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

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).

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 x 2to both sides and taking square roots thereof afterward generate

(2.569)

here only the plus sign was kept, since cosh y > 0 as per Fig. 2.14a. Based on Eq. (2.479)rewritten for y , one finds that

(2.570)

following combination with Eqs. (2.567)and (2.569)– or, after applying logarithms to both sides,

(2.571)

that is the same to write

(2.572)

as per Eq. (2.566); a plot of Eq. (2.572)is conveyed by Fig. 2.15a. Note that , so in view of Eq. (2.2); therefore, the argument of the logarithm in Eq. (2.572)is always positive, and sinh −1 x is defined for all real numbers.

Figure 2.15 Variation, with their argument x , of inverse hyperbolic functions, viz. (a) inverse hyperbolic sine (sinh −1) and cosine (cosh −1) and (b) inverse hyperbolic tangent (tanh −1) and cotangent (cotanh −1).

By the same token, if one sets

(2.573)

then hyperbolic cosine may be applied to both sides to produce

(2.574)

– again due to the inefficacy, with regard to its argument, of composing a function with its inverse. Upon combination of Eq. (2.496), rewritten for y , with Eq. (2.574), one obtains

(2.575)

where isolation of sinh y yields

(2.576)

– with both signs preceding the square root being now feasible, since sinh y may take either positive or negative values (see Fig. 2.14a); once in possession of Eqs. (2.574)and (2.576), one may resort to Eq. (2.479),with x relabeled as y , to write

(2.577)

Application of logarithms to both sides of Eq. (2.577)finally gives

(2.578)

or else

(2.579)

with the aid of Eq. (2.573), as depicted also in Fig. 2.15a. In this case, is defined only when x 2 > 1, or else x < − 1 ∨ x > 1; furthermore, , thus implying that when x < − 1, but for x > 1 – so only the latter can be taken as domain for cosh −1 x , so that logarithm thereof can be defined. However, , so two possible values actually arise, i.e. cosh −1 x is a double‐valued function; only the positive value of the logarithm is usually considered, thus implying use of the positive sign preceding the square root in its argument.