duality of Gudermannian and its inverse function
There are a lot of formulae concerning the Gudermannian function and its inverse function containing a hyperbolic function or a trigonometric function or both, such that if we change functions of one kind to the corresponding functions of the other kind, then the new formula also is true.
Some exemples:
(1) |
(2) |
(3) |
(4) |
(5) |
For proving (5) we can check that
and since both the expression in the brackets and the http://planetmath.org/node/11997Gudermannian vanish in the origin, we have
This equation implies (5).
The duality (http://planetmath.org/DualityInMathematics) of the formula pairs may be explained by the equality
(6) |
Title | duality of Gudermannian and its inverse function |
---|---|
Canonical name | DualityOfGudermannianAndItsInverseFunction |
Date of creation | 2013-03-22 19:06:41 |
Last modified on | 2013-03-22 19:06:41 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 7 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 33B10 |
Classification | msc 26E05 |
Classification | msc 26A09 |
Classification | msc 26A48 |
Related topic | InverseGudermannianFunction |
Related topic | IdealInvertingInPruferRing |