Category. Mathematics.
Abstract. Arc-hyperbolic secant: definition, plot, properties and identities.
Reference. This article is a part of Librow scientific formula calculator project.
Arc-hyperbolic secant is inverse of hyperbolic secant function. With the help of natural logarithm it can be represented as:
arsechx ≡ ln{[1 + √(1 − x2)] /x}Arc-hyperbolic secant is monotone function defined in the range (0, 1], point x = 0 is singular one. Its plot is depicted below — fig. 1.
[画像:Fig. 1. Plot of the arc-hyperbolic secant function y = arsech x.] Fig. 1. Plot of the arc-hyperbolic secant function y = arsechx.Function codomain is non-negative part of real axis: [0, +∞).
Reciprocal argument:
arsech(1/x) = arcoshxSum and difference:
arsechx + arsechy = arsech(xy /{1 + √[(1 − x2)(1 − y2)]})Arc-hyperbolic secant function arsech or arsch of the real argument is supported by free version of the Librow calculator.
Arc-hyperbolic secant function arsech or arsch of the complex argument is supported by professional version of the Librow calculator.
To calculate arc-hyperbolic secant of the number:
arsech(1);To calculate arc-hyperbolic secant of the current result:
arsech(rslt);To calculate arc-hyperbolic secant of the number x in memory:
arsech(mem[x]);