Laplace Transform

CHAPTER

6

6.1 Definition and Elementary Properties

The Laplace transform serves as a device for simplifying or mechanizing the
solution of ordinary and partial differential equations. It associates a function
f(t)with a function of another variableF(s)from which the original function
can be recovered.
Letf(t)be sectionally continuous in every interval 0≤t<T.TheLaplace
transform off, writtenL(f)orF(s),isdefinedbytheintegral

L(f)=F(s)=

∫∞

0

e−stf(t)dt. (1)

We use the convention that a function oftis represented by a lowercase letter
and its transform by the corresponding capital letter. The variablesmay be
real or complex, but in the computation of transforms by the definition,sis
usually assumed to be real. Two simple examples are

L( 1 )=

∫∞

0

e−st· 1 dt=

1

s,

L

(

eat

)

=

∫∞

0

e−steatdt=−e

−(s−a)t

s−a

∣∣

∣∣

∞

0

=^1

s−a

.

Not every sectionally continuous function ofthas a Laplace transform, for
the defining integral may fail to converge. For instance, exp(t^2 )has no trans-
form. However, there is a simple sufficient condition, as expressed in the fol-
lowing theorem.

363