# inverse Gudermannian function

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

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