The Laplace transform is defined as the integral of a function f(t) multiplied by e^-st from 0 to infinity, where s is a complex variable. Some basic Laplace transform properties and pairs are listed, including transforms of unit step functions, exponential functions, trigonometric functions, and their combinations. The document provides an introduction to basic Laplace transforms and lists some common transform pairs and properties.

1. Basic Laplace Transforms
The Laplace transform is defined by…
  




0
)
(
)
(
)
( dt
e
t
f
t
f
s
F st
L
…where s is an arbitrary complex variable chosen such that the integral converges.
A list of basic Laplace operations is shown below.
f (t) 



0
)
(
)
( dt
e
t
f
s
F st
)
(
)
( t
g
b
t
f
a  )
(
)
( s
G
b
s
F
a 
)
(t
f  )
0
(
)
( f
s
F
s 
)
(t
f 
 )
0
(
)
0
(
)
(
2
f
f
s
s
F
s 


)
(
)
(
t
f n
)
0
(
...
)
0
(
)
0
(
)
( )
1
(
2
1 






 n
n
n
n
f
f
s
f
s
s
F
s
)
(t
f
t )
(s
F

)
(t
f
tn
)
(
)
1
( )
(
s
F n
n

)
(t
f
e at

)
( a
s
F 
)
( a
t
f  )
(s
F
e as


t
du
u
f
0
)
(
s
s
F )
(

2. Basic Laplace Transforms
f (t) F(s)
)
(t
 1
)
( T
t 
 sT
e
)
(t
1
s
1
)
( T
t 
1 sT
e
s

1
)
(
)
( T
t
t 
1
1 )
1
(
1 sT
e
s


t 2
1
s
)!
1
(
1


n
tn
n
s
1
at
e
a
s 
1
at
n
e
t
n



1
)!
1
(
1
n
a
s )
(
1

t

sin 2
2



s
t

cos 2
2


s
s
t
e at


sin
1 
2
2
)
(
1


 a
s
t
e at

cos

2
2
)
( 



a
s
a
s

3. Basic Laplace Transforms
f (t) F(s)
)
(
1 bt
at
e
e
a
b



 )
)(
(
1
b
s
a
s 

)
(
1 bt
at
be
ae
b
a



 )
)(
( b
s
a
s
s


)
1
(
1 at
e
a


)
(
1
a
s
s 
)
1
(
1
2
at
at
ate
e
a



 2
)
(
1
a
s
s 
)
1
(
1
2
at
at
e
e
at
a





)
(
1
2
a
s
s 












a
b
ae
a
b
be
ab
bt
at
1
1
)
)(
(
1
b
s
a
s
s 

)
cos
1
(
1
2
t



)
(
1
2
2


s
s
t
e d
t
d
n



sin
1 
2
2
2
1
n
ns
s 
 

)
(
sin
1
1
2







 
t
e d
t
d
n
n
n
)
2
(
1
2
2
n
ns
s
s 
 

)
(
sin 
 
t 2
2
cos
sin






s
s
For a more extensive list of Laplace transforms, see Schaum's Mathematical Handbook, Murray
R. Spiegel, McGraw-Hill