Inverse Hyperbolic Secant

Min Max Re Im

The inverse hyperbolic secant (Beyer 1987, p. 181; Zwillinger 1995, p. 481), sometimes called the area hyperbolic secant (Harris and Stocker 1998, p. 271) and sometimes also denoted (Jeffrey 2000, p. 124), is the multivalued function that is the inverse function of the hyperbolic secant. The variants or (Harris and Stocker 1998, p. 263) are sometimes used to refer to explicit principal values of the inverse hyperbolic secant, 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). Note that in the notation , is the hyperbolic secant and the superscript denotes an inverse function, not the multiplicative inverse.

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

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

(1)

For real , it satisfies

(2)

The derivative of the inverse hyperbolic secant is given by

(3)

and its indefinite integral is

(4)

It has Maclaurin series

			(5)
			(6)

(OEIS A052468 and A052469).