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

      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 defining relationships, in agreement with Eqs. (2.288) and (2.290), complemented by Fig. 2.10 a; and after replacing ‖ u ‖ by ρ_i, by a_i, and by b_i . The product of two complex numbers reads

(2.363)

in view of Eqs. (2.359) and (2.360), or else

(2.364)

after lumping ρ₁ and ρ₂, and applying the distributive property to the product of sums of trigonometric functions; since ι² = −1 by definition, Eq. (2.364) becomes

(2.365)

along with convenient factoring out of ι. Insertion of Eqs. (2.325) and (2.328) supports transformation of Eq. (2.365) to

(2.366)

in the case of n complex numbers, Eq. (2.366) readily generalizes to

(2.367)

via consecutive application of the transformation of Eq. (2.363) to Eq. (2.366). Should, in addition, ρ₁ = ρ₂ = ⋯ = ρ_n = ρ and θ₁ = θ₂ = ⋯ = θ_n = θ, then Eq. (2.367) degenerates to

(2.368)

– in view of the functional form conveyed by either Eq. (2.359) or Eq. (2.360), and the definition of power; if ρ is further set equal to unity, then Eq. (2.368) simplifies to

(2.369)

usually known as Moivre's formula – and valid for any positive or negative integer n (as well as for rational numbers). For instance, Eq. (2.369) yields

(2.370)

      in the case of n = −1, which breaks down to merely

(2.371)

      in view of Eqs. (2.295) and (2.296) – and with z defined

(2.372)

combination of Eqs. (2.370) and (2.371) obviously looks like

(2.373)

Ordered addition of Eqs. (2.371) and (2.372) produces

      (2.374)