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Transformées de Laplace usuelles
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Intitulé : Transformées de Laplace usuelles
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Outils mathématiques et informatiques pour la physique
.
Fonction
Transformée de Laplace de la fonction
δ
(
t
)
{\displaystyle \delta (t)}
1
{\displaystyle 1}
1
{\displaystyle 1}
1
p
{\displaystyle {\frac {1}{p}}}
t
{\displaystyle t}
1
p
2
{\displaystyle {\frac {1}{p^{2}}}}
t
n
∀
n
≥
0
{\displaystyle t^{n}\qquad \forall n\geq 0}
n
!
p
n
+
1
{\displaystyle {\frac {n!}{p^{n+1}}}}
t
∀
t
∈
R
+
{\displaystyle {\sqrt {t}}\qquad \forall t\in R_{+}}
1
2
π
p
3
{\displaystyle {\frac {1}{2}}{\sqrt {\frac {\pi }{p^{3}}}}}
1
t
∀
t
∈
R
+
∗
{\displaystyle {\frac {1}{\sqrt {t}}}\qquad \forall t\in R_{+}^{*}}
π
p
{\displaystyle {\sqrt {\frac {\pi }{p}}}}
e
−
c
t
{\displaystyle e^{-ct}}
1
p
+
c
{\displaystyle {\frac {1}{p+c}}}
t
e
−
c
t
{\displaystyle te^{-ct}}
1
(
p
+
c
)
2
{\displaystyle {\frac {1}{(p+c)^{2}}}}
t
2
e
−
c
t
{\displaystyle t^{2}e^{-ct}}
2
(
p
+
c
)
3
{\displaystyle {\frac {2}{(p+c)^{3}}}}
t
n
e
−
c
t
∀
n
≥
0
{\displaystyle t^{n}e^{-ct}\qquad \forall n\geq 0}
n
!
(
p
+
c
)
n
+
1
{\displaystyle {\frac {n!}{(p+c)^{n+1}}}}
a
t
∀
a
>
0
{\displaystyle a^{t}\qquad \forall a>0}
1
p
−
ln
(
a
)
{\displaystyle {\frac {1}{p-\ln(a)}}}
sin
(
a
t
)
{\displaystyle \sin(at)}
a
p
2
+
a
2
{\displaystyle {\frac {a}{p^{2}+a^{2}}}}
t
sin
(
a
t
)
{\displaystyle t\sin(at)}
2
a
p
(
p
2
+
a
2
)
2
{\displaystyle {\frac {2ap}{(p^{2}+a^{2})^{2}}}}
t
2
sin
(
a
t
)
{\displaystyle t^{2}\sin(at)}
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(at)}
p
p
2
+
a
2
{\displaystyle {\frac {p}{p^{2}+a^{2}}}}
t
cos
(
a
t
)
{\displaystyle t\cos(at)}
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(at)}
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(at+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(at+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(at)}
a
p
2
−
a
2
{\displaystyle {\frac {a}{p^{2}-a^{2}}}}
t
sinh
(
a
t
)
{\displaystyle t\sinh(at)}
2
a
p
(
p
2
−
a
2
)
2
{\displaystyle {\frac {2ap}{(p^{2}-a^{2})^{2}}}}
cosh
(
a
t
)
{\displaystyle \cosh(at)}
p
p
2
−
a
2
{\displaystyle {\frac {p}{p^{2}-a^{2}}}}
t
cosh
(
a
t
)
{\displaystyle t\cosh(at)}
p
2
+
a
2
(
p
2
−
a
2
)
2
{\displaystyle {\frac {p^{2}+a^{2}}{(p^{2}-a^{2})^{2}}}}
sin
(
a
t
)
e
−
c
t
{\displaystyle \sin(at)e^{-ct}}
a
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {a}{(p+c)^{2}+a^{2}}}}
cos
(
a
t
)
e
−
c
t
{\displaystyle \cos(at)e^{-ct}}
p
+
c
(
p
+
c
)
2
+
a
2
{\displaystyle {\frac {p+c}{(p+c)^{2}+a^{2}}}}
sin
(
a
t
+
b
)
e
−
c
t
{\displaystyle \sin(at+b)e^{-ct}}
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}}}}
cos
(
a
t
+
b
)
e
−
c
t
{\displaystyle \cos(at+b)e^{-ct}}
(
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}}}}
sinh
(
a
t
)
e
−
c
t
{\displaystyle \sinh(at)e^{-ct}}
a
(
p
+
c
)
2
−
a
2
{\displaystyle {\frac {a}{(p+c)^{2}-a^{2}}}}
cosh
(
a
t
)
e
−
c
t
{\displaystyle \cosh(at)e^{-ct}}
p
+
c
(
p
+
c
)
2
−
a
2
{\displaystyle {\frac {p+c}{(p+c)^{2}-a^{2}}}}
sin
2
(
a
t
)
{\displaystyle \sin ^{2}(at)}
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}(at)}
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}(at)}
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}(at)}
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
)
∀
p
≠
2
{\displaystyle {\frac {2}{p(p^{2}-4)}}\qquad \forall p\neq 2}
cosh
2
(
t
)
{\displaystyle \cosh ^{2}(t)}
p
2
−
2
p
(
p
2
−
4
)
∀
p
≠
2
{\displaystyle {\frac {p^{2}-2}{p(p^{2}-4)}}\qquad \forall p\neq 2}
sin
(
a
t
)
sin
(
b
t
)
{\displaystyle \sin(at)\sin(bt)}
2
a
b
p
[
(
p
2
+
(
a
−
b
)
2
]
[
(
p
2
+
(
a
+
b
)
2
]
{\displaystyle {\frac {2abp}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}}
cos
(
a
t
)
cos
(
b
t
)
{\displaystyle \cos(at)\cos(bt)}
p
(
p
2
+
a
2
+
b
2
)
[
(
p
2
+
(
a
−
b
)
2
]
[
(
p
2
+
(
a
+
b
)
2
]
{\displaystyle {\frac {p(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(at)\cos(bt)}
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]}}}
![{\displaystyle {\frac {2abp}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe618b7fe772d47c2c258010fdf1fac5dc1795ab)
![{\displaystyle {\frac {p(p^{2}+a^{2}+b^{2})}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c58380de05e1c2d09862ded1606043fb5419b2bc)
![{\displaystyle {\frac {a(p^{2}+a^{2}-b^{2})}{\left[(p^{2}+(a-b)^{2}\right]\left[(p^{2}+(a+b)^{2}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/813df7662340417eb8412c7b335954e650bfd482)
