Laplace Table

Table of Laplace Transforms
(
)
(
)
{
}
1
ftFs
-
=L
(
)
(
)
{
}
Fsft
=L
(
)
(
)
{
}
1
ftFs
-
=L
(
)
(
)
{
}
Fsft
=L
1. 1
1
s
2.
at
e
1
sa
-
3.
,1,2,3,
n
tn=
K
1
!
n
n
s
+
4.
p
t
, p > -1
(
)
1
1
p
p
s
+
G+
5.
t
3
2
2
s
p
6. 1
2
,1,2,3,
n
tn
-=
K
(
)
1
2
13521
2
n
n
n
s
p
+
××-L
7.
(
)
sin
at
22
a
sa
+
8.
(
)
cos
at
22
s
sa
+
9.
(
)
sin
tat
( )
2
22
2
as
sa
+ 10.
(
)
cos
tat
( )
22
2
22
sa
sa
-
+
11.
(
)
(
)
sincos
atatat
-
( )
3
2
22
2a
sa
+ 12.
(
)
(
)
sincos
atatat
+
( )
2
2
22
2as
sa
+
13.
(
)
(
)
cossin
atatat
-
(
)
( )
22
2
22
ssa
sa
-
+ 14.
(
)
(
)
cossin
atatat
+
(
)
( )
22
2
22
3
ssa
sa
+
+
15.
(
)
sin
atb
+
(
)
(
)
22
sincos
sbab
sa
+
+
16.
(
)
cos
atb
+
(
)
(
)
22
cossin
sbab
sa
-
+
17.
(
)
sinh
at
22
a
sa
-
18.
(
)
cosh
at
22
s
sa
-
19.
(
)
sin
at
bt
e
( )
2
2
b
sab
-+
20.
(
)
cos
at
bt
e
( )
2
2
sa
sab
-
-+
21.
(
)
sinh
at
bt
e
( )
2
2
b
sab
--
22.
(
)
cosh
at
bt
e
( )
2
2
sa
sab
-
--
23.
,1,2,3,
nat
tn=e
K
( )
1
!
n
n
sa
+
- 24.
(
)
fct
1
s
F
cc
æö
ç÷
èø
25.
(
)
(
)
c
ututc
=-
Heaviside Function
cs
s
-
e
26.
(
)
tc
d
-
Dirac Delta Function
cs
-
e
27.
(
)
(
)
c
utftc
-
(
)
cs
Fs
-
e 28.
(
)
(
)
c
utgt
(
)
{
}
cs
gtc
-+eL
29.
(
)
ct
ft
e
(
)
Fsc
-
30.
(
)
,1,2,3,
n
tftn=
K
( )
()
()
1
nn
Fs
-
31.
()
1
ft
t
()
s
Fudu
¥
ò 32.
()
0
t
fvdv
ò
(
)
Fs
s
33. ( ) ()
0
t
ftgd
ttt
-
ò
(
)
(
)
FsGs
34.
(
)
(
)
ftTft
+=
()
0
1
Tst
sT
ftdt
-
-
-
òe
e
35.
(
)
ft
¢
(
)
(
)
0
sFsf- 36.
(
)
ft
¢¢
(
)
(
)
(
)
2
00
sFssff
¢
--
37.
(
)
(
)
n
ft
(
)
(
)
(
)
(
)
(
)
(
)
(
)
21
12
0000
nn
nnn
sFssfsfsff
--
--
¢
----L

Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.
() ()
coshsinh
22
tttt
tt
--
+-
==
eeee
3. Be careful when using “normal” trig function vs. hyperbolic functions. The only
difference in the formulas is the “+ a2” for the “normal” trig functions becomes a
“- a2” for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as
()
1
0
xt
txdx
¥--
G=
ò
e
If n is a positive integer then,
(
)
1!
nn
G+=
The Gamma function is an extension of the normal factorial function. Here are a
couple of quick facts for the Gamma function
(
)
(
)
()( ) ( ) ( )
( )
1
121
1
2
ppp
pn
ppppn
p
p
G+=G
G+
+++-=G
æö
G=
ç÷
èø
L