ENGS 22 — Systems
Laplace Table
Page 1
Laplace Transform Table
Largely modeled on a table in D’Azzo and Houpis, Linear Control Systems Analysis and Design, 1988
F (s)
f (t)
t
≤
0
1. 1
)
(t
δ
unit impulse at t = 0
2.
s
1
1 or
)
(t
u
unit step starting at t = 0
3.
2
1
s
)
(t
u
t
⋅
or t ramp function
4.
n
s
1
1
)!
1
(
1
−
−
n
t
n
n = positive integer
5.
as
e
s
−
1
)
(
a
t
u
−
unit step starting at t = a
6.
)
1
(
1
as
e
s
−
−
)
(
)
(
a
t
u
t
u
−
−
rectangular pulse
7.
a
s
+
1
at
e
−
exponential decay
8.
n
a
s
)
(
1
+
at
n
e
t
n
−
−
−
1
)!
1
(
1
n = positive integer
9.
)
(
1
a
s
s
+
)
1
(
1
at
e
a
−
−
10.
)
)(
(
1
b
s
a
s
s
+
+
)
1
(
1
bt
at
e
a
b
a
e
a
b
b
ab
−
−
−
+
−
−
11.
)
)(
(
b
s
a
s
s
s
+
+
+
α
]
)
(
)
(
[
1
bt
at
e
a
b
b
a
e
a
b
a
b
ab
−
−
−
−
+
−
−
−
α
α
α
12.
)
)(
(
1
b
s
a
s
+
+
)
(
1
bt
at
e
e
a
b
−
−
−
−
13.
)
)(
(
b
s
a
s
s
+
+
)
(
1
bt
at
be
ae
b
a
−
−
−
−
