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 . The two definitions of differ for real values of with . The ones of differ for real values of with . Principal value of the inverse hyperbolic cosecant For the inverse hyperbolic cosecant, the principal value is defined as . It is defined when the arguments of the logarithm and the square root are not non-positive real numbers. The principal value of the square root is thus defined outside the interval of the imaginary line. If the argument of the logarithm is real, then is a non-zero real number, and this implies that the argument of the logarithm is positive. Thus, the principal value is defined by the above formula outside the branch cut, consisting of the interval of the imaginary line. For , there is a singular point that is included in the branch cut. Principal value of the inverse hyperbolic secant Here, as in the case of the inverse hyperbolic cosine, we have to factorize the square root. This gives the principal value If the argument of a square root is real, then is real, and it follows that both principal values of square roots are defined, except if is real and belongs to one of the intervals and . If the argument of the logarithm is real and negative, then is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two branch cuts, the real intervals and . For , there is a singular point that is included in one of the branch cuts. Graphical representation In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color. The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains. In other words, the above defined branch cuts are minimal. See also Complex logarithm Hyperbolic secant distribution ISO 80000-2 List of integrals of inverse hyperbolic functions References Bibliography Herbert Busemann and Paul J. Kelly (1953) Projective Geometry and Projective Metrics, page 207, Academic Press. External links