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

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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 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

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