Inverse Hyperbolic Functions

The inverse hyperbolic functions, sometimes also called the area hyperbolic functions (Spanier and Oldham 1987, p. 263) are the multivalued function that are the inverse functions of the hyperbolic functions. They are denoted, , , , , and . Variants of these notations beginning with a capital letter are commonly used to denote their principal values (e.g., Harris and Stocker 1998, p. 263).

These functions are multivalued, and hence require branch cuts in the complex plane. Differing branch cut conventions are possible, but those adopted in this work follow those used by the Wolfram Language, summarized below.

function name	function	the Wolfram Language	branch cut(s)
inverse hyperbolic cosecant		ArcCsch[z]
inverse hyperbolic cosine		ArcCosh[z]
inverse hyperbolic cotangent		ArcCoth[z]
inverse hyperbolic secant		ArcSech[z]	and
inverse hyperbolic sine		ArcSinh[z]	and
inverse hyperbolic tangent		ArcTanh[z]	and

The inverse hyperbolic functions as defined in this work have the following ranges for domains on the real line , again following the convention of the Wolfram Language.

function name	function	domain	range
inverse hyperbolic cosecant
inverse hyperbolic cosine
inverse hyperbolic cotangent		or
inverse hyperbolic secant
inverse hyperbolic sine
inverse hyperbolic tangent

They are defined in the complex plane by

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). "Inverse Hyperbolic Functions." §4.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 86-89, 1972.

Beyer, W. H. "Inverse Hyperbolic Functions." CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 181-186, 1987.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, 2000.

Harris, J. W. and Stocker, H. "Area Hyperbolic Functions." Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 263-273, 1998.

Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions." §2.7 in Handbook of Mathematical Formulas and Integrals, 2nd ed.Orlando, FL: Academic Press, pp. 124-128, 2000.

Spanier, J. and Oldham, K. B. "The Inverse Hyperbolic Functions." Ch. 31 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 285-293, 1987.

Trott, M. "Inverse Trigonometric and Hyperbolic Functions." §2.2.5 in The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 180-191, 2004. http://www.mathematicaguidebooks.org/.

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