Laplace Transform
From Department of Mathematics at UTSA
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Caption text
f
(
t
)
{\displaystyle f(t)}
F
(
s
)
=
L
|
f
(
t
)
|
{\displaystyle F(s)={\mathcal {L}}|f(t)|}
1
{\displaystyle 1}
1
s
{\displaystyle {\frac {1}{s}}}
s
>
0
{\displaystyle s>0}
e
a
t
{\displaystyle e^{at}}
1
s
−
a
{\displaystyle {\frac {1}{s-a}}}
t
{\displaystyle t}
1
s
2
{\displaystyle {\frac {1}{s^{2}}}}
t
n
{\displaystyle t^{n}}
n
!
s
n
+
1
{\displaystyle {\frac {n!}{s^{n+1}}}}
t
n
e
a
t
{\displaystyle t^{n}e^{at}}
n
!
s
n
+
1
{\displaystyle {\frac {n!}{s^{n+1}}}}
sin
a
t
{\displaystyle \sin {at}}
a
s
2
+
a
2
{\displaystyle {\frac {a}{s^{2}+a^{2}}}}
cos
a
t
{\displaystyle \cos {at}}
s
s
2
+
a
2
{\displaystyle {\frac {s}{s^{2}+a^{2}}}}
sinh
a
t
{\displaystyle \sinh {at}}
a
s
2
−
a
2
{\displaystyle {\frac {a}{s^{2}-a^{2}}}}
cosh
a
t
{\displaystyle \cosh {at}}
s
s
2
−
a
2
{\displaystyle {\frac {s}{s^{2}-a^{2}}}}
e
a
t
sin
b
t
{\displaystyle e^{at}\sin {bt}}
b
(
s
−
a
)
2
+
b
2
{\displaystyle {\frac {b}{(s-a)^{2}+b^{2}}}}
e
a
t
cos
b
t
{\displaystyle e^{at}\cos {bt}}
s
−
a
(
s
−
a
)
2
+
b
2
{\displaystyle {\frac {s-a}{(s-a)^{2}+b^{2}}}}
e
a
t
sinh
b
t
{\displaystyle e^{at}\sinh {bt}}
b
(
s
−
a
)
2
−
b
2
{\displaystyle {\frac {b}{(s-a)^{2}-b^{2}}}}
e
a
t
cosh
b
t
{\displaystyle e^{at}\cosh {bt}}
s
−
a
(
s
−
a
)
2
−
b
2
{\displaystyle {\frac {s-a}{(s-a)^{2}-b^{2}}}}
Resources
Laplace Transforms
, Paul's Online Notes
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