ទំព័រដើម
ចៃដន្យ
ជិតខាង
កត់ឈ្មោះចូល
ការកំណត់
បរិច្ចាគ
អំពីវិគីភីឌា
ការបដិសេធ
ស្វែងរក
តារាងបំលែងឡាប្លាស
ភាសា
តាមដាន
កែប្រែ
លរ
f(t)
F(s)
សំគាល់
១
u
(
t
)
=
{
0
(
t
<
0
)
1
2
(
t
=
0
)
1
(
t
>
0
)
{\displaystyle u(t)={\begin{cases}0\quad &(t<0)\\{\frac {1}{2}}\quad &(t=0)\\1\quad &(t>0)\end{cases}}}
1
s
{\displaystyle {\frac {1}{s}}}
អនុគមន៍កាំជណ្តើរហេវីសាយ
២
δ
(
t
)
=
{
∞
(
t
=
0
)
0
(
t
≠
0
)
{\displaystyle \delta (t)={\begin{cases}\infty \quad &(t=0)\\0\quad &(t\neq 0)\end{cases}}}
1
{\displaystyle \,1}
អនុគមន៍ដែលតា
៣
α
{\displaystyle \alpha \,}
α
s
{\displaystyle {\frac {\alpha }{s}}}
៤
d
n
d
t
n
δ
(
t
)
{\displaystyle {\frac {d^{n}}{dt^{n}}}\delta (t)\,}
s
n
{\displaystyle s^{n}\,}
δ
(
t
)
{\displaystyle \delta (t)\,}
:
អនុគមន៍ដែលតា
៥
e
−
α
t
{\displaystyle e^{-{\alpha }t}\,}
1
s
+
α
{\displaystyle {\frac {1}{s+\alpha }}\,}
អនុគមន៍អិចស្ប៉ូណង់ស្យែល
៦
δ
(
t
)
−
α
e
−
α
t
{\displaystyle \,\delta (t)-{\alpha }e^{-{\alpha }t}}
s
s
+
α
{\displaystyle {\frac {s}{s+\alpha }}\,}
δ
(
t
)
{\displaystyle \delta (t)\,}
:
អនុគមន៍ដែលតា
៧
t
n
n
!
{\displaystyle {\frac {t^{n}}{n!}}\,}
1
s
n
+
1
{\displaystyle {\frac {1}{s^{n+1}}}\,}
n
{\displaystyle n\,}
:
ចំនួនគត់ធម្មជាតិ
៨
e
−
α
t
−
e
−
β
t
β
−
α
{\displaystyle {\frac {e^{-{\alpha }t}-e^{-{\beta }t}}{\beta -\alpha }}\,}
1
(
s
+
α
)
(
s
+
β
)
{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )}}\,}
៩
(
a
−
α
)
e
−
α
t
−
(
a
−
β
)
e
−
β
t
β
−
α
{\displaystyle {\frac {(a-\alpha )e^{-{\alpha }t}-(a-\beta )e^{-{\beta }t}}{\beta -\alpha }}\,}
s
+
a
(
s
+
α
)
(
s
+
β
)
{\displaystyle {\frac {s+a}{(s+\alpha )(s+\beta )}}\,}
១០
1
α
sin
α
t
{\displaystyle {\frac {1}{\alpha }}\sin {\alpha }t\,}
1
s
2
+
α
2
{\displaystyle {\frac {1}{s^{2}+{\alpha }^{2}}}\,}
១១
1
α
sinh
α
t
{\displaystyle {\frac {1}{\alpha }}\sinh {\alpha }t\,}
1
s
2
−
α
2
{\displaystyle {\frac {1}{s^{2}-{\alpha }^{2}}}\,}
១២
cos
α
t
{\displaystyle \cos {\alpha }t\,}
s
s
2
+
α
2
{\displaystyle {\frac {s}{s^{2}+{\alpha }^{2}}}\,}
១៣
cosh
α
t
{\displaystyle \cosh {\alpha }t\,}
s
s
2
−
α
2
{\displaystyle {\frac {s}{s^{2}-{\alpha }^{2}}}\,}
១៤
t
e
−
α
t
{\displaystyle te^{-{\alpha }t}\,}
1
(
s
+
α
)
2
{\displaystyle {\frac {1}{(s+\alpha )^{2}}}\,}
១៥
[
(
a
−
α
)
t
+
1
]
e
−
α
t
{\displaystyle [(a-\alpha )t+1]e^{{-\alpha }t}\,}
s
+
a
(
s
+
α
)
2
{\displaystyle {\frac {s+a}{(s+\alpha )^{2}}}\,}
១៦
1
β
e
−
α
t
sin
β
t
{\displaystyle {\frac {1}{\beta }}e^{-{\alpha }t}\sin {\beta }t\,}
1
(
s
+
α
)
2
+
β
2
{\displaystyle {\frac {1}{(s+\alpha )^{2}+{\beta }^{2}}}\,}
១៧
e
−
α
t
sin
β
t
{\displaystyle e^{-{\alpha }t}\sin {\beta }t\,}
β
(
s
+
α
)
2
+
β
2
{\displaystyle {\frac {\beta }{(s+\alpha )^{2}+{\beta }^{2}}}\,}
β
2
>
0
{\displaystyle {\beta }^{2}>0\,}
១៨
e
−
α
t
cos
β
t
{\displaystyle e^{-{\alpha }t}\cos {\beta }t\,}
s
+
α
(
s
+
α
)
2
+
β
2
{\displaystyle {\frac {s+\alpha }{(s+\alpha )^{2}+{\beta }^{2}}}\,}
β
2
>
0
{\displaystyle {\beta }^{2}>0\,}
១៩
1
β
[
(
a
−
α
)
2
+
β
2
]
1
2
e
−
α
t
sin
(
β
t
+
φ
)
{\displaystyle {\frac {1}{\beta }}[(a-\alpha )^{2}+{\beta }^{2}]^{\frac {1}{2}}e^{-{\alpha }t}\sin({\beta }t+\varphi )}
s
+
a
(
s
+
α
)
2
+
β
2
{\displaystyle {\frac {s+a}{(s+\alpha )^{2}+{\beta }^{2}}}\,}
φ
=
tan
−
1
β
a
−
α
{\displaystyle \varphi =\tan ^{-1}{\frac {\beta }{a-\alpha }}\,}
២០
e
−
α
t
sinh
β
t
{\displaystyle e^{-{\alpha }t}\sinh {\beta }t\,}
β
(
s
+
α
)
2
−
β
2
{\displaystyle {\frac {\beta }{(s+\alpha )^{2}-{\beta }^{2}}}\,}
២១
e
−
α
t
cosh
β
t
{\displaystyle e^{-{\alpha }t}\cosh {\beta }t\,}
s
+
α
(
s
+
α
)
2
−
β
2
{\displaystyle {\frac {s+\alpha }{(s+\alpha )^{2}-{\beta }^{2}}}\,}
២២
1
α
β
+
β
e
−
α
t
−
α
e
−
β
t
α
β
(
α
−
β
)
{\displaystyle {\frac {1}{{\alpha }{\beta }}}+{\frac {{\beta }e^{-{\alpha }t}-{\alpha }{e^{{-\beta }t}}}{{\alpha }{\beta }(\alpha -\beta )}}}
1
s
(
s
+
α
)
(
s
+
β
)
{\displaystyle {\frac {1}{s(s+\alpha )(s+\beta )}}\,}
២៣
a
α
β
+
a
−
α
α
(
α
−
β
)
e
−
α
t
+
a
−
β
β
(
α
−
β
)
e
−
β
t
{\displaystyle {\frac {a}{{\alpha }{\beta }}}+{\frac {a-\alpha }{{\alpha }(\alpha -\beta )}}e^{-{\alpha }t}+{\frac {a-\beta }{{\beta }(\alpha -\beta )}}e^{-{\beta }t}}
s
+
a
s
(
s
+
α
)
(
s
+
β
)
{\displaystyle {\frac {s+a}{s(s+\alpha )(s+\beta )}}\,}
២៤
e
−
α
t
(
β
−
α
)
(
γ
−
α
)
+
e
−
β
t
(
α
−
β
)
(
γ
−
β
)
+
e
−
γ
t
(
α
−
γ
)
(
β
−
γ
)
{\displaystyle {\frac {e^{-{\alpha }t}}{(\beta -\alpha )(\gamma -\alpha )}}+{\frac {e^{-{\beta }t}}{(\alpha -\beta )(\gamma -\beta )}}+{\frac {e^{-{\gamma }t}}{(\alpha -\gamma )(\beta -\gamma )}}}
1
(
s
+
α
)
(
s
+
β
)
(
s
+
γ
)
{\displaystyle {\frac {1}{(s+\alpha )(s+\beta )(s+\gamma )}}\,}
២៥
(
a
−
α
)
e
−
α
t
(
β
−
α
)
(
γ
−
α
)
+
(
a
−
β
)
e
−
β
t
(
α
−
β
)
(
γ
−
β
)
+
(
a
−
γ
)
e
−
γ
t
(
α
−
γ
)
(
β
−
γ
)
{\displaystyle {\frac {(a-\alpha )e^{-{\alpha }t}}{(\beta -\alpha )(\gamma -\alpha )}}+{\frac {(a-\beta )e^{-{\beta }t}}{(\alpha -\beta )(\gamma -\beta )}}+{\frac {(a-\gamma )e^{-{\gamma }t}}{(\alpha -\gamma )(\beta -\gamma )}}}
s
+
a
(
s
+
α
)
(
s
+
β
)
(
s
+
γ
)
{\displaystyle {\frac {s+a}{(s+\alpha )(s+\beta )(s+\gamma )}}\,}
២៦
1
α
2
(
1
−
cos
α
t
)
{\displaystyle {\frac {1}{{\alpha }^{2}}}(1-\cos {\alpha }t)\,}
1
s
(
s
2
+
α
2
)
{\displaystyle {\frac {1}{s(s^{2}+{\alpha }^{2})}}\,}
២៧
a
α
2
−
(
a
2
+
α
2
)
1
2
α
2
cos
(
α
t
+
φ
)
{\displaystyle {\frac {a}{{\alpha }^{2}}}-{\frac {(a^{2}+{\alpha }^{2})^{\frac {1}{2}}}{{\alpha }^{2}}}\cos({\alpha }t+\varphi )}
s
+
a
s
(
s
2
+
α
2
)
{\displaystyle {\frac {s+a}{s(s^{2}+{\alpha }^{2})}}\,}
φ
=
tan
−
1
α
a
{\displaystyle \varphi =\tan ^{-1}{\frac {\alpha }{a}}\,}
២៨
t
α
−
1
α
2
(
1
−
e
−
α
t
)
{\displaystyle {\frac {t}{\alpha }}-{\frac {1}{{\alpha }^{2}}}(1-e^{-{\alpha }t})}
1
s
2
(
s
+
α
)
{\displaystyle {\frac {1}{s^{2}(s+\alpha )}}\,}
២៩
a
−
α
α
2
e
−
α
t
+
a
α
t
−
α
−
a
α
2
{\displaystyle {\frac {a-\alpha }{{\alpha }^{2}}}e^{-{\alpha }t}+{\frac {a}{\alpha }}t-{\frac {\alpha -a}{{\alpha }^{2}}}}
s
+
a
s
2
(
s
+
α
)
{\displaystyle {\frac {s+a}{s^{2}(s+\alpha )}}\,}
៣០
1
−
(
1
+
α
t
)
e
−
α
t
α
2
{\displaystyle {\frac {1-(1+{\alpha }t)e^{-{\alpha }t}}{{\alpha }^{2}}}\,}
1
s
(
s
+
α
)
2
{\displaystyle {\frac {1}{s(s+\alpha )^{2}}}\,}
៣១
a
α
2
{
1
−
[
1
+
(
1
−
α
a
)
α
t
]
e
−
α
t
}
{\displaystyle {\frac {a}{{\alpha }^{2}}}\{1-[1+(1-{\frac {\alpha }{a}}){\alpha }t]e^{-{\alpha }t}\}}
s
+
a
s
(
s
+
α
)
2
{\displaystyle {\frac {s+a}{s(s+\alpha )^{2}}}\,}
៣២
ω
2
>
α
2
{\displaystyle \,{\omega }^{2}>{\alpha }^{2}}
1
ω
2
[
1
−
ω
0
ω
e
−
α
t
sin
(
ω
t
+
φ
)
]
{\displaystyle {\frac {1}{{\omega }^{2}}}[1-{\frac {{\omega }_{0}}{\omega }}e^{-{\alpha }t}\sin({\omega }t+\varphi )]}
ω
2
=
α
2
{\displaystyle \,{\omega }^{2}={\alpha }^{2}}
1
ω
2
[
1
−
e
−
α
t
(
1
+
α
t
)
]
{\displaystyle {\frac {1}{{\omega }^{2}}}[1-e^{-{\alpha }t}(1+{\alpha }t)]}
ω
2
<
α
2
{\displaystyle \,{\omega }^{2}<{\alpha }^{2}}
1
ω
2
[
1
−
ω
0
2
n
−
m
(
e
−
m
t
m
−
e
−
n
t
n
)
]
{\displaystyle {\frac {1}{{\omega }^{2}}}[1-{\frac {{\omega }_{0}^{2}}{n-m}}({\frac {e^{-mt}}{m}}-{\frac {e^{-nt}}{n}})]}
1
s
(
s
+
2
α
s
+
ω
0
2
)
{\displaystyle {\frac {1}{s(s+2{\alpha }s+{\omega }_{0}^{2})}}\,}
φ
=
tan
−
1
ω
α
ω
2
=
ω
0
2
−
α
2
{\displaystyle {\begin{aligned}\varphi &=\tan ^{-1}{\frac {\omega }{\alpha }}\\{\omega }^{2}&={\omega }_{0}^{2}-{\alpha }^{2}\end{aligned}}}
m
{\displaystyle \,m}
និង
n
{\displaystyle \,n}
ជារឹសនៃសមីការ
s
2
+
2
α
s
+
ω
0
2
=
0
{\displaystyle \,s^{2}+2{\alpha }s+{\omega }_{0}^{2}=0}
សូមមើលផងដែរ
កែប្រែ
បំលែងឡាប្លាស
: អនុគមន៍ដែលតា
: អនុគមន៍ដែលតា
: ចំនួនគត់ធម្មជាតិ
![{\displaystyle [(a-\alpha )t+1]e^{{-\alpha }t}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a0f1ac7e24381a5aa2e3bc2921bf72708d424a6)
![{\displaystyle {\frac {1}{\beta }}[(a-\alpha )^{2}+{\beta }^{2}]^{\frac {1}{2}}e^{-{\alpha }t}\sin({\beta }t+\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e809b50cb022e3919608682549c1ed4428b9be3)
![{\displaystyle {\frac {a}{{\alpha }^{2}}}\{1-[1+(1-{\frac {\alpha }{a}}){\alpha }t]e^{-{\alpha }t}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c564ed574042548a00599b13ffd655e27a21725f)
![{\displaystyle {\frac {1}{{\omega }^{2}}}[1-{\frac {{\omega }_{0}}{\omega }}e^{-{\alpha }t}\sin({\omega }t+\varphi )]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25478b45ffa200164731a67c06c176b4f68c2683)
![{\displaystyle {\frac {1}{{\omega }^{2}}}[1-e^{-{\alpha }t}(1+{\alpha }t)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3da9130baba7480b1715a86ad46768527f71e054)
![{\displaystyle {\frac {1}{{\omega }^{2}}}[1-{\frac {{\omega }_{0}^{2}}{n-m}}({\frac {e^{-mt}}{m}}-{\frac {e^{-nt}}{n}})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/773252f0a289162b25c79c892a04734db20cce21)
និង 
ជារឹសនៃសមីការ 
