Читать книгу Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata - Страница 26

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 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)

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.

The question still remains as how to calculate the Α’s in Eq. (2.205); to do so, one should start by multiplying both sides by (or , for that matter), thus generating

(2.216)

After splitting the outer summation, Eq. (2.216) becomes

(2.217)

one may further write

(2.218)

if the extended products are, in turn, splitted as and . Equation (2.218) may be rewritten as

(2.219)

after lumping powers in the argument of the first summation, or else

(2.220)

following explicitation of the independent term in the first summation; since Eq. (2.220) is universally valid, it should hold in particular when x = r₁ – in which case one obtains

(2.221)

that breaks down to

(2.222)

due to r₁ − r₁ being nil, as well as any (significant) power thereof. Isolation of Α_1,1 in Eq. (2.222) finally unfolds

(2.223)

A similar reasoning may be followed with regard to any other root r_k – provided that x − r_k is singled out in Eq. (2.216) instead of x − r₁ as done in Eq. (2.217), and eventually setting x = r_k; one accordingly finds

(2.224)

In attempts to determine Α_1,2 (should s₁ ≥ 2), one may to advantage differentiate both sides of Eq. (2.220) with regard to x so as to obtain an independent relationship, viz.

(2.225)

– by resorting to the rule of differentiation of a product; the said rule, coupled with the rule of differentiation of a sum, allows subsequent transformation of Eq. (2.225) to

(2.226)

The rule of differentiation of a power may now be invoked to transform Eq. (2.226) to

(2.227)

after setting x = r₁ again, Eq. (2.227) becomes

(2.228)

which dramatically simplifies to

(2.229)

due to the nil value of r₁ − r₁ and any significant powers thereof. After splitting as the ratio of to r₁ − r_r, and lumping the former to the extended product, Eq. (2.229) will take the form

(2.230)

– where may, in turn, be taken off the summation to yield

(2.231)

for being independent of r as counting variable; A_1,1 may now be eliminated via Eq. (2.223), viz.

(2.232)

where cancelation of identical extended products between numerator and denominator gives rise to

(2.233)

Isolation of Α_1,2 from Eq. (2.233) is finally in order, according to

(2.234)

generalization then ensues as

(2.235)

Although seldom of relevance, higher order constants, say A_l,3, A_l,4, …, may be calculated in a similar fashion – by resorting to higher order derivatives of Eq. (2.220), but accordingly requiring more cumbersome algebraic manipulations.