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
so application of hyperbolic sine to both sides gives rise to
– 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
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
as per Eq. (2.566); a plot of Eq. (2.572) is conveyed by Fig. 2.15a. Note that
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
– 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
– 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
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,
