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APPLIED MATHEMATICS
7th Chapter
Laplace Transform
3rd Semester, SBTE BIHAR
(As Per New Syllabus Effective from 2016-2019 Batch)
7th Chapter
Laplace Transform
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CHAP 1
7_LAPLACE TRANSFORM
PAGE 1
CHAPTER 7
7_LAPLACE TRANSFORM
1.1
INTRODUCTION
The Laplace transform is a very versatile tool to the engineer and scientist. It
enables one to find solutions to initial-value problems involving homogeneous
and non-homogeneous equations. The method can be extended to solve systems
of differential equations, partial differential equations and integral equations.
Its importance lies in the fact that its application is considerably easier than
other available techniques. In this chapter, we will start with a definition of the
transform and state some sufficient conditions for its existence. We then derive
its general properties and develop a table of transforms of some functions which
are usually encountered in the solution of linear differential equations.
1.2
DEFINITION OF LAPLACE TRANSFORM
Let f ^ t h be a given function defined for all positive values of t , i.e. t $ 0 . Then,
the Laplace transform of f ^ t h is defined as the function F ^s h such that
L 6f ^ t h@ = F ^s h =
#e
3-st
0
f ^ t h dt
...(1)
where L is referred to as the Laplace transform operator. Generally, the transform
will exist for more than one real value of s , and hence L 6f ^ t h@ defines a function
of s when it exists.
1.3
SUFFICIENT CONDITION FOR EXISTENCE
While finding the Laplace transforms of elementary function, it can be noticed
that the integral exists under certain conditions, such as s > 0 as s > a etc. In
general, the function f (t) must satisfy the following conditions for the existence
of the Laplace transform.
1. The function f (t) must be piece-wise continuous or sectionally continuous
in any limited interval 0 < a # t # b .
2. The function f (t) is of exponential order.
Piece-wise Continuous Function
A function f (t) is said to be piece-wise (or sectionally) continuous over the closed
interval [a , b] if it is defined on that interval and is such that the interval can be
divided into a finite number of subintervals, in each of which f (t) is continuous
and has both right and left hand limits at every end point of the subinterval.
e.g. :
1. The function f (t) = t2 , 0 < t < 5 = 2t + 3 , t > 5 is sectionally continuous
for t > 0 .
2. The function f (t) = 1t is not sectionally continuous in any interval containing
t = 0.
Functions of Exponential Order
A function f (t) is said to be of exponential order a if,

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PAGE 2
7_LAPLACE TRANSFORM
CHAP 1
lim e-at f (t) = a finite quantity
t"3
i.e., for a given positive number T , there exists a real number M > 0 such that,
e-at f (t) < M , 6t $ T
or
f (t) < Meat , 6t $ T
For example, f (t) = t2 , sin at , eat etc. are all of exponential order and also
continuous. But f (t) = et is not of exponential order and as such its Laplace
transform does not exist.
2
1.4
LAPLACE TRANSFORM OF SOME ELEMENTARY FUNCTIONS
We have just discussed that Eq(1.2) is the generalised form of Laplace
transformation for the causal function that assumes zero value for t < 0 . Using
the transformation, let us obtain the Laplace transforms of some elementary
functions (causal functions).
1.4.1
Unit Impulse Function
Unit impulse function is defined as
0, t ! 0
d^t h = *
3, t = 0
So, the Laplace transform of unit impulse function is given by
L 6d^ t h@ =
#e
3-st
0
d^ t h dt = lim 6e-st d^ t h@
t"0
=1
1.4.2
Unit Step Function
Unit step function is defined as
1,
u^t h = *
0,
t$0
t<0
So, the Laplace transform of unit step function is given by
L 61@ =
# ^e
3 -st
0
-st 3
h^1 h dt = :e- s D = :0 - -1s D
0
= 1 , for s > 0
s
1.4.3
Exponential Function
Let the exponential function,
f ^ t h = eat
Its Laplace transform is obtained as
L 6eat@ =
=
#
0
3-st at
e
e dt =
#
3 - s-a t
^
h
0
1 , for s > a
s-a
e
dt =
e-^s - a ht
-^s - a h
3
0

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CHAP 1
1.4.4
7_LAPLACE TRANSFORM
PAGE 3
Trigonometric Function
Let the sine function,
f ^ t h = sin at
Its Laplace transform is obtained as
iat
-iat
L 6sin at @ = L ;e - e E
2i
iat
e-iat E
;sin at = e 2i
= 1 8L ^eiat h - L ^e-iat hB
2i
= 1 : 1 - 1 D = 1 c 22ia 2 m :L ^eiat h = 1 D
2i s - ia s + ia
2i s + a
s - ia
= 2 a 2 , for s > 0
s +a
Again, let the cosine function,
f ^ t h = cos at
Its Laplace transform is obtained as
iat
-iat
L 6cos at @ = L ;e + e E
2
iat
-iat
;cos at = e +2 e E
= 1 8L ^eiat h + L ^e-iat hB
2
= 1 : 1 + 1 D = 1 c 2 2s 2 m
2 s - ia s + ia
2 s +a
= 2 s 2 , for s > 0
s +a
1.4.5
1
iat
:L ^e h = s - ia D
Hyperbolic Function
Let the sine hyperbolic function,
f ^ t h = sinh at
Its Laplace transform is obtained as
at
-at
L 6sinh at @ = L ;e - e E
2
at
-at
;sinh at = e -2 e E
= 1 8L ^eat h - L ^e-at hB
2
= 1 : 1 - 1 D = 1 b 2 2a 2 l
2 s-a s+a
2 s -a
= 2 a 2 , for s > a
s -a
1
at
:L ^e h = s - a D
Again, let the cosine hyperbolic function,
f ^ t h = cosh at
Its Laplace transform is obtained as
at
-at
L 6cosh at @ = L ;e + e E
2
at
-at
;cosh at = e +2 e E
= 1 8L ^eat h + L ^e-at hB
2
= 1 : 1 + 1 D = 1 b 2 2s 2 l
2 s-a s+a
2 s -a
1
at
:L ^e h = s - a D

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