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

(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)

Equation (2.383) may again be invoked to transform Eq. (2.411) to

      (2.412)

      while the denominator was rewritten as a product of composite powers – where ι² = −1 can be used to generate

      (2.413)

cancelation of (−1)ⁿ between numerator and denominator, and factoring out of 2 in the former unfold

(2.414)

      since (−1)ⁱ⁻ⁿ coincides with (−1)ⁿ⁻ⁱ and x is more generally used as argument than angle θ (as long as rad is employed as units).

      In the case of an odd exponent, Eq. (2.407) may be rephrased as

(2.415)

based on change of n to 2n + 1, and likewise of i to 2i + 1, complemented by splitting of the summation in two halves and rearrangement of exponents wherever appropriate; recalling Eq. (2.399), one obtains

(2.416)

as alternative form for the second summation in Eq. (2.415), where condensation of terms alike meanwhile took place. In view of Eq. (2.391) pertaining to the symmetry of binomial coefficients, one may proceed to

      (2.417)

as new version of Eq. (2.416) – which may be inserted in Eq. (2.415) to generate

(2.418)

where (−1)²ⁿ = (−1²)ⁿ = 1ⁿ = 1 was taken into account after factoring out, (−1)^−2j was rewritten as (−1)(−1)^−2j−1 = −(−1)^{−(2j + 1)}, and was in turn factored out; once 2j + 1 is replaced by i as (dummy) variable in the outstanding summation and ι² is replaced by −1, Eq. (2.418) simplifies to

(2.419)

along with rewriting of exponent 2n + 1 − 2i as 2(n − i) + 1 and 2²ⁿ⁺¹ as twice 2²ⁿ, as well as lumping of powers of −1 between numerator and denominator. After retrieving Eq. (2.384) and replacing n by 2(n − i) + 1, one may reformulate Eq. (2.419) to

(2.420)

      in view of (−1)ⁱ⁻ⁿ = 1/(−1)ⁱ⁻ⁿ = (−1)ⁿ⁻ⁱ and 2²ⁿ = (2²)ⁿ, which may instead look like

(2.421)

as more usual form – using x (expressed in rad) as independent variable rather than θ; Eqs. (2.414) and (2.421) accordingly permit calculation of an (integer) power of sine of any argument as a linear combination of cosines or sines of multiples of said argument.

      The converse problem of expressing sines and cosines of nθ in terms of powers of sin θ and cos θ may also be solved via de Moivre’s theorem; one should accordingly retrieve Eq. (2.369), and expand its left‐hand side via Newton’s binomial formula as

(2.422)

– where advantage was meanwhile taken of ι² = −1, ι³ = −ι, ι⁴ = 1, and Eq. (2.236). Following inspection of the forms of the terms in the right‐hand side of Eq. (2.422), one realizes that ι may be factored out to get

      (2.423)

a more condensed notation is, however, possible according to

(2.424)