Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

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

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