The document discusses the unit step function (also called the Heaviside function) and provides its definition and Laplace transform. It also discusses properties related to the Laplace transform of the unit step function, including: 1) The Laplace transform of the unit step function u(t-a) is 1/s when t ≥ a and 0 when t < a. 2) Using the shifting property, the Laplace transform of f(t)u(t-a) is e-asL[f(t+a)], where L[f(t)] is the Laplace transform of f(t). 3) An example calculates the Laplace transform of t2u(t-2) using the

1. SUBMITTED BY: SAAHIL R.
KSHATRIYA
ENROLLMENT NO.: 150120119164
BATCH: 3C1
SUBMITTED TO:
PROF. PRIYA JANI

2. 7. UNIT STEP FUNCTION (OR
HEAVISIDE’S FUNCTION
 The unit step function u(t - a) is defined as
u(t – a) =0 if t < a (a ≥ 0)
=1 if t ≥ a figure.
2

3. The unit step function is also called the Heaviside
function. In particular if a = 0, we have
u(t) = 0 if t < 0
= 1 if t ≥ 0
3

4. 0
0
0
{u( )} . ( )
(0) (1)
1
st
st st
a
st
st as
a
L t a e u t a dt
e dt e dt
e
e dt e
s s


 
 
 
 
   
 
 
    

 

 Laplace transform of unit step function
By definition of Laplace transform,
4

5. 1
1
1 1
{ ( )} { ( )}
in particular, if a 0
1 1
{ ( )} ( )
as as
L u t a e L e u t a
s s
L u t L u t
s s
  

 
       

 
     
5

6. Example: find the laplace
transform of t2 u(t-2)
 solution:
 L[f(t) u(t-a)]= e-as L[f(t+a)]
 L [t2 u(t-2)] = e-2s L [(t+2)2 ]
 = e-2s L [t2 + 4t+4]
 = e-2s ( 2/s3 + 4/s2 + 4/s)

7. Second shifting theorem
If
7
0
0
{f( )} ( ),
{f(t a) u(t a)} (s)
{ ( ) u(t a)} . ( ) ( )
(u ) (0)dt ( ) (1)
as
st
a
st st
a
L t f s then
L e f
L f t a e f t a u t a dt
e f a e f t a dt




 

  
    
   

 
Proof : by definition of Laplace transform, we have,

8. ( )
0
0
1
(u )) ,
(r)
(r) ( )
{ ( ) u(t a)} ( ) [ ( )]
[ ( )] ( ) ( )
st
a
s a r
as sr as
as as
as
e f a dt Substituting t a r
e f dr
e e f dr e f s
L f t a e f s e L f t
L e f s f t a u t a



 

  
 
 
   

 
    
   



8

9. .1: { ( ) ( )} [ ( )]
.2: [ ( )] [1. ( )] (1)
.3: [ ( ) ( )]
.4: [ ( ) { ( ) ( )] { ( )} { ( )}
as
as
as
as bs
as bs
Cor L f t u t a e L f t a
e
Cor L u t a L u t a e L
s
e e
Cor L u t a u t b
s
Cor L f t u t a u t b e L f t a e L f t b



 
 
  
    

   
      
(alternative form of second shifting theorem)
9

10. 2. The Laplace transform of the Dirac delta
To solve initial value problems involving the Dirac delta, we need to know its Laplace
transform. By the third property of the Dirac delta,
We look into an example below

12. 12