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).

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.

Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998.

Jeffrey, A. Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, 2000.

Sloane, N. J. A. Sequences A052468 and A052469 in "The On-Line Encyclopedia of Integer Sequences."

Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.