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Principal value of the inverse hyperbolic cosecant

For the inverse hyperbolic cosecant, the principal value is defined as

It is defined except when the arguments of the logarithm and the square root are 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 z 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.

(At z = 0, there is a singular point that is included in the branch cut.)