Table of Laplace Transforms
function (of t)
Laplace transform (a function of s)
1
,s > a
s−a
1.
eat
2.
1
3.
tn , n = 0, 1, . . .
4.
cos(bt)
s
s2 + b2
5.
sin(bt)
b
s2 + b2
6.
eat cos(bt)
s−a
(s − a)2 + b2
7.
eat sin(bt)
b
(s − a)2 + b2
8.
eat f (t)
F (s − a)
9.
δ(t − a)
e−as
1
,s > 0
s
n!
sn+1
e−as
s
10.
u(t − a), a ≥ 0
11.
f (t − a)u(t − a), a ≥ 0
12.
g(t)u(t − a), a ≥ 0
13.
tn f (t)
(−1)n F (n) (s)
14.
f 0 (t)
sF (s) − f (0)
15.
f (n) (t)
e−as F (s)
e−as L {g(t + a)}(s)
sn F (s) − sn−1 f (0) − sn−2 f 0 (0) − · · · sf (n−2) (0) − f (n−1) (0)
In this table F (s) = L {f (t)}(s).
u(t) is the step function defined on page 384 of the textbook and δ(t) is the Dirac delta function
discussed in Section 7.9 of the textbook.