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Transformées de Laplace usuelles
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Bibliothèque wikiversitaire
Intitulé : Transformées de Laplace usuelles
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Outils mathématiques et informatiques pour la physique
.
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]}}}
![{\displaystyle {\frac {2a.b.p}{\left[(p^{2}+(a-b)^{2}\right].\left[(p^{2}+(a+b)^{2}\right]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7407e072262349820d269c407b03574dd9f6ac9)
![{\displaystyle {\frac {p^{2}(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/4031885fab2094dbaa86bc4b3ef8a34867231318)
![{\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/623a9361b1aad4479179fca0f1d4e8efc0528dd6)
