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

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multiplicity s₁, s₂, …, s_s, respectively (or s for short), may be expanded as a sum of partial fractions bearing a constant in numerator, as well as x − r, (x − r)², …, (x − r)^s sequentially in denominator – irrespective of the mathematical nature of such roots.

      To avoid emergence of complex numbers – and taking advantage of the fact that if a polynomial with real coefficients has complex roots then they always appear as conjugate pairs (otherwise its coefficients would necessarily be complex numbers), one may lump pairs of complex partial fractions as

      (2.206)

      upon elimination of parentheses in numerator, and rearrangement of inner parentheses in denominator, one gets

(2.207)

After condensation of terms alike in numerator, and recalling Eq. (2.140), i.e. the product of two conjugate binomials equals the difference of their squares, Eq. (2.207) becomes

(2.208)

since, by definition, ι² = −1, one may simplify Eq. (2.208) to

(2.209)

Once the square of the binomial in denominator is expanded as per Newton’s rule, Eq. (2.209) becomes

      (2.210)

      which may be rewritten as

      (2.211)

      the new constants are defined as

      (2.212)

      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

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