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

(2.197)

In view of Eq. (2.159), r 1being a root of Y supports

(2.198)

where ς { x } denotes an ( m − 1)th degree polynomial of x ; insertion in Eq. (2.193)unfolds

(2.199)

Equation (2.199)degenerates to

(2.200)

after dropping x − r 1from both numerator and denominator of the first term in the right‐hand side; ς / is again a regular rational fraction, because the degree of ς { x } is lower than m − 1 (as indicated by the subscript utilized) – while the degree of the corresponding polynomial in denominator equals s 1– 1 + ( m − s 1) = m − 1, on account of the degree s 1− 1 of and the degree m − s 1of . Therefore, one may proceed to another splitting step of the type

(2.201)

with Α 1,2denoting a second constant to be replaced by

(2.202)

in parallel to Eq. (2.196)obtained from Eq. (2.193); insertion of Eq. (2.201)transforms Eq. (2.200)to

(2.203)

as long as , following Eqs. (2.199)and (2.200)as template. This method may undergo up to s 1iterations, to eventually produce

(2.204)

The same rationale may then be applied to the second root r 2, of multiplicity s 2, and so on, until one gets

(2.205)

therefore, any proper rational fraction with poles r 1, r 2, …, r s(or r , for short) of multiplicity s 1, s 2, …, 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 ) ssequentially 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, ι 2 = −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.