Higher Engineering Mathematics

Laplace transforms

K

64

Introduction to Laplace transforms

64.1 Introduction

The solution of most electrical circuit problems
can be reduced ultimately to the solution of differ-
ential equations. The use ofLaplace transforms
provides an alternative method to those discussed
in Chapters 46 to 51 for solving linear differential
equations.

64.2 Definition of a Laplace transform

The Laplace transform of the functionf(t) is defined
by the integral

∫∞

0 e

−stf(t)dt, wheresis a parameter

assumed to be a real number.

Common notations used for the Laplace transform

There are various commonly used notations for the
Laplace transform off(t) and these include:

(i)L{f(t)}orL{f(t)}

(ii)L(f)orLf

(iii)f(s)orf(s)

Also, the letterpis sometimes used instead ofsas
the parameter. The notation adopted in this book will
bef(t) for the original function andL{f(t)}for its
Laplace transform.

Hence, from above:

L{f(t)}=

∫∞

0

e−stf(t)dt (1)

64.3 Linearity property of the

Laplace transform

From equation (1),

L{kf(t)}=

∫∞

0

e−stkf(t)dt

=k

∫∞

0

e−stf(t)dt

i.e L{kf(t)}=kL{f(t)} (2)

wherekis any constant.

Similarly,

L{af(t)+bg(t)}=

∫∞

0

e−st(af(t)+bg(t)) dt

=a

∫∞

0

e−stf(t)dt

+b

∫∞

0

e−stg(t)dt

i.e. L{af(t)+bg(t)}=aL{f(t)}+bL{g(t)}, (3)

whereaandbare any real constants.

The Laplace transform is termed alinear opera-

torbecause of the properties shown in equations (2)

and (3).

64.4 Laplace transforms of elementary

functions

Using the definition of the Laplace transform in

equation (1) a number of elementary functions may

be transformed. For example:

(a)f(t)= 1. From equation (1),

L{ 1 }=

∫∞

0

e−st(1) dt=

[

e−st

−s

]∞

0

=−

1

s

[e−s(∞)−e^0 ]=−

1

s

[0−1]

=

1

s

(provideds>0)

(b)f(t)=k. From equation (2),

L{k}=kL{ 1 }

HenceL{k}=k

(

1

s

)

=

k

s

, from (a) above.

(c)f(t)=eat(whereais a real constant=0).