inverse Gudermannian function

Since the real Gudermannian functionDlmfPlanetmath gd is strictly increasing and forms a bijection from onto the open intervalDlmfPlanetmath(-π2,π2),  it has an inverse functionMathworldPlanetmath


The functionMathworldPlanetmath gd-1 is denoted also arcgd.

If  x=gdy, which may be explicitly written e.g.


one can solve this for y, getting first  tanhy=sinx  and then


(see the area functions).  Hence the inverse GudermannianMathworldPlanetmath is expressed as

gd-1(x)=arcgdx=artanh(sinx) (1)

It has other equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( expressions, such as

gd-1(x)=arsinh(tanx)=12ln1+sinx1-sinx=0xdtcost. (2)

Thus its derivativePlanetmathPlanetmath is

ddxgd-1(x)=1cosx. (3)

Cf. the formulae (1)–(3) with the corresponding ones of gd.

Title inverse Gudermannian function
Canonical name InverseGudermannianFunction
Date of creation 2013-03-22 19:06:28
Last modified on 2013-03-22 19:06:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Definition
Classification msc 33B10
Classification msc 26E05
Classification msc 26A48
Classification msc 26A09
Synonym inverse Gudermannian
Related topic HyperbolicFunctions
Related topic AreaFunctions
Related topic MercatorProjection
Related topic EulerNumbers2
Related topic DualityOfGudermannianAndItsInverseFunction