inverse Gudermannian function

Since the real Gudermannian function gd is strictly increasing and forms a bijection from $ℝ$ onto the open interval  $(-\frac{\pi }{2},\frac{\pi }{2})$,  it has an inverse function

$${\text{gd}}^{-1}:(-\frac{\pi }{2},\frac{\pi }{2})\to ℝ.$$

The function ${\text{gd}}^{-1}$ is denoted also arcgd.

If  $x=\text{gd}y$, which may be explicitly written e.g.

$$x=\arcsin (\tanh y),$$

one can solve this for $y$, getting first  $\tanh y=\sin x$  and then

$$y=\text{artanh}(\sin x)$$

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

${\text{gd}}^{-1}(x)=\text{arcgd}x=\text{artanh}(\sin x)$ (1)

It has other equivalent (http://planetmath.org/Equivalent3) expressions, such as

${\text{gd}}^{-1}(x)=\text{arsinh}(\tan x)={\displaystyle \frac{1}{2}}\ln {\displaystyle \frac{1+\sin x}{1-\sin x}}={\displaystyle {\int }_0^x}{\displaystyle \frac{dt}{\cos t}}.$ (2)

Thus its derivative is

${\displaystyle \frac{d}{dx}}{\text{gd}}^{-1}(x)={\displaystyle \frac{1}{\cos x}}.$ (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