# inverse hyperbolic functions

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##### 2: 4.1 Special Notation
 $k,m,n$ integers. …
The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. … ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\operatorname{csch}z$, $\operatorname{sech}z$, $\coth z$; the inverse hyperbolic functions $\operatorname{arcsinh}z$, $\operatorname{Arcsinh}z$, etc. …
##### 3: 4.29 Graphics
###### §4.29(i) Real Arguments Figure 4.29.6: Principal values of arccsch ⁡ x and arcsech ⁡ x . … Magnify
###### §4.29(ii) Complex Arguments
The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …
##### 6: 19.10 Relations to Other Functions
19.10.2 $(\sinh\phi)R_{C}\left(1,{\cosh^{2}}\phi\right)=\operatorname{gd}\left(\phi% \right).$
##### 7: 4.39 Continued Fractions
###### §4.39 Continued Fractions
4.39.2 $\frac{\operatorname{arcsinh}z}{\sqrt{1+z^{2}}}=\cfrac{z}{1+\cfrac{1\cdot 2z^{2% }}{3+\cfrac{1\cdot 2z^{2}}{5+\cfrac{3\cdot 4z^{2}}{7+\cfrac{3\cdot 4z^{2}}{9+% \cdots}}}}},$
4.39.3 $\operatorname{arctanh}z=\cfrac{z}{1-\cfrac{z^{2}}{3-\cfrac{4z^{2}}{5-\cfrac{9z% ^{2}}{7-\cdots}}}},$
For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569–571). …
##### 8: 4.38 Inverse Hyperbolic Functions: Further Properties
###### §4.38(ii) Derivatives
4.38.19 $\operatorname{Arctanh}u\pm\operatorname{Arccoth}v=\operatorname{Arctanh}\left(% \frac{uv\pm 1}{v\pm u}\right)=\operatorname{Arccoth}\left(\frac{v\pm u}{uv\pm 1% }\right).$
4.23.40 $\operatorname{gd}\left(x\right)=2\operatorname{arctan}\left(e^{x}\right)-% \tfrac{1}{2}\pi\\ =\operatorname{arcsin}\left(\tanh x\right)=\operatorname{arccsc}\left(\coth x% \right)\\ =\operatorname{arccos}\left(\operatorname{sech}x\right)=\operatorname{arcsec}% \left(\cosh x\right)\\ =\operatorname{arctan}\left(\sinh x\right)=\operatorname{arccot}\left(% \operatorname{csch}x\right).$
4.23.42 ${\operatorname{gd}^{-1}}\left(x\right)=\ln\tan\left(\tfrac{1}{2}x+\tfrac{1}{4}% \pi\right)=\ln\left(\sec x+\tan x\right)=\operatorname{arcsinh}\left(\tan x% \right)=\operatorname{arccsch}\left(\cot x\right)=\operatorname{arccosh}\left(% \sec x\right)=\operatorname{arcsech}\left(\cos x\right)=\operatorname{arctanh}% \left(\sin x\right)=\operatorname{arccoth}\left(\csc x\right).$