Page 44 - 4811

Partial Fractions

Theorem 2.14È

(Heaviside Expansion Theorem) If F(s) is the quotient P(s)/Q(s) of two
polynomials in s such that Q(s) has the higher degree and contains simple

poles the factor s − s , which are not repeated, then the term in f(t)
k
P(s k ) s k t
corresponding to this factor can be written e
Q (1) (s k )
. . . . . . .
.

Example 2.35, Find the inverse Laplace transform of the
following function using the Heaviside expansion theorem:

( )
s − 3
−1
L .
2
s + 5s + 6
. . . . .
.
Solution. We write in form

P(s) s − 3 s − 3
F(s) = = = .
2
Q(s) s + 5s + 6 (s + 2)(s + 3)
The derivative of the denominator is

(1)
Q (s) = 2s + 5

from which, for the roots of this equation,

(1)
(1)
Q (−2) = 1, Q (−3) = −1.
Hence, P(−2) = −5, P(−3) = −6. The final value for f(t) is f(t) =

. . . . −5e −2t + 6e −3t .
.

Example 2.36, Find the inverse Laplace transform of the
following function using the Heaviside expansion theorem:

( )
2s + 3
−1
L .
2
s + 4s + 7
. . . . .
.
Solution. The roots of the denominator are

√ √
2
s + 4s + 7 = (s + 2 + i 3)(s + 2 − i s).

That is, the roots of the denominator are complex. The derivative of the
denominator is
(1)
. . . . Q (s) = 2s + 4.

43