Inverse hyperbolic functions explained

In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.

For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle measure, for example

\operatorname{arsinh}(\sinh a) = a and \sinh(\operatorname{arsinh} x) = x. Hyperbolic angle measure is the length of an arc of a unit hyperbola x ^{2} - y ^{2} = 1 as measured in the Lorentzian plane (not the length of a hyperbolic arc in the Euclidean plane), and twice the area of the corresponding hyperbolic sector. 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 xy = 1 . Some authors call the inverse hyperbolic functions hyperbolic area functions.^[1]

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

Notation

The earliest and most widely adopted symbols use the prefix arc- (that is:,,,,,), by analogy with the inverse circular functions (etc.). For a unit hyperbola ("Lorentzian circle") in the Lorentzian plane (pseudo-Euclidean plane of signature)^[2] or in the hyperbolic number plane,^[3] the hyperbolic angle measure (argument to the hyperbolic functions) is indeed the arc length of a hyperbolic arc.

Also common is the notation

\sinh ^{-1} , \cosh ^{-1} , etc.,^{[4] [5]} although care must be taken to avoid misinterpretations of the superscript −1 as an exponent. The standard convention is that \sinh ^{-1} x or \sinh ^{-1} (x) means the inverse function while (\sinh x) ^{-1} or \sinh(x) ^{-1} means the reciprocal 1 / \sinh x. Especially inconsistent is the conventional use of positive integer superscripts to indicate an exponent rather than function composition, e.g. \sinh ^{2} x conventionally means (\sinh x) ^{2} and not \sinh(\sinh x).

Because the argument of hyperbolic functions is not the arclength 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:,,,,,).

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

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

Definitions in terms of logarithms

Since the hyperbolic functions are quadratic rational functions of the exponential function

\exp x, they may be solved using the quadratic formula and then written in terms of the natural logarithm.\begin{align} \operatorname{arsinh} x & = ln \left(x + \sqrt{x ^{2} + 1 }\right) & -inf ty &< x < inf ty, \\[ 10 mu] \operatorname{arcosh} x & = ln \left(x + \sqrt{x ^{2} - 1 }\right) & 1 &\leq x < inf ty, \\[ 10 mu] \operatorname{artanh} x & = (12ln (1+ x)/(1- x))/(&) -1 &< x < 1 , \\[ 10 mu] \operatorname{arcsch} x & = ln \left( (1 x)/(\sqrt{ +) (1 {x)/(^{2}}) + 1 }\right) & -inf ty &< x < inf ty, x ≠ 0 , \\[ 10 mu] \operatorname{arsech} x & = ln \left( (1 x)/(\sqrt{ +) (1 {x)/(^{2}}) - 1 }\right) & 0 &< x \leq 1 , \\[ 10 mu] \operatorname{arcoth} x & = (12ln (x +1)/(x -1))/(&) -inf ty &< x < -1 or 1 < x < inf ty. \end{align}

For complex arguments, the inverse circular and hyperbolic functions, the square root, and the natural logarithm are all multi-valued functions.

Addition formulae

\operatorname{arsinh} u \pm \operatorname{arsinh} v = \operatorname{arsinh} \left(u \sqrt{ 1 + v ^{2}} \pm v \sqrt{ 1 + u ^{2}\right)} \operatorname{arcosh} u \pm \operatorname{arcosh} v = \operatorname{arcosh} \left(u v \pm \sqrt{(u ^{2} - 1 ) (v ^{2} - 1 )}\right) \operatorname{artanh} u \pm \operatorname{artanh} v = \operatorname{artanh} \left( (u \pm v)/(1 \pm uv) \right) \operatorname{arcoth} u \pm \operatorname{arcoth} v = \operatorname{arcoth} \left( (1 \pm uv)/(u \pm v) \right) \begin{align}\operatorname{arsinh} u + \operatorname{arcosh} v & = \operatorname{arsinh} \left(u v + \sqrt{( 1 + u ^{2)} (v ^{2} - 1 )}\right) \\ & = \operatorname{arcosh} \left(v \sqrt{ 1 + u ^{2}} + u \sqrt{v ^{2} - 1 }\right) \end{align}

Other identities

\begin{align} 2 \operatorname{arcosh}x& = \operatorname{arcosh}( 2 x ^{2-1)} & \hbox{ for }x\geq 1 \\ 4 \operatorname{arcosh}x& = \operatorname{arcosh}( 8 x ^{4-8x} ^{2+1)} & \hbox{ for }x\geq 1 \\ 2 \operatorname{arsinh}x& = \operatorname{arcosh}( 2 x ^{2+1)} & \hbox{ for }x\geq 0 \\ 4 \operatorname{arsinh}x& = \operatorname{arcosh}( 8 x ^{4+8x} ^{2+1)} & \hbox{ for }x\geq 0 \end{align} ln (x) = \operatorname{arcosh} \left( (x ^{2} +1)/(2 x) \right) = \operatorname{arsinh} \left( (x ^{2} -1)/(2 x) \right) = \operatorname{artanh} \left( (x ^{2} -1)/(x ^{2} +1) \right)

Composition of hyperbolic and inverse hyperbolic functions

\begin{align} &\sinh(\operatorname{arcosh}x) = \sqrt{x ^{2} - 1 } for |x| > 1 \\ &\sinh(\operatorname{artanh}x) = (x)/(\sqrt{ 1- x ^{2}) } \quad \text \quad -1 < x < 1 \\ &\cosh(\operatornamex) = \sqrt \\ &\cosh(\operatornamex) = \frac \quad \text \quad -1 < x < 1 \\ &\tanh(\operatornamex) = \frac \\ &\tanh(\operatornamex) = \frac \quad \text \quad |x| > 1\end

Composition of inverse hyperbolic and circular functions

\operatorname{arsinh} \left( \tan \alpha \right) = \operatorname{artanh} \left( \sin \alpha \right) = ln \left( (1+ \sin \alpha)/(\cos \alpha) \right) = \pm \operatorname{arcosh} \left( (1)/(\cos \alpha) \right) ln \left( \left| \tan \alpha \right|\right) = - \operatorname{artanh} \left( \cos 2 \alpha \right) ^[7]

Conversions

ln x = \operatorname{artanh} \left( (x ^{2-1})/(x ^{2+1}) \right) = \operatorname{arsinh} \left( (x ^{2-1})/(2 x) \right) = \pm \operatorname{arcosh} \left( (x ^{2+1})/(2 x) \right) \operatorname{artanh} x = \operatorname{arsinh} \left( (x)/(\sqrt{ 1- x ^{2}) }\right)= \pm \operatorname \left(\frac\right)\operatorname{arsinh} x = \operatorname{artanh} \left( (x)/(\sqrt{ 1+ x ^{2}) }\right)= \pm \operatorname \left(\sqrt\right)\operatorname{arcosh} x = \left| \operatorname{arsinh} \left( \sqrt{x ^{2-1}\right)} \right| = \left| \operatorname{artanh} \left( (\sqrt{x ^{2-1})/() }\right) \right|

Derivatives

\begin{align} (d)/(dx) \operatorname{arsinh} x & {} = (1)/(\sqrt{x ^{2+1}) }, \text x\\\frac \operatorname x & = \frac, \text x>1\\\frac \operatorname x & = \frac, \text |x|<1\\\frac \operatorname x & = \frac, \text |x|>1\\\frac \operatorname x & = \frac, \text x \in (0,1)\\\frac \operatorname x & = \frac

\sqrt

, \text x\text 0\\\end

These formulas can be derived in terms of the derivatives of hyperbolic functions. For example, if

x = \sinh \theta , then $dx/d\backslash theta\; =\; \backslash cosh\; \backslash theta\; =\; \backslash sqrt,$ so(d)/(dx) \operatorname{arsinh}(x) = (d \theta)/(dx) = (1)/(dx/d\theta) = (1)/(\sqrt{ 1+ x ^{2}) }.

Series expansions

Expansion series can be obtained for the above functions:

\begin{align}\operatorname{arsinh} x & = x - \left( (1)/(2) \right) (x ^{3})/(3) + \left( (1 ⋅ 3)/(2 ⋅ 4) \right) (x ^{5})/(5) - \left( (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) \right) (x ^{7})/(7) \pm … \\ & = \sum ^{inf ty} _{n =0} \left( (( -1 ) ^{n(2n)!})/(2 ^{2n} (n!) ^{2}) \right) (x ^{2n+1})/(2 n +1) , \left| x \right| < 1 \end{align} \begin{align}\operatorname{arcosh} x & = ln ( 2 x) - \left( \left( (1)/(2) \right) (x ^{-2})/(2) + \left( (1 ⋅ 3)/(2 ⋅ 4) \right) (x ^{-4})/(4) + \left( (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) \right) (x ^{-6})/(6) + … \right) \\ & = ln ( 2 x) - \sum ^{inf ty} _{n =1} \left( (( 2 n)!)/(2 ^{2n} (n!) ^{2}) \right) (x ^{-2n})/(2 n) , \left| x \right| > 1 \end{align} \begin{align}\operatorname{artanh} x & = x + (x ^{3})/(3) + (x ^{5})/(5) + (x ^{7})/(7) + … \\ & = \sum ^{inf ty} _{n =0} (x ^{2n+1})/(2 n +1) , \left| x \right| < 1 \end{align} \begin{align}\operatorname{arcsch} x = \operatorname{arsinh} (1 x)/(&) = x ^{-1} - \left( (1)/(2) \right) (x ^{-3})/(3) + \left( (1 ⋅ 3)/(2 ⋅ 4) \right) (x ^{-5})/(5) - \left( (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) \right) (x ^{-7})/(7) \pm … \\ & = \sum ^{inf ty} _{n =0} \left( (( -1 ) ^{n(2n)!})/(2 ^{2n} (n!) ^{2}) \right) (x ^{-(2n+1)})/(2 n +1) , \left| x \right| > 1 \end{align} \begin{align}\operatorname{arsech} x = \operatorname{arcosh} (1 x)/(&) = ln (2)/(x) - \left( \left( (1)/(2) \right) (x ^{2})/(2) + \left( (1 ⋅ 3)/(2 ⋅ 4) \right) (x ^{4})/(4) + \left( (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) \right) (x ^{6})/(6) + … \right) \\ & = ln (2)/(x) - \sum ^{inf ty} _{n =1} \left( (( 2 n)!)/(2 ^{2n} (n!) ^{2}) \right) (x ^{2n})/(2 n) , 0 < x \le 1 \end{align} \begin{align}\operatorname{arcoth} x = \operatorname{artanh} (1 x)/(&) = x ^{-1} + (x ^{-3})/(3) + (x ^{-5})/(5) + (x ^{-7})/(7) + … \\ & = \sum ^{inf ty} _{n =0} (x ^{-(2n+1)})/(2 n +1) , \left| x \right| > 1 \end{align} An asymptotic expansion for arsinh is given by\operatorname{arsinh} x = ln ( 2 x) + \sum\limits ^{inf ty} _{n =1} {\left( { - 1 } \right) ^{n} ({\left( { 2 n -1)/(\right)!!}}{{ 2 n\left() { 2 n} \right)!!}}} (1)/({x ^{2n}) }

Principal values in the complex plane

As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic, except at a finite number of points. 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. These arcs are called branch cuts. For specifying the branch, that is, defining which value of the multivalued function is considered at each point, one generally define it at a particular point, and deduce the value everywhere in the domain of definition of the principal value by analytic continuation. When possible, it is better to define the principal value directly—without referring to analytic continuation.

For example, for the square root, the principal value is defined as the square root that has a positive real part. 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

\sqrt x in what follows. Similarly, the principal value of the logarithm, denoted \operatorname{Log} in what follows, is defined as the value for which the imaginary part has the smallest absolute value. It is defined everywhere except for non-positive real values of the variable, for which two different values of the logarithm reach the minimum.

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 do not give a correct principal value, as giving a domain of definition which is too small and, in one case non-connected.

Principal value of the inverse hyperbolic sine

The principal value of the inverse hyperbolic sine is given by

\operatorname{arsinh} z = \operatorname{Log}(z + \sqrt{z ^{2} + 1 } ).

The argument of the square root is a non-positive real number, if and only if belongs to one of the intervals and of the imaginary axis. If the argument of the logarithm is real, then it is positive. Thus this formula defines a principal value for arsinh, with branch cuts and . This is optimal, as the branch cuts must connect the singular points and to infinity.

Principal value of the inverse hyperbolic cosine

The formula for the inverse hyperbolic cosine given in 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 . Thus the square root has to be factorized, leading to

\operatorname{arcosh} z = \operatorname{Log}(z + \sqrt{z +1 } \sqrt{z -1 } ).

The principal values of the square roots are both defined, except if belongs to the real interval . If the argument of the logarithm is real, then is real and has the same sign. Thus, the above formula defines a principal value of arcosh outside the real interval, which is thus the unique branch cut.

Principal values of the inverse hyperbolic tangent and cotangent

The formulas given in suggests

\begin{align} \operatorname{artanh} z & = (12 \operatorname{Log}\left( (1+ z)/(1- z))/(\right)) \\ \operatorname{arcoth} z & = (12 \operatorname{Log}\left( (z +1)/(z -1))/(\right)) \end{align} for the definition of the principal values of the inverse hyperbolic tangent and cotangent. In these formulas, the argument of the logarithm is real if and only if is real. For artanh, this argument is in the real interval, if belongs either to or to . For arcoth, the argument of the logarithm is in, if and only if belongs to the real interval .

Therefore, these formulas define convenient principal values, for which the branch cuts are and for the inverse hyperbolic tangent, and for the inverse hyperbolic cotangent.

In view of a better numerical evaluation near the branch cuts, some authors use the following definitions of the principal values, although the second one introduces a removable singularity at . The two definitions of

\operatorname {artanh} differ for real values of z with z > 1 . The ones of \operatorname {arcoth} differ for real values of z with z \in [ 0 , 1 ) .\begin{align} \operatorname{artanh} z & = \tfrac 12 \operatorname{Log}\left({ 1+ z}\right) - \tfrac 12 \operatorname{Log}\left({ 1- z}\right) \\ \operatorname{arcoth} z & = \tfrac 12 \operatorname{Log}\left({ 1+ (1)/(z) }\right) - \tfrac 12 \operatorname{Log}\left({ 1- (1)/(z) }\right) \end

Principal value of the inverse hyperbolic cosecant

For the inverse hyperbolic cosecant, the principal value is defined as

\operatorname{arcsch} z = \operatorname{Log}\left( (1)/(z) + \sqrt{ (1)/(z ^{2}) +1 } \right) .

It is defined except when the arguments of the logarithm and the square root are non-positive real numbers. The principal value of the square root is thus defined outside the interval of the imaginary line. If the argument of the logarithm is real, then is a non-zero real number, and this implies that the argument of the logarithm is positive.

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

(At, there is a singular point that is included in the branch cut.)

Principal value of the inverse hyperbolic secant

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

\operatorname{arsech} z = \operatorname{Log}\left( (1)/(z) + \sqrt{ (1)/(z) + 1 } \sqrt{ (1)/(z) -1 } \right).

If the argument of a square root is real, then is real, and it follows that both principal values of square roots are defined, except if is real and belongs to one of the intervals and . If the argument of the logarithm is real and negative, then is also real and negative. It follows that the principal value of arsech is well defined, by the above formula outside two branch cuts, the real intervals and .

For, there is a singular point that is included in one of the branch cuts.

Graphical representation

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. In other words, the above defined branch cuts are minimal.

See also

• Complex logarithm
• Hyperbolic secant distribution
• ISO 80000-2
• List of integrals of inverse hyperbolic functions

Bibliography

• Herbert Busemann and Paul J. Kelly (1953) Projective Geometry and Projective Metrics, page 207, Academic Press.

Notes and References

1. For example: Book: Weltner . Klaus . John . Sebastian . Weber . Wolfgang J. . Schuster . Peter . Grosjean . Jean . 1 . Mathematics for Physicists and Engineers . Springer . 2014 . 2009 . 2nd . 978-364254124-7 . Book: Durán, Mario . 2012 . Mathematical methods for wave propagation in science and engineering . 1 . Ediciones UC . 9789561413146 . 89.
2. Trigonometry in Lorentzian Geometry . Birman . Graciela S. . Nomizu . Katsumi . American Mathematical Monthly . 1984 . 91 . 9 . 543–549 . 10.1080/00029890.1984.11971490 . 2323737 .
3. Sobczyk . Garret . The hyperbolic number plane . College Mathematics Journal . 26 . 4 . 1995 . 268–280. 10.1080/07468342.1995.11973712 .
4. Web site: Weisstein . Eric W.. Inverse Hyperbolic Functions . 2020-08-30 . Wolfram Mathworld . Web site: Inverse hyperbolic functions . 2020-08-30 . Encyclopedia of Mathematics.
5. Book: Press . W.H. . Teukolsky . S.A. . Vetterling . WT . Flannery . B.P. . 1992 . Numerical Recipes in FORTRAN . 2nd . Cambridge University Press . 0-521-43064-X . § 5.6. Quadratic and Cubic Equations. Book: Woodhouse, N.M.J. . N. M. J. Woodhouse . Special Relativity . Springer . 2003 . 71 . 1-85233-426-6.
6. Book: Gullberg, Jan . Jan Gullberg . Mathematics: From the Birth of Numbers . W. W. Norton . 1997 . 039304002X . 539 . Another form of notation,,, etc., is a practice to be condemned as these functions have nothing whatever to do with arc, but with area, as is demonstrated by their full Latin names, ¶   area sinus hyperbolicus ¶   area cosinus hyperbolicus, etc.. Book: Zeidler . Eberhard . Eberhard Zeidler (mathematician) . Hackbusch . Wolfgang . Wolfgang Hackbusch . Schwarz . Hans Rudolf . Hunt . Bruce . . Oxford University Press . 2004 . 0198507631 . § 0.2.13 The inverse hyperbolic functions . 68 . The Latin names for the inverse hyperbolic functions are area sinus hyperbolicus, area cosinus hyperbolicus, area tangens hyperbolicus and area cotangens hyperbolicus (of)..... . Zeidler & al. use the notations, etc.; note that the quoted Latin names are back-formations, invented long after Neo-Latin ceased to be in common use in mathematical literature. Book: Bronshtein . Ilja N. . Ilja N. Bronshtein . Semendyayev . Konstantin A. . Konstantin A. Semendyayev . Musiol . Gerhard . Heiner . Mühlig . . Springer . 5th . 2007 . 978-3540721215 . 10.1007/978-3-540-72122-2 . § 2.10: Area Functions . 91 . The area functions are the inverse functions of the hyperbolic functions, i.e., the inverse hyperbolic functions. The functions,, and are strictly monotone, so they have unique inverses without any restriction; the function has two monotonic intervals so we can consider two inverse functions. The name area refers to the fact that the geometric definition of the functions is the area of certain hyperbolic sectors .... Book: Bacon, Harold Maile . 1942 . Differential and Integral Calculus . McGraw-Hill . 203 .
7. Web site: Identities with inverse hyperbolic and trigonometric functions. math stackexchange. stackexchange. 3 November 2016.