Mathematical Methods for Physics and Engineering - Matematica.NET

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS42sech −1 xcosh −1 x123 4x−2−4sech −1 xcosh −1 xFigure 3.14 Graphs of cosh −1 x and sech −1 x.◮Find a closed-form expression for the inverse hyperbolic function y =tanh −1 x.First we write x as a function of y, i.e.y =tanh −1 x ⇒ x =tanhy.Now, using the definition of tanh y and rearranging, we findx = ey − e −ye y + e −y ⇒ (x +1)e −y =(1− x)e y .Thus, it follows thate 2y = 1+x1 − x√1+x⇒ e y =1 − x ,√1+xy =ln1 − x ,tanh −1 x = 1 ( ) 1+x2 ln . ◭1 − xGraphs of the inverse hyperbolic functions are given in figures 3.14–3.16.3.7.6 Calculus of hyperbolic functionsJust as the identities of hyperbolic functions closely follow those of their trigonometriccounterparts, so their calculus is similar. The derivatives of the two basic106