Inverse Hyperbolic Cosecant

The inverse hyperbolic cosecant (Zwillinger 1995, p. 481), sometimes called the area hyperbolic cosecant (Harris and Stocker 1998, p. 271) and sometimes denoted (Beyer 1987, p. 181) or (Abramowitz and Stegun 1972, p. 87; Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic cosecant. The variants (Abramowitz and Stegun 1972, p. 87) and (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values. Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). Note that in the notation , is the hyperbolic cosecant and the superscript denotes an inverse function, not the reciprocal.

The principal value of is implemented in the Wolfram Language as ArcCsch[z].

It has special value

(1)

where is the golden ratio.

The inverse hyperbolic cosecant is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at the line segment . This follows from the definition of as

(2)

The derivative of the inverse hyperbolic cosecant is

(3)

and the indefinite integral is

(4)

For a real number , it satisfies

(5)

For , the inverse hyperbolic cosecant has the logarithmic series expansion

			(6)
			(7)

(OEIS A052468 and A052469) about 0. For , it has the Laurent series

			(8)
			(9)
			(10)

(OEIS A055786 and A002595), where is a Legendre polynomial and is a Pochhammer symbol.