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

(2.185)

By the same token, the terms in x n−1of Eq. (2.135)are accounted for by the product of x in x − x 1, x − x 2, …, x − x i, x − x i+1, …, x − x n, respectively, of Eq. (2.182)by – r iin 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−jin general, the x ’s of n − j factors of the x − r iform are to be picked up, and multiplied by the remaining j roots of the r iform – 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).

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 mwith the corresponding polynomial in numerator,

(2.188)

The roots r kof the polynomial in denominator may, in general, take real or complex values (i.e. of the form α + ιβ ); when s lroots are equal to r k, one may lump the corresponding binomials as instead of multiplying x − r lby itself s ltimes, 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 s 1was arbitrarily chosen among the (multiple) roots, and denotes an ( m − s 1)th degree polynomial defined as

(2.191)

and not holding r 1as root, while U { x } is defined as

(2.192)

and does not also have r 1as 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 1,1denoting 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 { x } nor have r 1as root – otherwise would not explicitly appear in Eq. (2.190); in fact, U { x } having r 1as root would permit factoring out of x – r 1in numerator, so power s 1of in denominator would be reduced – whereas having r 1as root would allow factoring out of x – r 1in denominator, so power s 1of in denominator would be increased, and in either case the multiplicity of r 1would not equal s 1(as postulated). Hence, one may arbitrarily define (the still unknown) constant Α 1,1as

(2.196)

note that both and are themselves constants, i.e. polynomials of x taken at a specific value of x , viz. r 1. If Eq. (2.196)is valid, then r 1becomes a root of Y as per Eq. (2.195), i.e.