to the same denominator the partial fraction decomposition, of the given rational
function. Second, since the denominators on both sides are the same, the numerators
must also be the same. Then, we equalize the similar coeﬃcients (corresponding to
the same power of
x
) of the two polynomials of the numerators on either side of the
equality
i
. Therefore, the scalars
A
(k)
i
can be found by solving a system of linear equations.
The second approach is based on the application of the Heaviside’s "cover-up method",
which necessitate substitutions to establish the scalars
A
(k)
i
, of the partial fraction
decomposition, in the case with single poles
γ
j
(1
≤ j ≤ deg
(
Q
)). For multiple poles
case
γ
j
(1
≤ j ≤ s
), with
m
j
≥
2 for some
j
, successive diﬀerentiation are applied, for
calculating the scalars
A
(k)
i
. Despite that, this topic continue to attract much attention,
and there has been recent developments in the computation aspect of the scalars
A
(k)
i
,
for general rational functions (see for example, [3, 4, 6, 15]) as well as for some special
cases (see, for example, [
5
,
6
,
12
–
14
]). Meanwhile, the approaches and methods for
decomposing a rational function into partial fractions are computationally intensive,
especially when the multiplicities of roots of the denominator are higher.
In this paper we establish another approach for providing the explicit formulas for the
scalars
A
(k)
i
of the partial fraction decomposition of the rational functions
F
(
x
) =
R(x)
Q(x)
,
where
R
(
x
),
Q
(
x
) are polynomials in
R
[
X
] or
C
[
X
], such that (without loss of generality)
the degree of
R
is less than the degree of
Q
and are mutually prime. The essence of our
approach requires a computational process, based on two known results of the literature.
More precisely, suppose that
R
and
Q
are mutually prime and
Q
(
x
) =
Q
s
j=1
(
x − γ
j
)
m
j
,
where each root
γ
j
is of multiplicity
m
j
≥
1. We develop a computational process,
which allows us to present a new method, for exhibiting compact explicit formulas of
the partial fraction decomposition,
F (x) =
R(x)
Q(x)
=
R(x)
Q
s
i=1
(x − γ
i
)
m
i
=
s
X
i=1
m
i
X
k=1
A
(k)
i
(x − γ
i
)
k
.
Our main goal is to give a new compact explicit formulas for the scalars
A
(k)
i
(1
≤ i ≤ s
,
1
≤ k ≤ m
j
). As a consequence, some applications and several illustrative examples are
presented, in order to show the eﬃciency of our approach.
This study is organized as follows. For reason of clarity and conciseness, Section 2
is devoted to the two fundamental results, representing the basic tools of our method.
Section 3 is concerned with the generic case, where a compact explicit formula of the
i
It is known that two polynomials are equal, if and only if, the coefﬁcients at the corresponding powers of x are equal.
L. B. Lima; R. Mustapha INTERMATHS, 4(1), 48–66, June 2023 | 49