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Principal values of the inverse hyperbolic tangent and cotangent

The formulas given in § Definitions in terms of logarithms suggests

for the definition of the principal values of the inverse hyperbolic tangent and cotangent. In these formulas, the argument of the logarithm is real if and only if z is real. For artanh, this argument is in the real interval (−∞, 0], if z belongs either to (−∞, −1] or to, if and only if z belongs to the real interval.

Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and for the inverse hyperbolic cotangent.

In view of a better numerical evaluation near the branch cuts, some authors use the following definitions of the principal values, although the second one introduces a removable singularity at z = 0. The two definitions of artanh {\displaystyle \operatorname {artanh} } differ for real values of z {\displaystyle z} with z > 1 {\displaystyle z>1}. The ones of arcoth {\displaystyle \operatorname {arcoth} } differ for real values of z {\displaystyle z} with z ∈ [ 0, 1) {\displaystyle z\in [0,1)}.