Section 4.3 - Rimmer
∞
Laplace Transform L { f ( t )} = ∫ f ( t ) e − st dt = F (s )
0
{
}
∞
∞
L e f (t ) = ∫ e f (t )e dt = ∫ f (t )e −( s − a )t dt = F (s − a )
at
a any real
number
at
− st
0
0
First
Translation
Theorem
laplace of f (t )
with s − a replacing s
shift on the s − axis
Section 4.3 - Rimmer
We’ve seen this translation theorem in action already when we derived both
L {t n } =
{ }
L t n e at =
n ! for integer n > 0
s n +1 s > 0
for integer n > 0
s>a
n!
(s − a )n+1
We derived
L{sin (bt )} =
b
,s > 0
s + b2
2
Now, instead of using the definition, we can use the translation theorem to find
{
}
L e at sin (bt ) =
b
(s − a )2 + b 2
,s > a
1
Section 4.3 - Rimmer
Working backwards, we have:
L−1{F (s − a )} = e at f (t )
example 1:
1


L−1  2

 s − 6 s + 13 
We need to complete the
square in the denominator.
 −1 

1

 −1 
1
2
L−1  2
= L 
= L  12

=
2
2
 s − 6 s + 9 + 13 − 9 
 ( s − 3) + 4 
 ( s − 3) + 4 
1
2
e3t sin 2t
( 1 2 of 6)2
example 2:
s


L−1  2

 s + 4s + 5 
L{cos(bt )} =
s
,s > 0
s 2 + b2

s



s
= L−1  2
 = L−1 

2
 s + 4s + 4 + 5 − 4 
 (s + 2 ) + 1 
need s + 2 in
the numerator
 s+2−2 

s+2
1
−1 
−2 t
−2 t
= L−1 
−2
=L 
 = e cos t − 2e sin t
2
2
2
(
s
+
2
)
+
1
(
s
+
2
)
+
1
(
s
+
2
)
+
1




Section 4.3 - Rimmer
Unit Step Function
on
0, 0 ≤ t < a
U (t − a ) = 
t≥a
1,
off
 0, 0 ≤ t < a
f (t )U (t − a ) = 
t≥a
 f (t ),
 f (t ) OFF, 0 ≤ t < a
=
t≥a
 f (t ) ON,
 0, 0 ≤ t < π2
sin tU (t − π2 ) = 
t ≥ π2
sin t ,
0 ≤ t < π2
 0,
sin (t − π2 )U (t − π2 ) = 
π
t ≥ π2
sin (t − 2 ),
0 up to π2 and then
picks up sin t at π2
0 up to π2 and then
shift sin t and start
it at π2
shift on the t − axis
2
Section 4.3 - Rimmer
Second
Translation
Theorem
L { f ( t − a ) U ( t − a )} = e − as L { f ( t )}
a any real
number
∞
L{ f (t − a )U (t − a )} = ∫ f (t − a )U (t − a )e − st dt
0
a
∞
= ∫ f ( t − a ) U ( t − a ) e dt + ∫ f ( t − a ) U ( t − a ) e − st dt
− st
0
a
0 for 0 ≤ t < a
∞
= ∫ f ( t − a ) e − st dt
a
1 for t ≥ a
w=t −a
t = w+a
dw = dt
t =a⇒w=0
∞
∞
= e − as L { f ( t )}
= ∫ f ( w ) e − s( w+ a ) dw = e − as ∫ f ( w ) e − sw dw
0
0 L { f ( t )}
Section 4.3 - Rimmer
−π
s
e2
π
π
L {sin ( t − 2 )U ( t − 2 )} = 2
s +1
 g ( t ) , 0 ≤ t < a
f (t ) = 
⇒ f ( t ) = g ( t ) − g ( t )U ( t − a ) + h ( t )U ( t − a )
t≥a
 h ( t ) ,
0≤t <a
 0,

f (t ) =  g (t ) , a ≤ t < b
 0,
t ≥b

⇒ f ( t ) = g ( t ) U ( t − a ) − U ( t − b ) 
Alternative form of 2nd Translation Theorem
L { f ( t ) U ( t − a )} = e − as L { f ( t + a )}
a any real
number
3
Section 4.3 - Rimmer
−π
s
L {sin ( t ) U ( t − π2 )} = e 2 L {sin ( t + π2 )}
−π
s
= e 2 L {cos ( t )}
−π
s
e2 s
= 2
s +1
4