Unit Vl: Transforms
PTER
21)
Laplace Transforms
functions. 4. Properties of
I 1. Introduction. 2. Definition; Conditions for existence. 3. Transforms of elementary
Laplace transforms. 5. Transforms of Periodic functions. 6. Transforms of Special functions. 7. Transforms of
derivatives. 8. Transforms of integrals. 9. Multiplication by ". 10. Division byt. 11. Evaluation of integrals by
Laplace transforms. 12. Inverse transforms. 13. Other methods of finding inverse transforms. 14. Convolution
theorem. 15. Application to diferential equations. 16. Simultaneous linear equations with constant co-efficients,
17. Unit step function. 18. Unit impulse function. 19. Objective Type of QuestionsS.
21.1 INTRODUCTION
The knowledge of Laplace transforms has in recent years become an essential part of mathematical
background required of engineers and scientists. This is because the transform methods provide an easy and
effective means for the solution of many problems arising in engineering.
This subject originated from the operational methods applied by the English engineer Oliver Heaviside
(1850-1925), to problems in electrical engineering. Unfortunately, Heaviside's treatment was unsystematic and
lacked rigour, which was placed on sound mathematical footing by Bromwich and Carson during 1916-17. It
was found that Heaviside's operational calculus is best introduced by means of a particular type of definite
integrals called Laplace transforms.*
21.2 (1) DEFINITION
Let ft) be a function oft defined for all positive values of t. Then the
by Lif)) is defined by
Laplace transforms of ft), denoted
Lif) =
e ft) dt ...(1)
provided that the integral exists. s is a parameter which may be a real or complex number.
LifO) being clearly a function of s is briefly written as f(s) i.e.,
Lif ()) f(s), =
which can also be written as ft)
L-11f(s).
=
Then ft) is called the inverse
Laplace transform of f (s). The symbol L, which transforms into
ft)
f (s), is called the Laplace transformation operator.
(2) Conditions for the existence
The Laplace transform of f() i.e., J e f(t) dt exists for s > a, if
*Pierre de Laplace (1749-1827) (See footnote p. 18) used such
of probability. transforms, much earlier in 1799, while
developing the theory
726
, LAPLACE TRANSFORMS
727
() f) is continuous (i) Lt le a f)) is finite
t ShOuld however, be noted that the above conditions are sufficient and not necessary. (U.P.T.U., 2012)
For example, L(1//t )exists, though 1/t is infinite at t = 0,
(3) Advantages
)
Particular solution of
than
a diferential
equations is obtained directly without first finding the general solution and
finding the constants by substituting the initial conditions.
on-homogenous differential equations can be solved without the necessity of first solving the corresponding
homogeneous differential equations.
)
APplicable not to continuous functions but also to discontinuous functions, periodic functions, step runclOns,
impulse functions.
(U)
Laplace
reduces
transforms of various functions are readily available in the form of tables see Appendix 2 Table l l which
the problem of
solving differential equations to mere algebraic manupulations.
(0)
Laplace transforms are very useful for finding the solutions of both ordinary and
partial differential equations,
solutions of simultaneous differential
equations, solution of linear difference equations (Chap. 31), solution ot
integral equations and in evaluating complicated definite
integrals.
21.3 TRANSFORMS OF ELEMENTARY FUNCTIONS
The direct application of the definition gives the following formulae:
(1) L 1) =
(s>0)
n!
(2) L (t") T when 0, 1, 2, 3,
Otherwise Tn+1)1
=
n
+1
=
..
(3) L (e") =
S-a (s >a)
a
4) L (sin at) =
a (s>0)
(5) L (cos at) =
a? (s 0 )
6) L (sinh at) =
(s>a D
S
(7) L (cosh at) =
(s>a )
a?
Proofs. (1) LD= J*.1dt= -ifa>0.
0
(2) L.(P) =
J, ".t" dt =J, e") , onputingst =p
TP.pf dp=Tn+1if n>-1ands>0. [Page 302]
7/2) ;LG)= T3/2) Vt
In particular L (-1/2) s / 2 3/223/2
Inn be positiveL integer,
a T(n +1) =n! (o) p. 3021,
(t") = n !/s" +1,
therefore, (V.T.U., 2013)
es-a)t 1
(3) L(et)= e,ett dt = - dt= , if s >a.
--a)ls-a
PTER
21)
Laplace Transforms
functions. 4. Properties of
I 1. Introduction. 2. Definition; Conditions for existence. 3. Transforms of elementary
Laplace transforms. 5. Transforms of Periodic functions. 6. Transforms of Special functions. 7. Transforms of
derivatives. 8. Transforms of integrals. 9. Multiplication by ". 10. Division byt. 11. Evaluation of integrals by
Laplace transforms. 12. Inverse transforms. 13. Other methods of finding inverse transforms. 14. Convolution
theorem. 15. Application to diferential equations. 16. Simultaneous linear equations with constant co-efficients,
17. Unit step function. 18. Unit impulse function. 19. Objective Type of QuestionsS.
21.1 INTRODUCTION
The knowledge of Laplace transforms has in recent years become an essential part of mathematical
background required of engineers and scientists. This is because the transform methods provide an easy and
effective means for the solution of many problems arising in engineering.
This subject originated from the operational methods applied by the English engineer Oliver Heaviside
(1850-1925), to problems in electrical engineering. Unfortunately, Heaviside's treatment was unsystematic and
lacked rigour, which was placed on sound mathematical footing by Bromwich and Carson during 1916-17. It
was found that Heaviside's operational calculus is best introduced by means of a particular type of definite
integrals called Laplace transforms.*
21.2 (1) DEFINITION
Let ft) be a function oft defined for all positive values of t. Then the
by Lif)) is defined by
Laplace transforms of ft), denoted
Lif) =
e ft) dt ...(1)
provided that the integral exists. s is a parameter which may be a real or complex number.
LifO) being clearly a function of s is briefly written as f(s) i.e.,
Lif ()) f(s), =
which can also be written as ft)
L-11f(s).
=
Then ft) is called the inverse
Laplace transform of f (s). The symbol L, which transforms into
ft)
f (s), is called the Laplace transformation operator.
(2) Conditions for the existence
The Laplace transform of f() i.e., J e f(t) dt exists for s > a, if
*Pierre de Laplace (1749-1827) (See footnote p. 18) used such
of probability. transforms, much earlier in 1799, while
developing the theory
726
, LAPLACE TRANSFORMS
727
() f) is continuous (i) Lt le a f)) is finite
t ShOuld however, be noted that the above conditions are sufficient and not necessary. (U.P.T.U., 2012)
For example, L(1//t )exists, though 1/t is infinite at t = 0,
(3) Advantages
)
Particular solution of
than
a diferential
equations is obtained directly without first finding the general solution and
finding the constants by substituting the initial conditions.
on-homogenous differential equations can be solved without the necessity of first solving the corresponding
homogeneous differential equations.
)
APplicable not to continuous functions but also to discontinuous functions, periodic functions, step runclOns,
impulse functions.
(U)
Laplace
reduces
transforms of various functions are readily available in the form of tables see Appendix 2 Table l l which
the problem of
solving differential equations to mere algebraic manupulations.
(0)
Laplace transforms are very useful for finding the solutions of both ordinary and
partial differential equations,
solutions of simultaneous differential
equations, solution of linear difference equations (Chap. 31), solution ot
integral equations and in evaluating complicated definite
integrals.
21.3 TRANSFORMS OF ELEMENTARY FUNCTIONS
The direct application of the definition gives the following formulae:
(1) L 1) =
(s>0)
n!
(2) L (t") T when 0, 1, 2, 3,
Otherwise Tn+1)1
=
n
+1
=
..
(3) L (e") =
S-a (s >a)
a
4) L (sin at) =
a (s>0)
(5) L (cos at) =
a? (s 0 )
6) L (sinh at) =
(s>a D
S
(7) L (cosh at) =
(s>a )
a?
Proofs. (1) LD= J*.1dt= -ifa>0.
0
(2) L.(P) =
J, ".t" dt =J, e") , onputingst =p
TP.pf dp=Tn+1if n>-1ands>0. [Page 302]
7/2) ;LG)= T3/2) Vt
In particular L (-1/2) s / 2 3/223/2
Inn be positiveL integer,
a T(n +1) =n! (o) p. 3021,
(t") = n !/s" +1,
therefore, (V.T.U., 2013)
es-a)t 1
(3) L(et)= e,ett dt = - dt= , if s >a.
--a)ls-a
