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

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having the original n replaced by 2(n − i), as well as 2 taken off the outstanding summation; θ was also relabeled as x, provided that rad is used.

      If the exponent of cosine in Eq. (2.388) is an odd integer, say, 2n+1, then one gets

(2.398)

upon replacement of n by 2n + 1, and accordingly of i by 2i + 1; the summation was meanwhile rewritten as two consecutive summations, while z in the first summation gave the floor to its reciprocal at the expense of taking the negative of the exponent. A new change of variable, viz.

(2.399)

– inspired on Eq. (2.392), proves useful to transform the second summation in Eq. (2.398) to

(2.400)

where condensation of terms alike meanwhile took place; Eq. (2.391) may then be used to transform Eq. (2.400) to

(2.401)

– while insertion of Eq. (2.401) in Eq. (2.398) further leads to

(2.402)

with summations pooled together on the basis of their similarity, and factored out afterward. Since 2j + 1 is a dummy (counting) variable, it may be swapped for just i, thus allowing reformulation of Eq. (2.402) to

(2.403)

      where 2 was meanwhile factored out in the exponents; in view of Eq. (2.383) again – after exchanging exponent n by 2(n − i) + 1, one finds

(2.404)

as alternative version of Eq. (2.403) – also with the aid of 2²ⁿ being identical to the nth power of 2². Since 2 can cancel out between numerator and denominator, Eq. (2.404) becomes simply

(2.405)

after renaming angle θ to x (expressed in rad); all in all, an integer power of the cosine of an angle x may be expressed as a (finite) sum of cosines of integer multiples of x – see Eqs. (2.397) and (2.405).

With regard to the power of any sine, Eq. (2.386) may be similarly revisited with the aid of Newton’s binomial, labeled as Eq. (2.236), to get

      (2.406)

      lumping of the powers of z and 1/z supports transformation to

(2.407)

Should the exponent of sine be an even integer, Eq. (2.407) transforms to

(2.408)

      once n and i are replaced by 2n and 2i, respectively – where terms were deliberately grouped according to 2i < n, 2i = n and 2i > n, and variable z swapped for 1/z at the expense of a minus sign in the original exponent; variable 2j as per Eq. (2.392) may now be retrieved to rewrite the second summation as

(2.409)

after taking Eq. (2.391) into account and performing elementary algebraic rearrangement. Insertion of Eq. (2.409), together with realization that (−1)^2n−2j = (−1)²ⁿ (−1)^−2j = (−1)^−2j = (−1)^2j (since 2n and 2j are even integers) and z⁰ = 1, allow transformation of Eq. (2.408) to

(2.410)

with convenient factoring out of , after having lumped the two summations; upon replacement of 2j by i for being a dummy counting variable, factoring out 2 in the exponents afterward, and splitting the power in denominator, Eq. (2.410) becomes

(2.411)

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