 The
Argument to the Hyperbolic Functions 
The unit hyperbola is the curve satisfying the
equation
(1)
The
hyperbolic cosine and sine are given by:
(2)
and they satisfy the identity
(
3)
which implies that the point (
cosh(u),
sinh(u)) lies on the unit hyperbola.
As we stated on the
hyperbolic rotations
page, the argument,
u, to the
cosh() and sinh() functions is twice the lightbrown area shown in
figure 1 (which we have copied
from our hyperbolic rotations page). We'll prove that here;
the proof consists of a simple integration.
Figure 1  The
unit hyperbola, showing u:
In
figure
2 we show the area we need to find, broken down.
Figure 2
 Area shown in figure 1 is the area on the left,
minus the area on the right: 
 Minus  


The area shown in
figure 2 is:
(4)
We substitute:
(5)
which, applying equation (
3), produces:
(6)
After
multiplying out the first term and collecting the pieces, and making a
quick trip to the handydandy CRC Math Tables for the integral, we
obtain:
(7)
which, after collecting terms, is just:
(8)
which was to be shown.
Page created on 11/15/06