Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier MalcataЧитать онлайн книгу.

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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) equation

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

(2.571) equation

that is the same to write

(2.572) equation

as per Eq. (2.566); a plot of Eq. (2.572) is conveyed by Fig. 2.15a. Note that images , so images 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.

Left: Graph with 2 curves connecting curves labeled sinh−1 x and cosh−1 x. Right: Graph displaying a fluctuating curves with 3 segments labeled cotanh−1 x, tanh−1 x, and cotanh−1 x.

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) equation

then hyperbolic cosine may be applied to both sides to produce

(2.574) equation

– 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) equation

where isolation of sinh y yields

(2.576) equation

– with both signs preceding the square root being now feasible, since sinh y may take either positive or negative values (see Fig. 2.14 a); 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) equation

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

(2.578) equation

or else

(2.579) equation

with the aid of Eq. (2.573), as depicted also in Fig. 2.15a. In this case, images is defined only when x² > 1, or else x < − 1 ∨ x > 1; furthermore, images , thus implying that images when x < −1, but images for x > 1 – so only the latter can be taken as domain for cosh⁻¹ x, so that logarithm thereof can be defined. However, images , 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.

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

(2.580) equation

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

(2.581) equation

as per composition of functions; consequently,

(2.582) equation

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

(2.583) equation

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

(2.584) equation

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