Some combinations of =powers are so common that they have been given special names. They are the so-called hyperbolic functions.
From the definitions it immediately follows that
is an odd function,
is an even function, and that
is an odd function. The graphs of
and
can be quickly sketched as the graph of the sum or difference of two exponential functions. It also follows from the definitions of
and
that the graphs of these functions are approaching each other for large values of
.
The graph of is less easily sketched on the basis of the graphs of exponential functions and it is shown below.
The graphs above also illustrate the domain and range of the hyperpbolic functions: they are defined for all real numbers and the range of , and is , and , respectively.
Some pronounce the function names and as ''shine'' and "kosh", respectively.
The naming "hyperbolic functions" suggests that these have to do with a hyperbola. Also, the names and notations of the functions resemble those of the trigonometric functions , and . There is indeed a strong analogy: just as the coordinates of a point on the unit circle with equation can be written as , the coordinates of a point on the hyperbola with equation can be written as . But the relationship is actually more of algebraic nature, as becomes clear from the special formulas for hyperbolic functions.
The cosine hyperbolic is the function used to mathematically describe the shape of a uniform chain hanging under gravity.
Some properties of the trigonometric functions follow almost directly from the definitions.
By taking the difference of the above equations you arrive at the identity to be proved.
Adding the above equations gives
Adding the above equations gives
These are special cases of the addition formulas.
Expressed in the natural logarithm, the inverse hyperbolic functions are defined as follows:
The domain of
,
, and
is
,
, and
, respectively. The graphs of the inverse hyperbolic functions are shown below. They can be constructed by reflection of the graphs of the hyperbolic functions in the line with equation
. This can be seen in the diagrams because the graph of the corresponding hyperbolic function is also shown as a dotted line..
graph of arsinh
graph of arcosh
graph of artanh
The convention of at the beginning of the name indicates that we are dealing with area: is equal to the area of the region enclosed by the hyperbola with equation and the two halflines through the origin and the points on the hyperbola with vertical coordinates . The interactive diagram below illustrates that the shaded area has an area equal to . Likewise, equals the area of the region enclosed by the hyperbola with equations and the two halflines through the origin and the points on the hyperbola with horizontal coordinate .
As an example, we prove that
Suppose
, i.e.,
, then from
and the fact that
for all
follows that
So, with the definitions of
and
in terms of exponential functions we get: