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) |