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Properties of Laplace Transform

Theorem 2.13È

p(s)
(Partial Fraction Expansion) Let be a ratio of polynomials with real
q(s)
coefficients. If the denominator q(s) factors as

2
n 1
q(s) = k(s − a ) (s + b s + c ),
1
1
1
where the quadratic factor is irreducible then the following identity holds
p(s) A 1 A 2 A B s + C 1
1
≡ + + · · · + n 1 + + R(s),
2
q(s) s − a 1 (s − a ) 2 (s − a ) n 1 s + b s + c 1
1
1
1
where R(s) is a polynomial that is equal to zero if the degree of the
polynomial p(s) is less than the degree of the polynomial q(s) and
A , A , . . . , A , B , C are real constants.
2
1
1
1
n 1
. . . . . . .
.
Example 2.31, The partial fraction expansion of

2
3s + 4s + 5
F(s) =
2
2
(s + 1)(s − 2) (s + 2s + 2)
takes the form

2
3s + 4s + 5 A B C Ds + E
= + + + ,
2
2
2
(s + 1)(s − 2) (s + 2s + 2) s + 1 s − 2 (s − 2) 2 s + 2s + 2
where A, B, C, D and E are constants to be determined.
. . . . .
.

Example 2.32, Find the inverse Laplace transform of F(s) =
2
s − s + 9
.
3
s + 9s
. . . . .
.
2
s − s + 9
Solution. We do not immediately recognize the expression as the
3
s + 9s
Laplace transform of a function that we know. So we use partial fractions
2
s − s + 9
to write as a sum of expressions in s that we do recognize. We
3
s + 9s
first factorize the denominator

2
2
s − s + 9 s − s + 9
= ,
3
2
. . . . s + 9s s(s + 9)
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