Laplace Transforms
 
f (s )  Lf (t )
f (t )  L1 f (s )
1
a
1
s a
1
s a
1
e at
e at
t
tn ,
1
s
a
s
n = a positive integer
u(t )
s2
n!
s n 1
1
s
u(t  a)
e  as
s
u(t  a)f (t  a)
e as f (s)
Pulse of β in time a
1  e 

 as
s
 (t )
1
sin bt
2
cos bt
sinh bt
cosh bt
b
s  b2
s
s2  b2
b
s2  b2
s
s2  b2
e  at f (t )
f (s  a)
f (at )
1 s
f  ,
a  a
f (t)
sf (s)  f (0)
f (n ) (t )
s n f (s )  s n 1f (0)  s n 2f (0)    f (n 1) (0)
tf (t )

t 2 f (t )
t n f (t)
t
 f (t)dt
0
a0
df (s)
ds
d 2 f (s)
ds 2
(1)n
f (s)
s
d n f (s)
dsn