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Article 11 — Appendix A.6

arcsch — arc-hyperbolic cosecant function

Category. Mathematics.

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

arcschx ≡ ln[1/x + √(1/x2 + 1)]

2. Graph

Arc-hyperbolic cosecant is antisymmetric function defined everywhere on real axis, except its singular point 0 — so, its domain is (−∞, 0)∪(0, +∞). Function graph is depicted below — fig. 1.

Fig. 1. Graph of the arc-hyperbolic cosecant function y = arcsch x Fig. 1. Graph of the arc-hyperbolic cosecant function y = arcschx.

Function codomain is entire real axis, except 0: (−∞, 0)∪(0, +∞).

3. Identities

Property of antisymmetry:

arcsch−x = −arcschx

Reciprocal argument:

arcsch(1/x) = arsinhx

Sum and difference:

arcschx + arcschy = arcsch{xy / [x√(1 + 1 /x2) + y√(1 + 1 /y2)]}
arcschx − arcschy = arcsch{xy / [y√(1 + 1 /y2) − x√(1 + 1 /x2)]}

4. Support

Arc-hyperbolic cosecant function arcsch is supported in:

Arc-hyperbolic cosecant function of the complex argument arcsch is supported in:

5. How to use

To calculate arc-hyperbolic cosecant of the number:

arcsch(-1);

To calculate arc-hyperbolic cosecant of the current result:

arcsch(Rslt);

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

arcsch(Mem[x]);