Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

      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

(2.216)

After splitting the outer summation, Eq. (2.216) becomes

(2.217)

one may further write

(2.218)

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

(2.220)

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

(2.222)

due to r₁ − r₁ being nil, as well as any (significant) power thereof. Isolation of Α_1,1 in Eq. (2.222) finally unfolds

(2.223)

A similar reasoning may be followed with regard to any other root r_k – provided that x − r_k is singled out in Eq. (2.216) instead of x − r₁ as done in Eq. (2.217), and eventually setting x = r_k; one accordingly finds

      (2.224)

      In attempts to determine Α_1,2 (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.

(2.225)

– 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

(2.226)

The rule of differentiation of a power may now be invoked to transform Eq. (2.226) to

(2.227)

after setting x = r₁ again, Eq. (2.227) becomes

      (2.228)

      which dramatically simplifies to

(2.229)

due to the nil value of r₁ − r₁ and any significant powers thereof. After splitting as the ratio of to r₁ − r_r, 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; A_1,1 may now be eliminated via Eq. (2.223), viz.

      (2.232)