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

after replacement of n by 2 n as upper limit, and concomitant replacement of i by 2 i as counting variable of the summation – with subsequent splitting of the said summation, so as to make the median term appear explicitly. At this stage, it is convenient to revisit Eq. (2.240)and realize that

(2.390)

following straightforward algebraic manipulation; in other words,

(2.391)

– i.e. the row entries of Pascal’s triangle are symmetrical relative to its median (see Table 2.1). On the other hand, one may introduce a new counting variable satisfying

(2.392)

so that the second summation in Eq. (2.389)can be algebraically converted to

(2.393)

In view of Eq. (2.391), one may reformulate Eq. (2.393)to

(2.394)

– which, in turn, supports conversion of Eq. (2.389)to

(2.395)

upon lumping of summations for sharing the same lower and upper boundaries, coupled with factoring out of in their kernel; one may now replace dummy variable 2 j by merely i , and factor out 2 in the exponents of z and 1/ z to get

(2.396)

along with definition of a composite power. Equation (2.383)may finally be invoked to rewrite Eq. (2.396)as

(2.397)

after 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, 2 n+ 1, then one gets

(2.398)

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