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| The
Argument to the Hyperbolic Functions |
The unit hyperbola is the curve satisfying the
equation
(1)
![](formulas/hyperbolic-angle-1-formulas_html_2be4193a.png)
The
hyperbolic cosine and sine are given by:
(2)
![](formulas/hyperbolic-angle-1-formulas_html_50b9b7e3.png)
and they satisfy the identity
(
3)
![](formulas/hyperbolic-angle-1-formulas_html_4e9ecb03.png)
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 light-brown 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:
![Hyperbola](images/hyperbola_1.png)
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: |
![Area under cosh-sinh triangle](images/hyperbola_u_area.png) | Minus | ![Area under hyperbola](images/hyperbola_sub_area.png) |
|
|
The area shown in
figure 2 is:
(4)
![](formulas/hyperbolic-angle-1-formulas_html_m5d024907.png)
We substitute:
(5)
![](formulas/hyperbolic-angle-1-formulas_html_5b156f00.png)
which, applying equation (
3), produces:
(6)
![](formulas/hyperbolic-angle-1-formulas_html_1dea9ba0.png)
After
multiplying out the first term and collecting the pieces, and making a
quick trip to the handy-dandy CRC Math Tables for the integral, we
obtain:
(7)
![](formulas/hyperbolic-angle-1-formulas_html_m41c73e19.png)
which, after collecting terms, is just:
(8)
![](formulas/hyperbolic-angle-1-formulas_html_m3123d412.png)
which was to be shown.
Page created on 11/15/06