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

(2.548), since it involves only hyperbolic cosines.

      2.4.3 Euler’s Form of Complex Numbers

      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 ι² = −1 and ι⁴ = 1 as algebraic rules (supporting switch between −1 and 1 along consecutive terms), and ι and ι³ = −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 a² + b² = 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)

      whereas ordered subtraction thereof gives rise to

(2.561)

      division of both sides by 2 yields

(2.562)

      and

(2.563)

from Eqs. (2.560) and (2.561), respectively – or else

(2.564)

from Eq. (2.562) after combination with Eq. (2.473), as well as

(2.565)

from Eq. (2.563) following combination with Eq. (2.472). The two basic relationships between hyperbolic and circular functions are consequently conveyed by Eqs. (2.564) and (2.565).

      2.4.4 Inverse Functions

      Inverse hyperbolic functions are often useful, and will accordingly be discussed next; in the case of sinh⁻¹ x, one should start by setting

(2.566)

      so application of hyperbolic sine to both sides gives rise to

(2.567)

– since composition of a function with its inverse cancels out their mutual effects. Insertion of Eq. (2.567) allows transformation of Eq. (2.496) – rewritten in (dummy) variable y, to

      (2.568)

      so adding x² to both sides and taking square roots thereof afterward generate

(2.569)