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

By the same token, the hyperbolic secant, sech x , abides to

(2.487)

after resorting to Eq. (2.473); while the corresponding hyperbolic cosecant, cosech x , looks like

(2.488)

once Eq. (2.472)is retrieved – and as plotted in Fig. 2.14c. When x approaches zero, the hyperbolic cosecant is driven by

(2.489)

arising from Eq. (2.488), so x = 0 plays the role of vertical asymptote for this function; as for the remaining values, cosech x monotonically decreases within] −∞, 0[ and also within]0 ,∞ [, as can be grasped in Fig. 2.14c. Conversely,

(2.490)

based on Eq. (2.487)– so the horizontal axis serves as horizontal asymptote toward −∞, whereas

(2.491)

indicates that the very same straight line serves as (horizontal) asymptote toward ∞. In this case, there is a maximum at x = 0 – since d sech x / dx = − 2(e x− e −x)/(e x + e −x) 2(to be fully proven at a later stage) equals zero when e x = e −x, or else at x = 0; this critical point is easily perceived in Fig. 2.14c. Finally, note the resemblance between the functional form of Eqs. (2.482)and (2.483)with Eqs. (2.299)and (2.304), respectively – as well as between Eqs. (2.487)and (2.488), on the one hand, and Eqs. (2.309)and (2.314), on the other; this contributes to justify the denomination of (hyperbolic) trigonometric functions.

After squaring both sides of Eqs. (2.472)and (2.473), and then performing ordered subtraction of the result, one obtains

(2.492)

– where Newton’s binomial as per Eqs. (2.237)and (2.238)may be invoked to write

(2.493)

or, equivalently,

(2.494)

because e xe −x = e x−x = e 0 = 1; after canceling symmetrical terms, Eq. (2.494)becomes

(2.495)

that readily simplifies to

(2.496)

– which reminds of Eq. (2.442)pertaining to circular functions proper (except for the minus sign). If Eqs. (2.472)and (2.473)are instead multiplied by one another, i.e.

(2.497)

one finds that

(2.498)

with the aid of the distributive property – or else

(2.499)

after lumping factors alike and canceling out symmetrical terms; if Eq. (2.499)is rewritten as

(2.500)

then comparison with Eq. (2.472)allows further reformulation to

(2.501)

that is equivalent to

(2.502)

– identical in form to Eq. (2.328), after setting x = y . This similarity further accounts for the extra labeling of trigonometric ascribed to the hyperbolic functions.

If Eqs. (2.472)and (2.473)are instead employed in parametric form, viz.

(2.503)

coupled with

(2.504)

one may square both sides of Eqs. (2.503)and (2.504)– and then proceed to ordered subtraction thereof to get

(2.505)

after factoring a 2out, Eq. (2.505)becomes

(2.506)

while insertion of Eq. (2.496)supports simplification to

(2.507)

Equation (2.507)is but the analytical equation of a hyperbola – thus backing up the hyperbolic designation for the functions under scrutiny.

Once in possession of Eq. (2.496), one may divide both its sides by sinh 2 x to get

(2.508)

where insertion of Eqs. (2.482), (2.483), and (2.488)gives rise to

(2.509)