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

Following a similar rationale, one may calculate the inverse hyperbolic tangent – by, once again, setting

(2.580)

at startup, in parallel to Eq. (2.566)– thus implying that

(2.581)

as per composition of functions; consequently,

(2.582)

owing to Eqs. (2.482)and (2.581), which becomes

(2.583)

after elimination of denominator. Upon insertion of Eq. (2.581), one obtains

(2.584)

from Eq. (2.511),with x replaced by y – which readily becomes

(2.585)

only the plus sign preceding the square root was taken here, because sech y only takes positive values (see Fig. 2.14c). One may now revisit Eq. (2.479)as

(2.586)

at the expense of Eq. (2.583), along with factoring out of cosh y ; in view of Eq. (2.487), one may redo Eq. (2.586)to

(2.587)

where insertion of Eq. (2.585)permits further transformation to

(2.588)

Since a difference of squares is expressible as the product of two conjugate binomials, Eq. (2.588)unfolds

(2.589)

or else

(2.590)

after cancelation of between numerator and denominator; following application of logarithms to both sides, Eq. (2.590)turns to

(2.591)

where Eqs. (2.25)and (2.580)support final transformation to

(2.592)

– defined for ∣ x ∣ < 1 only, so as to guarantee a positive argument for the logarithm. Equation (2.592)is illustrated in Fig. 2.15b; note the monotonically increasing pattern of tanh −1 x , spanning] − 1,1[ as domain; at either x = − 1 or x = 1, a vertical asymptote arises – according to

(2.593)

that drives the curve toward −∞ at x = − 1, coupled with

(2.594)

that drives the curve toward ∞ at x = 1.

The inverse hyperbolic cotangent may be obtained after applying the hyperbolic tangent operator to both sides of Eq. (2.592), namely,

(2.595)

once reciprocals are taken of both sides, Eq. (2.595)becomes

(2.596)

– also with the aid of Eq. (2.483). Division of both numerator and denominator of the argument of the logarithm function by x converts Eq. (2.596)to

(2.597)

where a change of variable to is in order, i.e.

(2.598)

all is left is taking the inverse hyperbolic cotangent of both sides, according to

(2.599)

where retrieval of the original (dummy) variable x unfolds

(2.600)

Equation (2.600)is depicted in Fig. 2.15b; it is not defined within [ − 1,1] since x – 1 < 0 in that range would compromise existence of the logarithm. Outside said range, one notices that

(2.601)

based on Eq. (2.600)– so x = −1 drives the behavior of cotanh −1 x toward −∞, in the neighborhood of − 1; by the same token,

(2.602)