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

if both sides of Eq. (2.496)were instead divided by cosh 2 x , one would have gotten

(2.510)

that degenerates to

(2.511)

after taking Eqs. (2.482)and (2.487)on board; the similarity of Eqs. (2.509)and (2.511)with Eqs. (2.469)and (2.471), respectively, is again striking – except for the minus, instead of plus sign.

Equation (2.472), pertaining to sinh x , may be multiplied by Eq. (2.473), pertaining to cosh y , to get

(2.512)

– in much the same way said equations were combined to produce Eq. (2.497)when y = x ; after applying the distributive property of multiplication, Eq. (2.512)becomes

(2.513)

or, after recalling the rule of multiplication of exponential functions conveyed by Eq. (2.15),

(2.514)

Algebraic rearrangement permits transformation of Eq. (2.514)to

(2.515)

which is the same as writing

(2.516)

in view of the functional form of Eq. (2.472). A similar rationale may be applied to the product of sinh y by cosh x to get

(2.517)

again based on Eqs. (2.472)and (2.473)– where the two factors may be pooled together as

(2.518)

Eq. (2.518)may also appear as

(2.519)

After splitting its right‐hand side and factoring out − 1 in selected exponents, Eq. (2.519)becomes

(2.520)

so Eq. (2.472)may again be invoked to write

(2.521)

in view of Eq. (2.475), one may redo Eq. (2.521)as

(2.522)

Ordered addition of Eqs. (2.516)and (2.522)gives rise to

(2.523)

which simplifies to

(2.524)

following condensation of terms alike – while ordered subtraction unfolds

(2.525)

with combination of terms alike leading to

(2.526)

Note the similarity of Eqs. (2.524)and (2.526)to Eqs. (2.328)and (2.330), respectively.

If Eq. (2.472)is multiplied by itself – using x and y at a time as arguments, one gets

(2.527)

in which case the two fractions may be collapsed to produce

(2.528)

Eq. (2.528)may be rewritten as

(2.529)

with the aid of Eq. (2.15). After splitting the right‐hand side, Eq. (2.529)turns to

(2.530)

– which can be condensed to

(2.531)

at the expense of Eq. (2.473). Departure from Eq. (2.473)applied to x and y will instead lead to

(2.532)

where multiplication of the first by the second factor in the right‐hand side generates

(2.533)

after lumping exponential functions in the usual fashion, Eq. (2.533)transforms to

(2.534)

Algebraic reorganization of Eq. (2.534)gives rise to

(2.535)