duality of Gudermannian and its inverse function

There are a lot of formulae concerning the Gudermannian functionDlmfPlanetmath and its inverse function containing a hyperbolic functionDlmfMathworldPlanetmath or a trigonometric functionDlmfMathworldPlanetmath or both, such that if we change functionsMathworldPlanetmath of one kind to the corresponding functions of the other kind, then the new formula also is true.

Some exemples:

gdx=0xdtcosht,gd-1x=0xdtcost (1)
ddxgdx=1coshx,ddxgd-1x=1cosx (2)
tan(gdx)=sinhx,tanh(gd-1x)=sinx (3)
sin(gdx)=tanhx,sinh(gd-1x)=tanx (4)
tangdx2=tanhx2,tanhgd-1x2=tanx2 (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

gdx 2arctan(tanhx2).

This equation implies (5).

The duality (http://planetmath.org/DualityInMathematics) of the formula pairs may be explained by the equality

gdix=igd-1x. (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