Understanding Engineering Mathematics

Table 17.1

f(t) f(s) ̃ =F(s)=L[f(t)]

1

1

s

(s > 0 )

t^1

s^2

(s > 0 )

tn(na positive integer)

n!

sn+^1 (s >^0 )

eat

1

s−a

(s > a)

tneat

n!

(s−a)n+^1

(s > a)

sinωt ω

s^2 +ω^2

(s > 0 )

cosωt s

s^2 +ω^2

(s > 0 )

tsinωt^2 ωs

(s^2 +ω^2 )^2

(s > 0 )

tcosωt s

(^2) −ω 2
(s^2 +ω^2 )^2
(s > 0 )
eatsinωt
ω
(s−a)^2 +ω^2
(s > a)
eatcosωt
s−a
(s−a)^2 +ω^2
(s > a)
discontinuity must be finite. We can integrate such a function by integrating the continuous
sections separately.
A well known example of a piecewise continuous function is theunit (or Heaviside)
step function, shown in Figure 17.1, defined by
H(t)= 0 t< 0
= 1 t> 0
0 t
1
H(t)
Figure 17.1The step function.
This function is discontinuous att=0 and yet has a continuous Laplace transform, namely
1 /s. Integration effectively smoothes things out.