1.5 Trigonometric and Hyperbolic
Functions
1.5.1 Definitions
We recall Euler’s formula for the
sine and cosine,
where
is a real number. From this, we can express sine and
cosine as
We now
define the hyperbolic
functions ‘hyperbolic cosine’ and ‘hyperbolic
sine’ as
i.e. analogous to cosine and sine but without
the imaginary unit
. Using
, we recognise that
which means that trigonometric and hyperbolic
functions are closely related. Their behaviour as a function of
, however, is different: while sine and cosine are
oscillatory functions, the hyperbolic functions
and
are not oscillatory, because they are just linear
combinations of
and
which are not oscillatory. We have the following
properties:
from which we already can sketch the two
hyperbolic functions, see Fig.
1.2
.
In addition, one defines the hyperbolic
tangent and cotangent
1.5.2 Inverse hyperbolic functions
Inverting
we find the inverse hyperbolic sine
by setting
This is a quadratic equation in
with the solutions
Since
is positive, we must take the positive solution
and must discard the negative solution
. Therefore,
which means that
Similarly, one obtains
The
is a bit more tricky.
1.5.3 Derivatives
These are obtained by going back to the
definitions of the hyperbolic functions.
1.5.4 Hyperbolic Identities
These also are obtained by using the
definitions of
and
: