Properties of laplace transform

1. Properties of Laplace Transform
Name
Md. Mehedi Hasan
Student ID
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2. Laplace Transform Properties
 Definition of the Laplace transform
 A few simple transforms
 Rules
 Demonstrations

3. Definition
 The Laplace transform is.
 Common notation:
    
 s
dtett st
F
ffL
0

 



    
    st
st
Gg
Ff


L
L    
   st
st
Gg
Ff


   




0
fF dtets st
       



0
vuvu dttttt

4. Definition
 Notation:
 Variables in italics t, s
 Functions in time space f, g
 Functions in frequency space F, G
 Specific limits
   
   t
t
t
t
flim0f
flim0f
0
0









5. Example Transforms
 We will look at the Laplace transforms of:
 The impulse function d(t)
 The unit step function u(t)
 The ramp function t
 Sine and cosine
 Exponential Function
 Time differentiation

6. Impulse Function
 The easiest transform is that of the impulse function:
    
1
δδ
0
0







s
st
e
dtettL
  1δ  t

7. Unit Step Function
 Next is the unit step function
    
s
e
s
e
s
dte
dtett
s
st
st
st
1
1
0
1
uuL
0
0
0
0























 






11
00
u
t
t
t  
s
t
1
u 

8. Ramp Function
 The ramp function
  
2
0
0
00
0
111
1
0
1
1
1
u
s
e
ss
dte
s
dte
s
e
s
t
dttett
st
st
stst
st





































L
  2
1
u
s
tt 
t
t
ddf
f


st
st
e
s
te




1
g
ddg

9. The Sine Function
 Sine requires two integration by parts:
      
   
 
   
 
    tt
ss
dtet
ss
dtet
ss
stet
s
dtet
s
dtet
s
stet
s
dtettt
st
st
st
st
st
usin
11
sin
11
sin
11
cos
1
cos
1
0
cos
1
sin
1
sinusin
22
0
22
0
2
0
0
00
0
L
L





























10. The Sine Function
 Consequently:
         
      
    
1
1
usinL
1usinL1
usinL
11
usinL
2
2
22




s
tt
tts
tt
ss
tt
   
1
1
usin 2


s
tt

11. The Cosine Function
 As does cosine:
      
   
 
   
 
    tt
ss
dtet
ss
dtet
ss
stet
ss
dtet
ss
dtet
s
stet
s
dtettt
st
st
st
st
st
ucosL
11
cos
1
0
1
cos
11
sin
11
sin
11
sin
1
cos
1
cosucosL
2
0
22
0
2
0
0
00
0





































12. The Cosine Function
 Consequently:
         
      
    
1
ucos
ucos1
ucos
11
ucos
2
2
2




s
s
tt
stts
tt
ss
tt
L
L
LL
   
1
ucos 2


s
s
tt

13. Integration of Exponential Functions
 We now have the following commutative diagram
 
ass
s
e at



11
L
 








 
assa
e
a
at
111
1L
1
 ass 
1










t
a
de
0
L 

14. The Convolution
 Define the convolution to be
 Then
      
    

dt
dtt








gf
gfgf
      
       ss
j
tt
sst
GF
2
1
gf
GFgf




15. Integration
 As a special case of the convolution
         
   
s
s
s
s
sttsd
t
F1
F
uff
0











LL 

16. Summary
 We have seen these Laplace transforms:
 
 
 
  1
2
!
u
1
u
1
u
1δ





n
n
s
n
tt
s
tt
s
t
t
 
   
   
1
ucos
1
1
usin
1
1
u
2
2






s
s
tt
s
tt
s
tet

17. Summary
 In this topic:
 We defined the Laplace transform
 Looked at specific transforms
 Derived some properties
 Applied properties

18. References
 Spiegel, Laplace Transforms, McGraw-Hill, Inc., 1965.
 Wikipedia, http://en.wikipedia.org/wiki/Laplace_Transform

19. Thank You