# inverse Gudermannian function

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

 $\mbox{gd}^{-1}\!:\;(-\frac{\pi}{2},\,\frac{\pi}{2})\,\to\,\mathbb{R}.$

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

If  $x=\mbox{gd}\,y$, 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

 $y\;=\;\mbox{artanh}(\sin{x})$

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

 $\displaystyle\mbox{gd}^{-1}(x)\;=\;\mbox{arcgd}\,x\;=\;\mbox{artanh}(\sin{x})$ (1)

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

 $\displaystyle\mbox{gd}^{-1}(x)\;=\;\mbox{arsinh}(\tan{x})\;=\;\frac{1}{2}\ln% \frac{1+\sin{x}}{1-\sin{x}}\;=\;\int_{0}^{x}\!\frac{dt}{\cos{t}}.$ (2)

Thus its derivative is

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