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

which is equivalent to

(2.536)

due to Eq. (2.473). Ordered addition of Eq. (2.531)to Eq. (2.536)unfolds

(2.537)

that breaks down to

(2.538)

after condensation of similar terms; while ordered subtraction of Eq. (2.536)from Eq. (2.531)yields

(2.539)

– where negatives may be taken of both sides to get

(2.540)

Except for the change in sign, Eq. (2.538)matches Eq. (2.325), and Eq. (2.540)likewise matches Eq. (2.321).

Insertion of Eqs. (2.524)and (2.538)transforms Eq. (2.482)to

(2.541)

with division of both numerator and denominator by cosh x cosh y allowing simplification to (the more usual) form

(2.542)

again with the aid of Eq. (2.482); whereas divison of Eq. (2.526)by Eq. (2.540)supports

(2.543)

with division of both numerator and denominator by cosh x and cosh y justifying reformulation to

(2.544)

The similarity of Eqs. (2.542)and (2.544)to Eqs. (2.336)and (2.333), respectively, is again noteworthy – except for the change of sign in denominator.

A common form of Eq. (2.542)arises after setting y = x , viz.

(2.545)

– which breaks down to

(2.546)

A similar procedure, i.e. setting y = x , allows transformation of Eq. (2.538)to

(2.547)

or, equivalently,

(2.548)

elimination of sinh 2 x via Eq. (2.496)unfolds

(2.549)

which leads to

(2.550)

after isolation of cosh 2 x . Equation (2.550)is often preferable to Eq. (2.548), since it involves only hyperbolic cosines.

As will be derived later, the exponential of ξ may be exactly expressed as

(2.551)

as per Taylor’s series expanded around ξ = 0; the sine of ξ may likewise appear as

(2.552)

while cos ξ abides to

(2.553)

After setting ξ ≡ ιθ , Eq. (2.551)becomes

(2.554)

– where splitting of the summation between even and odd values of k was possible based on ι 2 = − 1 and ι 4 = 1 as algebraic rules (supporting switch between − 1 and 1 along consecutive terms), and ι and ι 3 = −ι (that switches between ι and −ι along consecutive terms). In view of

(2.555)

based on Eq. (2.552), and

(2.556)

stemming from Eq. (2.553), one may rewrite Eq. (2.554)as

(2.557)

– thus supporting Euler’s (exponential) form of a complex number a + ιb , as long as a 2 + b 2 = 1. By the same token,

(2.558)

based on Eq. (2.557)after replacing θ by −θ – where the even nature of cosine as per Eq. (2.296)and the odd nature of sine as per Eq. (2.295)allow further transformation to

(2.559)

Ordered addition of Eqs. (2.557)and (2.559)unfolds

(2.560)