Inverse Hyperbolic Cotangent

The inverse hyperbolic cotangent (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic cotangent (Harris and Stocker 1998, p. 267), is the multivalued function that is the inverse function of the hyperbolic cotangent.

The variants and (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic cotangent, although this distinction is not always made. Worse yet, the notation is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p. 87). The function is sometimes denoted (Jeffrey 2000, p. 124) or (Gradshteyn and Ryzhik 2000, p. xxx). Note that in the notation , is the hyperbolic cotangent and the superscript denotes an inverse function, not the reciprocal.

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

The inverse hyperbolic cotangent 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 . For , this follows from the definition of as

(1)

For , the inverse hyperbolic cotangent is given in terms of the inverse cotangent by

(2)

(Gradshteyn and Ryzhik 2000, p. xxx). For or , this simplifies to

(3)

The derivative is

(4)

and its indefinite integral is

(5)

It has the special values

			(6)
			(7)
			(8)
			(9)

It has series expansions

			(10)
			(11)

for . For , the constant term is . For , it has the expansion

			(12)
			(13)

(OEIS A005408).