Chapter 63

Inverse Laplace transforms

63.1 Definition of the inverse Laplace

transform

If the Laplace transform of a function f(t)isF(s),
i.e. L{f(t)}=F(s),thenf(t) is called theinverse
Laplace transform of F(s) and is written as
f(t)=L−^1 {F(s)}.

For example, sinceL{ 1 }=

1

s

thenL−^1

{

1

s

}

= 1.

Similarly, sinceL{sinat}=

a

s^2 +a^2

then

L−^1

{

a

s^2 +a^2

}

=sinat,and so on.

63.2 Inverse Laplace transforms of

simple functions

Tables of Laplace transforms, such as the tables in
Chapters 61 and 62 (see pages 584 and 587) may be
used to find inverse Laplace transforms.
However, for convenience, a summary of inverse
Laplace transforms is shown in Table 63.1.

Problem 1. Find the following inverse Laplace

transforms:

(a)L−^1

{

1

s^2 + 9

}

(b)L−^1

{

5

3 s− 1

}

(a) From (iv) of Table 63.1,

L−^1

{

a

s^2 +a^2

}

=sinat,

Table 63.1Inverse Laplace transforms

F(s)=L{f(t)} L−^1 {F(s)}=f(t)

(i)

1

s

1

(ii)

k

s

k

(iii)

1

s−a

eat

(iv)

a

s^2 +a^2

sinat

(v)

s

s^2 +a^2

cosat

(vi)

1

s^2

t

(vii)

2!

s^3

t^2

(viii)

n!

sn+^1

tn

(ix)

a

s^2 −a^2

sinhat

(x)

s

s^2 −a^2

coshat

(xi)

n!

(s−a)n+^1

eattn

(xii)

ω

(s−a)^2 +ω^2

eatsinωt

(xiii)

s−a

(s−a)^2 +ω^2

eatcosωt

(xiv)

ω

(s−a)^2 −ω^2

eatsinhωt

(xv)

s−a

(s−a)^2 −ω^2

eatcoshωt