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Properties of Laplace Transform

immediately recognizable as the Laplace transform of some elementary

function, such as
1
F(s) = ,
(s − 2)(s − 3)
for s confined to some region Ω (e.g., Re(s) > α). Just as in calculus (for s real),

where the goal is to integrate such a function, the procedure required here is to
decompose the function into partial fractions.
There are several different ways for computing constants in partial fractions
decomposition:
1. Method of undetermined coefficients;

2. Plug in particular values method;
3. Cover up method.
The last method is a quick way to determine the coefficients in a partial

fraction decomposition. The cover up method was popularized by the English

electrical engineer Oliver Heaviside (1850 - 1925).
Recall that to use partial fractions the degree of the polynomial in the
numerator must be strictly less than the degree of the polynomial in the
denominator. If it is not then you must first use long division to obtain a quotient

polynomial and a rational function remainder; the remainder will certainly have
a denominator of higher degree than the numerator and so partial fractions can
be used on it.
We will be concerned with the quotient of two polynomials, namely a rational

function
P(s)
F(s) = ,
Q(s)
where the degree of Q(s) is greater than the degree of P(s), and P(s) and Q(s)
have no common factors. Then F(s) can be expressed as a finite sum of partial
fractions.

(i) For each linear factor of the form as + b of Q(s), there corresponds a partial
fraction of the form
A
, A constant.
as + b
n
(ii) For each repeated linear factor of the form (as + b) , there corresponds a
partial fraction of the form

A 1 A 2 A n
+ + · · · + ,
as + b (as + b) 2 (as + b) n

A , A , . . . , A constants.
2
1
n
2
(iii) For every quadratic factor of the form as +bs+c, there corresponds a partial
fraction of the form
As + b
, A, B constants.
2
as + bs + c
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