystyle {\frac {1}{2a}}(\sin(a.t)+a.t.\cos(a.t))}
1
p
3
+
a
3
{\displaystyle {\frac {1}{p^{3}+a^{3}}}}
1
3
a
2
[
e
−
a
t
−
e
a
t
2
(
cos
(
3
2
a
t
)
−
3
sin
(
3
2
a
t
)
)
]
{\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]}
p
p
3
+
a
3
{\displaystyle {\frac {p}{p^{3}+a^{3}}}}
1
3
a
[
−
e
−
a
t
−
e
a
t
2
(
cos
(
3
2
a
t
)
+
3
sin
(
3
2
a
t
)
)
]
{\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]}
p
2
p
3
+
a
3
{\displaystyle {\frac {p^{2}}{p^{3}+a^{3}}}}
1
3
[
e
−
a
t
+
2
e
a
t
2
cos
(
3
2
a
t
)
]
{\displaystyle {\frac {1}{3}}\left[e^{-at}+2e^{\frac {at}{2}}\cos \left({\frac {\sqrt {3}}{2}}at\right)\right]}
1
(
τ
1
p
+
1
)
(
τ
2
p
+
1
)
p
2
{\displaystyle {\frac {1}{(\tau 1p+1)(\tau 2p+1)p^{2}}}}
t
−
(
τ
1
+
τ
2
)
+
1
(
τ
1
−
τ
2
)
.
(
τ
1
2
.
e
−
t
/
τ
1
−
τ
2
2
.
e
−
t
/
τ
2
)
{\displaystyle 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})}
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Transformée de Laplace
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