COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
3.7 Hyperbolic functionsThehyperbolic 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 DefinitionsThe 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 analogiesIn 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 findcosix=^12 (ex+e−x),sinix=^12 i(ex−e−x).