Formula:2069

Digital Library of Mathematical Functions ID 4.23.E42

${\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}\pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x\right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(\sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}\left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).}}$

Symbols List

• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\csc\NVar{z}}}$: cosecant function
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\cot\NVar{z}}}$: cotangent function
• ${\displaystyle{\displaystyle\operatorname{arccsch}\NVar{z}}}$: inverse hyperbolic cosecant function
• ${\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}$:
• ${\displaystyle{\displaystyle\operatorname{arccoth}\NVar{z}}}$: inverse hyperbolic cotangent function
• ${\displaystyle{\displaystyle\operatorname{arcsech}\NVar{z}}}$: inverse hyperbolic secant function
• ${\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}$:
• ${\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}$:
• ${\displaystyle{\displaystyle{\operatorname{gd}^{-1}}\left(\NVar{x}\right)}}$: inverse Gudermannian function
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle\sec\NVar{z}}}$: secant function
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle\tan\NVar{z}}}$: tangent function
• ${\displaystyle{\displaystyle x}}$: real variable
 

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