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

in denominator would be increased, and in either case the multiplicity of r₁ would not equal s₁ (as postulated). Hence, one may arbitrarily define (the still unknown) constant Α_1,1 as

(2.196)

note that both and are themselves constants, i.e. polynomials of x taken at a specific value of x, viz. r₁. If Eq. (2.196) is valid, then r₁ becomes a root of Y_<m as per Eq. (2.195), i.e.

      (2.197)

      In view of Eq. (2.159), r₁ being a root of Y_<m supports

      (2.198)

where ς_<m‐1{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 − r₁ from both numerator and denominator of the first term in the right‐hand side; ς_<m−1/ is again a regular rational fraction, because the degree of ς_<m−1{x} is lower than m − 1 (as indicated by the subscript utilized) – while the degree of the corresponding polynomial in denominator equals s₁ – 1 + (m − s₁) = m − 1, on account of the degree s₁ − 1 of and the degree m − s₁ of . Therefore, one may proceed to another splitting step of the type

(2.201)

      with Α_1,2 denoting 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 s₁ iterations, to eventually produce

      (2.204)

      The same rationale may then be applied to the second root r₂, of multiplicity s₂, and so on, until one gets

(2.205)

      therefore, any proper rational fraction with poles r₁, r₂, …, r_s (or r, for short) of 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)