Category. Mathematics.
Abstract. Arc-hyperbolic sine: definition, plot, properties and identities.
Reference. This article is a part of Librow scientific formula calculator project.
Arc-hyperbolic sine is inverse of hyperbolic sine function. With the help of natural logarithm it can be represented as:
arsinhx ≡ ln[x + √(x2 + 1)]Arc-hyperbolic sine is antisymmetric function defined everywhere on real axis. Its plot is depicted below — fig. 1.
[画像:Fig. 1. Plot of the arc-hyperbolic sine function y = arsinh x.] Fig. 1. Plot of the arc-hyperbolic sine function y = arsinhx.Function codomain is entire real axis.
Property of antisymmetry:
arsinh−x = −arsinhxReciprocal argument:
arsinh(1/x) = arcschxSum and difference:
arsinhx + arsinhy = arsinh[x√(y2 + 1) + y√(x2 + 1)]Arc-hyperbolic sine function arsinh or arsh of the real argument is supported by free version of the Librow calculator.
Arc-hyperbolic sine function arsinh or arsh of the complex argument is supported by professional version of the Librow calculator.
To calculate arc-hyperbolic sine of the number:
arsinh(-1);To calculate arc-hyperbolic sine of the current result:
arsinh(rslt);To calculate arc-hyperbolic sine of the number x in memory:
arsinh(mem[x]);