Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS


3.7 Hyperbolic functions

Thehyperbolic functionsare the complex analogues of the trigonometric functions.


The analogy may not be immediately apparent and their definitions may appear


at first to be somewhat arbitrary. However, careful examination of their properties


reveals the purpose of the definitions. For instance, their close relationship with


the trigonometric functions, both in their identities and in their calculus, means


that many of the familiar properties of trigonometric functions can also be applied


to the hyperbolic functions. Further, hyperbolic functions occur regularly, and so


giving them special names is a notational convenience.


3.7.1 Definitions

The two fundamental hyperbolic functions are coshxand sinhx, which, as their


names suggest, are the hyperbolic equivalents of cosxand sinx. They are defined


by the following relations:


coshx=^12 (ex+e−x), (3.38)
sinhx=^12 (ex−e−x). (3.39)

Note that coshxis an even function and sinhxis an odd function. By analogy


with the trigonometric functions, the remaining hyperbolic functions are


tanhx=

sinhx
coshx

=

ex−e−x
ex+e−x

, (3.40)

sechx=

1
coshx

=

2
ex+e−x

, (3.41)

cosechx=

1
sinhx

=

2
ex−e−x

, (3.42)

cothx=

1
tanhx

=

ex+e−x
ex−e−x

. (3.43)


All the hyperbolic functions above have been defined in terms of the real variable


x. However, this was simply so that they may be plotted (see figures 3.11–3.13);


the definitions are equally valid for any complex numberz.


3.7.2 Hyperbolic–trigonometric analogies

In the previous subsections we have alluded to the analogy between trigonometric


and hyperbolic functions. Here, we discuss the close relationship between the two


groups of functions.


Recalling (3.32) and (3.33) we find

cosix=^12 (ex+e−x),

sinix=^12 i(ex−e−x).
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