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

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href="#ulink_7dc46580-e83c-5f10-be3f-7ec47f94a215">(2.325), one gets

(2.343)

      whereas ordered subtraction thereof gives rise to

(2.344)

      by the same token, ordered addition of Eqs. (2.328) and (2.330) generates

(2.345)

      and ordered subtraction unfolds

(2.346)

      One may now define two auxiliary variables, X and Y, according to

(2.347)

      and

(2.348)

respectively; ordered addition of Eqs. (2.347) and (2.348) yields

      (2.349)

      or else

(2.350)

      upon isolation of x – whereas ordered subtraction of Eq. (2.348) from Eq. (2.347) leads to

      (2.351)

      which in turn conveys

(2.352)

when solved for y. Insertion of Eqs. (2.347), (2.348), (2.350), and (2.352) transforms Eq. (2.343) to

(2.353)

and Eq. (2.344) likewise supports

      (2.354)

one may similarly transform Eq. (2.345) to

      (2.355)

while Eq. (2.346) yields

(2.356)

Equations (2.353)–(2.356) may be useful whenever logarithms of their left‐hand sides are to be handled, because there is no formula to calculate the logarithm of an algebraic sum – unlike happens with the logarithm of a product, see Eq. (2.20).

      In attempts to generate further useful relationships involving transformation of arguments of trigonometric functions, it is convenient to resort to the complex domain – and recall that complex numbers, say z₁ and z₂, may be defined by Cartesian (or rectangular) coordinates as

(2.357)

      and

      (2.358)

with denoting imaginary unit, besides a_i ’s and b_i ’s denoting real numbers. Since z_i is represented by two coordinates in a plane – a_i as abscissa (on the real axis) and b_i as ordinate (on the imaginary axis), one may denote as θ_i the angle of vector of coordinates (a_i ,b_i) with the real axis, and by ρ_i its length; hence, Eq. (2.357) may appear alternatively as

(2.359)

      and

(2.360)

      respectively. Note that

      (2.361)

      and

      (2.362)

      were used as

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