and
(2.213)
pertaining to the numerator – complemented by
(2.214)
and
(2.215)
appearing in denominator. Therefore, any pair of partial fractions involving conjugate complex numbers in denominator may to advantage be replaced by a new type of (composite) partial fraction – constituted by a first‐order polynomial in numerator and a second‐order polynomial in denominator.
The question still remains as how to calculate the Α’s in Eq. (2.205); to do so, one should start by multiplying both sides by
After splitting the outer summation, Eq. (2.216) becomes
one may further write
if the extended products are, in turn, splitted as
(2.219)
after lumping powers in the argument of the first summation, or else
following explicitation of the independent term in the first summation; since Eq. (2.220) is universally valid, it should hold in particular when x = r1 – in which case one obtains
(2.221)
that breaks down to
due to r1 − r1 being nil, as well as any (significant) power thereof. Isolation of Α1,1 in Eq. (2.222) finally unfolds
A similar reasoning may be followed with regard to any other root rk – provided that x − rk is singled out in Eq. (2.216) instead of x − r1 as done in Eq. (2.217), and eventually setting x = rk; one accordingly finds
(2.224)
In attempts to determine Α1,2 (should s1 ≥ 2), one may to advantage differentiate both sides of Eq. (2.220) with regard to x so as to obtain an independent relationship, viz.
– by resorting to the rule of differentiation of a product; the said rule, coupled with the rule of differentiation of a sum, allows subsequent transformation of Eq. (2.225) to
The rule of differentiation of a power may now be invoked to transform Eq. (2.226) to
after setting x = r1 again, Eq. (2.227) becomes
(2.228)
which dramatically simplifies to
due to the nil value of r1 − r1 and any significant powers thereof. After splitting
(2.230)
– where
(2.231)
for being independent of r as counting variable; A1,1 may now be eliminated via Eq. (2.223), viz.
(2.232)