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Lecture notes by Mr. Zalak Patel Page: - 1 -
LAPLACE TRANSFORMS
Weightage for university exam: 22Marks
No. of lectures required to teach: 06hrs
Laplace Transforms:
Let f (t) be a given function of t, for all t 0
Then the Laplace transform of f (t) ,denoted by Lf (t) or f sF(s)or(s)
Can be defined by
Lf t f s F s o s e f t dt st ( ) ( ) ( ) ( )
0
Where s is a real or complex parameter.
Linearity of Laplace Transform:
Laplace transform is linear.
i.e. La f (t) b g(t) a Lf (t) b Lg(t .)
Formulae:
Sr.no f (t) Laplace of f (t)
1 f (t) =1
s
L 1 1
2 f (t) = at e s a
L e at
1
3 f (t) = at e
s a
L e at
1
4 f (t) =sin at 2 2 sin
s a
a L at
5 f (t) =cos at 2 2 cos
s a
s L at
6 f (t) =sinh at 2 2 sinh
s a
a L at
7 f (t) =cosh at 2 2 cosh
s a
s L at
8 f (t) = n
t 1 1
| 1 ! n n
n
s
n
s
n L t
First Shifting Theorem:
Statement: If Lf (t) = f (s) then Le f (t) at = f (s a)
Change The Scale Property:
Statement: If Lf (t) = f (s) then Le f (bt) at =
b
1
b
s a f , b>0
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Lecture notes by Mr. Zalak Patel Page: - 2 -
Formulae:
Sr.no f (t) Laplace of f (t)
1 f (t) = at e sin bt
2 2
at e sinbt
s a b
b L
2 f (t) = at e cosbt
2 2
cos
s a b
s a L e bt at
3 f (t) = at e sinh bt
2 2
sinh
s a b
b L e bt at
4 f (t) = at e cosh bt
2 2
cosh
s a b
s a L e bt at
5 f (t) = at e n
t 1 1
| 1 !
n n
at n
s a
n
s a
n L e t
Inverse Laplace Transform:
If Lf (t) = f (s) then f (t) is called the inverse Laplace transform of
f (s) and denoted by ( ) 1 L f s = f (t).
Sr.no Laplace of f (t) Inverse of Laplace
1
s
L 1 1 1 1 1
s
L
2 s a
L e at
1 at e
s a
L
1 1
3 s a
L e at
1 at e
s a
L
1 1
4 2 2 sin
s a
a L at
a
at
s a
L 1 sin
2 2
1
5 2 2 cos
s a
s L at
at
s a
s L cos 2 2
1
6 2 2 sinh
s a
a L at a
at
s a
L 1 sinh
2 2
1
7 2 2 cosh
s a
s L at at
s a
s L cosh 2 2
1
8 1 1
| 1 ! n n
n
s
n
s
n L t !1
1
!
1 1
1
1
1
n
t
s
or L
n
t
s
L
n
n
n
n
9
2 2
at e sinbt
s a b
b L
e sinbt 1 1 at
2 2
1
s a b b
L
10
2 2
cos
s a b
s a L e bt at
e bt
s a b
s a L at cos 2 2
1
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Lecture notes by Mr. Zalak Patel Page: - 3 -
11
2 2
sinh
s a b
b L e bt at
e bt
s a b b
L at sinh 1 1
2 2
1
12
2 2
cosh
s a b
s a L e bt at
e bt
s a b
s a L at cosh 2 2
1
13 1 1
| 1 !
n n
at n
s a
n
s a
n L e t
at n at n
n e t
n
e t
s a n
L
!
1
| 1
1 1
1
1
Laplace transform of Derivative of f(t):
Lf (t) s Lf (t) f )0(
( ) ( ) )0( )0( )0( .............. )0( ( ) 1 2 3 ( )1 n n n n n n L f t s L f t s f s f s f f
Laplace transform of the integration of f(t):
If . ( ) ( ) ( ) ( )
0 s
f s L f t f s then L f u du
t
Derivative of Laplace transform of f(t):
If ( ) ( ) ( ) ( )1 f (s) ds
d L f t f s then L t f t n
n
n n
Integration of Laplace transform of f(t):
If f u du
t
f t L f t f s then L
s
( ) ( ) ( ) ( )
Convolution:
Convolution of function f(t) and g(t) is defined and denoted by,
( *) )(
( ) ( )
( *) ( ) ( ) ( )
0
0
g t f t
g u f t u du
f t g t f u g t u du
t
t
Convolution Theorem:
If Lf (t) f (s) and Lgf (t) g (s) then
( .) ( ) 1 L f s g s f t *)( g(t)
t
f u g t u du
0
( ) ( )
*)( ( )
( ) ( )
0
g t f t
g u f t u du
t