Inverse hyperbolic functions

They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-, or with a superscript

Hyperbolic angle measure is the length of an arc of a unit hyperbola

This is analogous to the way circular angle measure is the arc length of an arc of the unit circle in the Euclidean plane or twice the area of the corresponding circular sector.

Alternately hyperbolic angle is the area of a sector of the hyperbola

Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

The earliest and most widely adopted symbols use the prefix arc- (that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).

Especially inconsistent is the conventional use of positive integer superscripts to indicate an exponent rather than function composition, e.g.

Because the argument of hyperbolic functions is not the arc length of a hyperbolic arc in the Euclidean plane, some authors have condemned the prefix arc-, arguing that the prefix ar- (for area) or arg- (for argument) should be preferred.

[6] Following this recommendation, the ISO 80000-2 standard abbreviations use the prefix ar- (that is: arsinh, arcosh, artanh, arsech, arcsch, arcoth).

In computer programming languages, inverse circular and hyperbolic functions are often named with the shorter prefix a- (asinh, etc.).

This article will consistently adopt the prefix ar- for convenience.

they may be solved using the quadratic formula and then written in terms of the natural logarithm.

For such a function, it is common to define a principal value, which is a single valued analytic function which coincides with one specific branch of the multivalued function, over a domain consisting of the complex plane in which a finite number of arcs (usually half lines or line segments) have been removed.

This defines a single valued analytic function, which is defined everywhere, except for non-positive real values of the variables (where the two square roots have a zero real part).

This principal value of the square root function is denoted

For all inverse hyperbolic functions, the principal value may be defined in terms of principal values of the square root and the logarithm function.

However, in some cases, the formulas of § Definitions in terms of logarithms do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.

The principal value of the inverse hyperbolic sine is given by The argument of the square root is a non-positive real number, if and only if z belongs to one of the intervals [i, +i∞) and (−i∞, −i] of the imaginary axis.

Thus this formula defines a principal value for arsinh, with branch cuts [i, +i∞) and (−i∞, −i].

This is optimal, as the branch cuts must connect the singular points i and −i to infinity.

The formula for the inverse hyperbolic cosine given in § Inverse hyperbolic cosine is not convenient, since similar to the principal values of the logarithm and the square root, the principal value of arcosh would not be defined for imaginary z.

Thus, the above formula defines a principal value of arcosh outside the real interval (−∞, 1], which is thus the unique branch cut.

For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1].

In view of a better numerical evaluation near the branch cuts, some authors[citation needed] use the following definitions of the principal values, although the second one introduces a removable singularity at z = 0.

The principal value of the square root is thus defined outside the interval [−i, i] of the imaginary line.

Thus, the principal value is defined by the above formula outside the branch cut, consisting of the interval [−i, i] of the imaginary line.

Here, as in the case of the inverse hyperbolic cosine, we have to factorize the square root.

It follows that the principal value of arsech is well defined, by the above formula outside two branch cuts, the real intervals (−∞, 0] and [1, +∞).

In the following graphical representation of the principal values of the inverse hyperbolic functions, the branch cuts appear as discontinuities of the color.

The fact that the whole branch cuts appear as discontinuities, shows that these principal values may not be extended into analytic functions defined over larger domains.

Graphs of the inverse hyperbolic functions
The hyperbolic functions sinh , cosh , and tanh with respect to a unit hyperbola are analogous to circular functions sin , cos , tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
A ray through the unit hyperbola in the point , where is twice the area between the ray, the hyperbola, and the -axis