Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

(iii) Once again using the definition (13.53) we have

f ̄n(s)=

∫∞

0

tne−stdt.

Integrating by parts we find

f ̄n(s)=

[

−tne−st

s

]∞

0

+

n

s

∫∞

0

tn−^1 e−stdt

=0+

n

s

f ̄n− 1 (s), ifs> 0.

We now have a recursion relation between successive transforms and by calculating

f ̄ 0 we can inferf ̄ 1 ,f ̄ 2 ,etc.Sincet^0 = 1, (i) above gives

f ̄ 0 =^1

s

, ifs> 0 , (13.55)

and

f ̄ 1 (s)=^1

s^2

, f ̄ 2 (s)=

2!

s^3

,...,f ̄n(s)=

n!

sn+1

ifs> 0.

Thus, in each case (i)–(iii), direct application of the definition of the Laplace transform
(13.53) yields the required result.

Unlike that for the Fourier transform, the inversion of the Laplace transform

is not an easy operation to perform, since an explicit formula forf(t), givenf ̄(s),

is not straightforwardly obtained from (13.53). The general method for obtaining

an inverse Laplace transform makes use of complex variable theory and is not

discussed until chapter 25. However, progress can be made without having to find

anexplicitinverse, since we can prepare from (13.53) a ‘dictionary’ of the Laplace

transforms of common functions and, when faced with an inversion to carry out,

hope to find the given transform (together with its parent function) in the listing.

Such a list is given in table 13.1.

When finding inverse Laplace transforms using table 13.1, it is useful to note

that for all practical purposes the inverse Laplace transform is unique§and linear

so that

L−^1

[

af ̄ 1 (s)+bf ̄ 2 (s)

]

=af 1 (t)+bf 2 (t). (13.56)

In many practical problems the method of partial fractions can be useful in

producing an expression from which the inverse Laplace transform can be found.

Using table 13.1 findf(t)if

f ̄(s)= s+3

s(s+1)

.

Using partial fractionsf ̄(s) may be written

f ̄(s)=^3

s

−

2

s+1

.

§This is not strictly true, since two functions can differ from one another at a finite number of

isolated points but have thesameLaplace transform.