Category:Gudermannian function

Gudermannian function 

function that relates the circular functions and hyperbolic functions without using complex numbers  image   schematic
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Instance of	function
Named after	Christoph Gudermann
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The Gudermannian function (gd).

${\displaystyle {\begin{aligned}\operatorname {gd} x&=\arcsin \left(\tanh x\right)=\arctan(\sinh x)=\operatorname {arccsc} (\coth x)\\&=\operatorname {sgn} (x)\cdot \arccos \left(\operatorname {sech} x\right)=\operatorname {sgn} (x)\cdot \operatorname {arcsec} (\cosh x)\\&=2\arctan \left[\tanh \left({\tfrac {1}{2}}x\right)\right]\\&=2\arctan(e^{x})-{\tfrac {1}{2}}\pi \end{aligned}}}$