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Article 11 — Appendix A.13

artanh or arth — arc-hyperbolic tangent function

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

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. Graph

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

Fig. 1. Graph of the arc-hyperbolic tangent function y = artanh x Fig. 1. Graph 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[(xy) /(1 − xy)]

4. Support

Arc-hyperbolic tangent function artanh or arth is supported in:

Arc-hyperbolic tangent function of the complex argument artanh or arth is supported in:

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]);