This document discusses properties of the Laplace transform. It begins by defining the Laplace transform and common notation used. It then provides examples of Laplace transforms for some common functions like the impulse function, unit step function, ramp function, sine and cosine functions, and exponential functions. It also discusses rules like linearity and time shifting. The document concludes by summarizing some of the key Laplace transforms and properties discussed.

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