MATH 334 LAPLACE TRANFORMATIONS
Definition: Let f(t) be a function defined for all t ≥ 0. Then the integral,
∞
− = ∫
0
{ ( )} ( ) st L f t
... [Show More] e f t dt
if exists, is called Laplace Transform of f(t). ‘s’ is a parameter, may be real or complex number.
Clearly L{f(t)} being a function of s is briefly written as f s ie L f t f s ( ) . . { ( )} ( ) = . Here the symbol
L which transforms f(t) into f (s) is called Laplace Transform Operator. Then f(t) is called
inverse Laplace transform of f (s) or simply inverse transform of 1 f s ie L f s ( ) . . { ( )} − .
Note: There are two types of laplace transforms. The above form of integral is known as one sided or unilateral transform.
However, if the transform is defined as
0
{ ( )} ( ) st L f t e f t dt
∞ − = ∫ , where s is complex variable, called two sided or bilateral
laplace of f(t), provided the integral exists. If in the first type transform, variable s is a complex number with a result that
756 Engineering Mathematics through Applications
laplace transform is defined over a portion of complex plane. If L{f(t)} exists for s real and then L{f(t)} exists in half of the
complex plane in which Re s>a (Fig.12.1). The transform f s( ) is an analytic function with properties:
(i) →∞
= Re.
() 0 s
Lt f s viz. the necessary condition for f s( ) to be a transform.
(ii) . () s
Lt sf s A
→∞ = , if the original function has a limit . ( )
t
Lt f t A
→∞
=
Imgn.s
o
a
Fig. 12.1: Real axis
12.2 EXISTENCE CONDITIONS
The laplace transform does not exist for all functions. If it exists, it is uniquely determined.
For existence of laplace, the given function has to be continuous on every finite interval and
of exponential order i.e. if there exist positive constants M and ‘a’ such that ( ) at f t Me ≤ for
all t ≥ 0. The function f(t) is some times termed as object function defined for all t ≥ 0 and f s( )
is termed as the resultant image function. Here the parameter should be sufficiently large to
make the integral convergent. In the above discussion, the condition for existence of f s( ) is
sufficient but not necessary, which precisely means that if the above conditions are satisfied,
the laplace transform of f(t) must exist.
But if these conditions are not satisfied, the laplace transform may or may not exist. For
e.g. in case of ( ) ( ) 1 ft ft t , as 0
t
= →∞ → , precisely means ( ) 1 f t t = is not piecewise
continuous on every finite interval in the range t ≥ 0. However, f(t) is integrable from 0 to
any positive value, say t0. Also ( ) at f t Me ≤ for all t > 1 with M = 1 and a = 0. Thus,
Lf t s () , 0
s
π
= > exists even if 1
t is not piecewise continuous in the range t ≥ 0.
12.3 EXISTENCE THEOREM ON LAPLACE TRANSFORM
If f(t) is a function which is piecewise* continuous on every finite interval in the range t ≥ 0
and satisfy ( ) at f t Me ≤ for all t ≥ 0 and for some positive constants ‘a’ and M means, f(t) is of
exponential order ‘a’, then the laplace transform of f(t) i.e.
∞ − ∫
0
( ) st e f t dt exists.
Laplace Transforms and their Application 757
Proof: We have { } ( ) () () () 0
0 0 0
t
st st st
t
L f t e f t dt e f t dt e f t dt
∞ ∞ −−− ==+ ∫∫∫ … (1)
Here ( ) − ∫
0
0
t
st e f t dt exists since f(t) is piecewise continuous on every finite interval 0 ≥ t ≥ t0.
Now
00 0
() () , st st st at
tt t
e f t dt e f t dt e Me dt
∞∞ ∞
− −− ∫∫ ∫ ≤ ≤ since ( ) at f t Me ≤
( ) ( )
( )
0
0
;
s at
s at
t
e e Mdt M s a
s a
∞ − − − − == > − ∫ … (2)
But ( )
( )
0 s at e M
s a
− −
− can be made as small as we please by making t0 sufficiently large. Thus,
from (1), we conclude that L{f(t)} exists for all s > a.
*Note: A function is said to be piecewise (sectionally) continuous on a closed interval [a, b], if this closed interval can
be divided into a finite number of subintervals in each of which f(t) is continuous and has finite left hand and right hand
limits.
A function is said to be of exponential order ‘a’ (a > 0) as t → ∞, if there exist finite positive constants t0 and M such
that ( ) at f t Me ≤ or ( ) at e ft M − ≤ for all t ≥ t0.
12.4 TRANSFORMS OF ELEMENTARY FUNCTIONS
By direct use of definition, we find the laplace transform of some of the simple functions:
1. 1 L(1)
s = , (s > 0)
2. 1 ( ) at L e
s a = − , (s > a)
3. 1
! ( ) , where 0,1, 2, 3, n
n
n Lt n
s + = =… otherwise ( )
1
1
n
n
s +
Γ +
4. 2 2 (sin ) , a L at
s a = +
(s > 0)
5. 2 2 (cos ) , s L at
s a = +
(s > 0)
6. = −2 2 (sinh ) , a L at
s a ( ) s a >
7. 2 2 (cosh ) , s L at
s a = − ( ) [Show Less]