Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS


cothx

cothx

tanhx
x

2


4


− 2


− 4


− 2 − 1 1 2


Figure 3.13 Graphs of tanhxand cothx.

metric functions transparent. The similarity in their calculus is discussed further


in subsection 3.7.6.


3.7.3 Identities of hyperbolic functions

The analogies between trigonometric functions and hyperbolic functions having


been established, we should not be surprised that all the trigonometric identities


also hold for hyperbolic functions, with the following modification. Wherever


sin^2 xoccurs it must be replaced by−sinh^2 x, and vice versa. Note that this


replacement is necessary even if the sin^2 xis hidden, e.g. tan^2 x=sin^2 x/cos^2 x


and so must be replaced by (−sinh^2 x/cosh^2 x)=−tanh^2 x.


Find the hyperbolic identity analogous tocos^2 x+sin^2 x=1.

Using the rules stated above cos^2 xis replaced by cosh^2 x,andsin^2 xby−sinh^2 x,andso
the identity becomes


cosh^2 x−sinh^2 x=1.

This can be verified by direct substitution, using the definitions of coshxand sinhx;see
(3.38) and (3.39).


Some other identities that can be proved in a similar way are

sech^2 x=1−tanh^2 x, (3.48)
cosech^2 x=coth^2 x− 1 , (3.49)

sinh 2x=2sinhxcoshx, (3.50)

cosh 2x=cosh^2 x+ sinh^2 x. (3.51)
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