TOPICS

# 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.

They are defined in the complex plane by

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

Hyperbolic Functions, Inverse Function, Inverse Hyperbolic Cosecant, Inverse Hyperbolic Cosine, Inverse Hyperbolic Cotangent, Inverse Hyperbolic Secant, Inverse Hyperbolic Sine, Inverse Hyperbolic Tangent, Inverse Trigonometric Functions

## 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.

## Referenced on Wolfram|Alpha

Inverse Hyperbolic Functions

## Cite this as:

Weisstein, Eric W. "Inverse Hyperbolic Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicFunctions.html