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

Eq. (2.135). For instance, the independent term of P_n {x} must result from the product of only (−r₁) × (−r₂) × … × (−r_n) of factors x − r₁, x − r₂, …, x − r_n, respectively, in Eq. (2.182), so one may state

      (2.184)

      similarly, one finds that the highest order term in Eq. (2.135) will necessarily result from the product of only x × x × ⋯ × x of factors x − r₁, x − r₂, …, x − r_n, respectively, in Eq. (2.182) – thus leading to the trivial result

      (2.185)

      By the same token, the terms in x_n−1 of Eq. (2.135) are accounted for by the product of x in x − x₁, x − x₂, …, x − x_i, x − x_i+1, …, x − x_n, respectively, of Eq. (2.182) by –r_i in x − r_i, thus generating ; this is then to be extended, via addition, to i = 1, 2, …, n, thus eventually giving rise to

      (2.186)

as apparent in Eq. (2.183). The coefficients of all remaining terms can be generated in a similar fashion; for power x^n−j in general, the x’s of n − j factors of the x − r_i form are to be picked up, and multiplied by the remaining j roots of the r_i form – and the results added up, thus unfolding a composite set of j summations that account for all such combinations. Special care is to be exercised to increment the lower limit of each summation by one unit relative to the lower limit of the previous summation, and decrement the upper limit of each summation by one unit relative to the upper limit of the next summation; in this way, each possible combination is counted just once (as it should).

      2.2.4 Splitting

      Once in possession of the equivalent result conveyed by Eq. (2.182) but applied to P_m {x}, one may revisit Eq. (2.141) as

(2.187)

      – or, after lumping constant b_m with the corresponding polynomial in numerator,

(2.188)

The roots r_k of the polynomial in denominator may, in general, take real or complex values (i.e. of the form α + ιβ); when s_l roots are equal to r_k, one may lump the corresponding binomials as instead of multiplying x − r_l by itself s_l times, i.e. Eq. (2.188) may alternatively appear as

(2.189)

as long as the r_l ’s represent the s distinct roots (or poles) of P_m {x}, each with multiplicity s_l, and . After splitting the denominator of the proper rational fraction in Eq. (2.189), one obtains

(2.190)

      – where s₁ was arbitrarily chosen among the (multiple) roots, and denotes an (m − s₁)th degree polynomial defined as

      (2.191)

      and not holding r₁ as root, while U_<m {x} is defined as

      (2.192)

and does not also have r₁ as root. If a partial fraction Α_1,1/ is deliberately separated from the right‐hand side of Eq. (2.190), then one may write

(2.193)

      with Α_1,1 denoting a (putative) constant; since the left‐hand side and the second term in the right‐hand side share their functional form, they may be pooled together as

      (2.194)

– which is equivalent to

(2.195)

      in view of the common denominators of left‐ and right‐hand sides. By hypothesis, neither U_<m {x} nor have r₁ as root – otherwise would not explicitly appear in Eq. (2.190); in fact, U_<m {x} having r₁ as root would permit factoring out of x – r₁ in numerator, so power s₁ of in denominator would be reduced – whereas having r₁ as root would allow factoring out of x – r₁ in denominator, so power s₁ of