The Art of Interface

Article 11 — Appendix A.13

artanh or arth — arc-hyperbolic tangent function

Category. Mathematics.

Abstract. Arc-hyperbolic tangent: definition, plot, properties and identities.

Reference. This article is a part of Librow scientific formula calculator project.

1. Definition

Arc-hyperbolic tangent is inverse of hyperbolic tangent function. With the help of natural logarithm it can be represented as:

artanhx ≡ ln[(1 + x) /(1 − x)] /2

2. Plot

Arc-hyperbolic tangent is antisymmetric function defined in the range (−1, 1), points x = ±1 are singular ones. Its plot is depicted below — fig. 1.

Fig. 1. Plot of the arc-hyperbolic tangent function y = artanh x. Fig. 1. Plot of the arc-hyperbolic tangent function y = artanhx.

Function codomain is entire real axis.

3. Identities

Property of antisymmetry:

artanh−x = −artanhx

Reciprocal argument:

artanh(1/x) = arcothx

Sum and difference:

artanhx + artanhy = artanh[(x + y) /(1 + xy)]
artanhx − artanhy = artanh[(x − y) /(1 − xy)]

4. Support

Arc-hyperbolic tangent function artanh or arth of the real argument is supported by free version of the Librow calculator.

Arc-hyperbolic tangent function artanh or arth of the complex argument is supported by professional version of the Librow calculator.

5. How to use

To calculate arc-hyperbolic tangent of the number:

artanh(-.5);

To calculate arc-hyperbolic tangent of the current result:

artanh(rslt);

To calculate arc-hyperbolic tangent of the number x in memory:

artanh(mem[x]);