Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier MalcataЧитать онлайн книгу.
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 (or
, for that matter), thus generating
After splitting the outer summation, Eq. (2.216) becomes
one may further write
if the extended products are, in turn, splitted as and
. Equation (2.218) may be rewritten 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 as the ratio of
to r1 − rr, and lumping the former to the extended product, Eq. (2.229) will take the form
(2.230)
– where may, in turn, be taken off the summation to yield
(2.231)
for being independent of r as counting variable; A1,1 may now be eliminated via Eq. (2.223), viz.
(2.232)