Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

sech−^1 x

sech−^1 x

cosh−^1 x

cosh−^1 x

x

2

4

− 2

− 4

1 2 3 4

Figure 3.14 Graphs of cosh−^1 xand sech−^1 x.

Find a closed-form expression for the inverse hyperbolic functiony=tanh−^1 x.

First we writexas a function ofy,i.e.

y=tanh−^1 x ⇒ x=tanhy.

Now, using the definition of tanhyand rearranging, we find

x=

ey−e−y

ey+e−y

⇒ (x+1)e−y=(1−x)ey.

Thus, it follows that

e^2 y=

1+x

1 −x

⇒ ey=

√

1+x

1 −x

,

y=ln

√

1+x

1 −x

,

tanh−^1 x=

1

2

ln

(

1+x

1 −x

)

.

Graphs of the inverse hyperbolic functions are given in figures 3.14–3.16.

3.7.6 Calculus of hyperbolic functions

Just as the identities of hyperbolic functions closely follow those of their trigono-

metric counterparts, so their calculus is similar. The derivatives of the two basic