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Transformées inverses de Laplace usuelles
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Intitulé : Transformées inverses de Laplace usuelles
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Outils mathématiques et informatiques pour la physique
.
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})}
![{\displaystyle {\frac {e^{-a.t}-\left[1+(b-a)t\right]e^{-b.t}}{(b-a)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2e93315befc20e6db2327a477d4fb31e2b7ca9)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e76ec5208e4abaaa4b457fbe184a7b65c076fdc)
![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32411573d93ffb6497352b55be1e0d9dc4abeb33)
![{\displaystyle {\frac {1}{3}}\left[e^{-at}+2e^{\frac {at}{2}}\cos \left({\frac {\sqrt {3}}{2}}at\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ae080768aa737dc0b0d5fb575752fc0bac091c)
