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

Figure 2.1 Variation of absolute value, |x|, as a function of a real number, x.

      It is easily proven that

(2.3)

based on the four possible combinations of signs of x and y, coupled with Eq. (2.2); by the same token,

      (2.4)

after replacing y by its reciprocal in Eq. (2.3). On the other hand, the definition conveyed by Eq. (2.2) allows one to conclude that

(2.5)

      or else

(2.6)

after taking negatives of both sides; based on the definition as per Eq. (2.2) and the corollary labeled as Eq. (2.5), one finds

(2.7)

upon replacement of x by x + y ≥ 0. Equations (2.2) and (2.6) similarly support the conclusion

(2.8)

      in general terms, one concludes that

(2.9)

after bringing Eqs. (2.7) and (2.8) together. On the other hand, one may depart from the definition of auxiliary variable z as

(2.10)

      to readily obtain

(2.11)

after recalling Eq. (2.9), one may redo Eq. (2.11) to

      (2.12)

with the aid of Eq. (2.10), where straightforward algebraic rearrangement unfolds

      (2.13)

      that complements Eq. (2.9).

Another essential function is the (natural) exponential, e^x – i.e. a power where Neper’s number (ca. 2.718 28) serves as basis; it is sketched in Fig. 2.2a. Note the exclusively positive values of this function – as well as its horizontal asymptote, viz.

      (2.14)

      The exponential function converts a sum into a product, i.e.

(2.15)

      based on the rule of multiplication of powers with the same base; one also realizes that

      (2.16)

pertaining to a difference as argument, and obtainable from Eq. (2.15) after replacement of y by −y (since e^−y is, by definition, 1/e^y). A generalization of Eq. (2.15) reads

      (2.17)

where x₁ = x₂ = ⋯ = x_n = x readily implies

(2.18)

      by virtue of the definition of multiplication as an iterated sum.