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Inverse Hyperbolic Functions

The inverse hyperbolic functions, sometimes also called the area hyperbolic functions (Spanier and Oldham 1987, p. 263) are the multivalued function that are the inverse functions of the hyperbolic functions. They are denoted cosh^(-1)z, coth^(-1)z, csch^(-1)z, sech^(-1)z, sinh^(-1)z, and tanh^(-1)z. Variants of these notations beginning with a capital letter are commonly used to denote their principal values (e.g., Harris and Stocker 1998, p. 263).

These functions are multivalued, and hence require branch cuts in the complex plane. Differing branch cut conventions are possible, but those adopted in this work follow those used by the Wolfram Language, summarized below.

function namefunctionthe Wolfram Languagebranch cut(s)
inverse hyperbolic cosecantcsch^(-1)zArcCsch[z](-i,i)
inverse hyperbolic cosinecosh^(-1)zArcCosh[z](-infty,1)
inverse hyperbolic cotangentcoth^(-1)zArcCoth[z][-1,1]
inverse hyperbolic secantsech^(-1)zArcSech[z](-infty,0] and (1,infty)
inverse hyperbolic sinesinh^(-1)zArcSinh[z](-iinfty,-i) and (i,iinfty)
inverse hyperbolic tangenttanh^(-1)zArcTanh[z](-infty,-1] and [1,infty)
InverseHyperbolicFunctions

The inverse hyperbolic functions as defined in this work have the following ranges for domains on the real line R, again following the convention of the Wolfram Language.

function namefunctiondomainrange
inverse hyperbolic cosecantcsch^(-1)x(-infty,infty)(-infty,infty)
inverse hyperbolic cosinecosh^(-1)x[1,infty)[0,infty)
inverse hyperbolic cotangentcoth^(-1)x(-infty,-1) or (1,infty)(-infty,infty)
inverse hyperbolic secantsech^(-1)x(0,1][0,infty)
inverse hyperbolic sinesinh^(-1)x(-infty,infty)(-infty,infty)
inverse hyperbolic tangenttanh^(-1)x(-1,1)(-infty,infty)

They are defined in the complex plane by

sinh^(-1)z=ln(z+sqrt(z^2+1))
(1)
cosh^(-1)z=ln(z+sqrt(z-1)sqrt(z+1))
(2)
tanh^(-1)z=1/2[ln(1+z)-ln(1-z)]
(3)
csch^(-1)z=ln(sqrt(1+1/(z^2))+1/z)
(4)
sech^(-1)z=ln(sqrt(1/z-1)sqrt(1+1/z)+1/z)
(5)
coth^(-1)z=1/2[ln(1+1/z)-ln(1-1/z)].
(6)

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