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Công thức hoán chuyển Laplace
Language
Watch
Edit
x
(
t
)
=
L
−
1
{
X
(
s
)
}
{\displaystyle x(t)={\mathcal {L}}^{-1}\left\{X(s)\right\}}
X
(
s
)
=
L
{
x
(
t
)
}
{\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}
1
1
2
π
j
∫
σ
−
j
∞
σ
+
j
∞
X
(
s
)
e
s
t
d
s
{\displaystyle {\frac {1}{2\pi j}}\int _{\sigma -j\infty }^{\sigma +j\infty }X(s)e^{st}ds}
∫
−
∞
∞
x
(
t
)
e
−
s
t
d
t
{\displaystyle \int _{-\infty }^{\infty }x(t)e^{-st}dt}
2
δ
(
t
)
{\displaystyle \delta (t)\,}
1
{\displaystyle 1\,}
3
δ
(
t
−
a
)
{\displaystyle \delta (t-a)\,}
e
−
a
s
{\displaystyle e^{-as}\,}
4
u
(
t
)
{\displaystyle u(t)\,}
1
s
{\displaystyle {\frac {1}{s}}}
5
u
(
t
−
a
)
{\displaystyle u(t-a)\,}
e
−
a
s
s
{\displaystyle {\frac {e^{-as}}{s}}}
6
t
u
(
t
)
{\displaystyle tu(t)\,}
1
s
2
{\displaystyle {\frac {1}{s^{2}}}}
7
t
n
u
(
t
)
{\displaystyle t^{n}u(t)\,}
n
!
s
n
+
1
{\displaystyle {\frac {n!}{s^{n+1}}}}
8
1
π
t
u
(
t
)
{\displaystyle {\frac {1}{\sqrt {\pi t}}}u(t)}
1
s
{\displaystyle {\frac {1}{\sqrt {s}}}}
9
e
a
t
u
(
t
)
{\displaystyle e^{at}u(t)\,}
1
s
−
a
{\displaystyle {\frac {1}{s-a}}}
10
t
n
e
a
t
u
(
t
)
{\displaystyle t^{n}e^{at}u(t)\,}
n
!
(
s
−
a
)
n
+
1
{\displaystyle {\frac {n!}{(s-a)^{n+1}}}}
11
cos
(
ω
t
)
u
(
t
)
{\displaystyle \cos(\omega t)u(t)\,}
s
s
2
+
ω
2
{\displaystyle {\frac {s}{s^{2}+\omega ^{2}}}}
12
sin
(
ω
t
)
u
(
t
)
{\displaystyle \sin(\omega t)u(t)\,}
ω
s
2
+
ω
2
{\displaystyle {\frac {\omega }{s^{2}+\omega ^{2}}}}
13
cosh
(
ω
t
)
u
(
t
)
{\displaystyle \cosh(\omega t)u(t)\,}
s
s
2
−
ω
2
{\displaystyle {\frac {s}{s^{2}-\omega ^{2}}}}
14
sinh
(
ω
t
)
u
(
t
)
{\displaystyle \sinh(\omega t)u(t)\,}
ω
s
2
−
ω
2
{\displaystyle {\frac {\omega }{s^{2}-\omega ^{2}}}}
15
e
a
t
cos
(
ω
t
)
u
(
t
)
{\displaystyle e^{at}\cos(\omega t)u(t)\,}
s
−
a
(
s
−
a
)
2
+
ω
2
{\displaystyle {\frac {s-a}{(s-a)^{2}+\omega ^{2}}}}
16
e
a
t
sin
(
ω
t
)
u
(
t
)
{\displaystyle e^{at}\sin(\omega t)u(t)\,}
ω
(
s
−
a
)
2
+
ω
2
{\displaystyle {\frac {\omega }{(s-a)^{2}+\omega ^{2}}}}
17
1
2
ω
3
(
sin
ω
t
−
ω
t
cos
ω
t
)
{\displaystyle {\frac {1}{2\omega ^{3}}}(\sin \omega t-\omega t\cos \omega t)}
1
(
s
2
+
ω
2
)
2
{\displaystyle {\frac {1}{(s^{2}+\omega ^{2})^{2}}}}
18
t
2
ω
sin
ω
t
{\displaystyle {\frac {t}{2\omega }}\sin \omega t}
s
(
s
2
+
ω
2
)
2
{\displaystyle {\frac {s}{(s^{2}+\omega ^{2})^{2}}}}
19
1
2
ω
(
sin
ω
t
+
ω
t
cos
ω
t
)
{\displaystyle {\frac {1}{2\omega }}(\sin \omega t+\omega t\cos \omega t)}
s
2
(
s
2
+
ω
2
)
2
{\displaystyle {\frac {s^{2}}{(s^{2}+\omega ^{2})^{2}}}}
