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

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or else

      (2.271)

      following straightforward algebraic manipulation and condensation afterward. If r is instead set equal to −1 and y set equal to 1, then Eq. (2.265) gives rise to

      (2.272)

      – where algebraic rearrangement supports dramatic simplification to

      (2.273)

      the right‐hand side is but a geometric series of first term equal to 1 and ratio between consecutive terms equal to −x, so one may retrieve Eq. (2.93) to write

      (2.274)

      since when ∣x ∣ < 1 – also consistent with (x + 1)⁻¹ representing the reciprocal of x + 1 in the first place, as obtained by long division of 1 by 1 + x following the algorithm depicted in Fig. 2.8. One also finds that

(2.275)

after replacement of x in Eq. (2.265) by its negative, y by 1, and exponent r by a general negative number −z; Eq. (2.275) eventually yields

(2.276)

      If a rising factorial, (z)_k, is defined as

      (2.277)

with evolution opposite to that entailed by Eq. (2.267), one may condense Eq. (2.276) to

      (2.278)

      In the case of a trinomial, its square may be calculated via

      (2.279)

      in agreement with Eq. (2.237) applied to x₁ + x₂ and x₃, rather than x and y; a second application of said formula to (x₁ + x₂)² generates

      (2.280)

      that may be rearranged to read

(2.281)

      upon elimination of parenthesis. The above reasoning may be applied to any (integer) exponent n, and to any number m of terms of polynomial x₁ + x₂ + ⋯ + x_m; the generalized formula looks indeed like

(2.282)

      where the summation in the right‐hand side is taken over all sequences of (nonnegative) integer indices k₁ through k_m, such that the sum of all k_i ’s is n. The multinomial coefficients are given by

(2.283)

      – and count the number of different ways an n‐element set can be partitioned into disjoint subsets of sizes k₁, k₂, …, k_m . For the example above, one would have been led to

(2.284)

after setting m = 3 and n = 2 in Eq. (2.282), while Eq. (2.283) would yield

(2.285)

the possibilities of integer values for (k₁,k₂,k₃) satisfying the condition placed at the bottom of the summation in Eq. (2.284) encompass (2,0,0), (0,2,0), (0,0,2), (1,1,0), (1,0,1), and (0,1,1) – so Eq. (2.284) becomes

(2.286)

with the aid of Eq. (2.285). After realizing that 0! = 1, 2!/2! = 1, and 2!/1! = 2 – besides – one eventually retrieves Eq. (2.281), using Eq. (2.286) as departure point.

2.3 Trigonometric Functions

      Trigonometry is the branch of mathematics that studies relationships involving lengths of sides and amplitudes of angles in triangles. This field emerged in the Hellenistic world during the third century, by the hand of Euclid and Archimedes – who studied the properties of chords and inscribed angles in circles, while proving theorems equivalent to most modern trigonometric formulae; Hipparchus

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