principal value of the logarithm, denoted in what follows, is defined as the value for which the imaginary part has the smallest absolute value. It is defined everywhere except for non-positive real values of the variable, for which two different values of the logarithm reach the minimum. For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function. However, in some cases, the formulas of do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected. Principal value of the inverse hyperbolic sine The principal value of the inverse hyperbolic sine is given by The argument of the square root is a non-positive real number, if and only if belongs to one of the intervals and of the imaginary axis. If the argument of the logarithm is real, then it is positive. Thus this formula defines a principal value for arsinh, with branch cuts and . This is optimal, as the branch cuts must connect the singular points and to the infinity. Principal value of the inverse hyperbolic cosine The formula for the inverse hyperbolic cosine given in is not convenient, since similar to the principal values of the logarithm and the square root, the principal value of arcosh would not be defined for imaginary . Thus the square root has to be factorized, leading to The principal values of the square roots are both defined, except if belongs to the real interval . If the argument of the logarithm is real, then is real and has the same sign. Thus, the above formula defines a principal value of arcosh outside the real interval , which is thus the unique branch cut. Principal values of the inverse hyperbolic tangent and cotangent The formulas given in 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 is real. For artanh, this argument is in the real interval , if belongs either to or to . For arcoth, the argument of the logarithm is in , if and only if belongs to the real interval . Therefore, these formulas define convenient principal values, for which the branch cuts are and for the inverse hyperbolic tangent, and for the inverse