area functions

The most usual area functions:

  • The inverse function of the hyperbolic sineMathworldPlanetmath (in Latin sinus hyperbolicus) is arsinh (area sini hyperbolici):

  • The inverse function of the hyperbolic cosine (in Latin cosinus hyperbolicus) is arcosh (area cosini hyperbolici):


    It is defined for  x1.

  • The inverse function of the hyperbolic tangent (in Latin tangens hyperbolica) is artanh (area tangentis hyperbolicae):


    It is defined for  -1<x<1.

  • The inverse function of the hyperbolic cotangent (in Latin cotangens hyperbolica) is arcoth (area cotangentis hyperbolicae):


    It is defined for  |x|>1.

These four functionsMathworldPlanetmath are denoted also by sinh-1x, cosh-1x, tanh-1x and coth-1x.



The functions  arsinh  and  artanh  have the Taylor seriesMathworldPlanetmath


Because the inverse tangentMathworldPlanetmath function (see the cyclometric functions) has the   arctanx=x-x33+x55-x77+-(|x|1), we see that


similarly we get


Some other formulae which may be obtained by means of the addition formulae of the hyperbolic functionsDlmfMathworld:


The classic abbreviations “arsinh” and “arcosh” are explained as follows:  The unit hyperbola  x2-y2=1 (its right half) has the parametric


here A means the area by the hyperbola and the straight line segments OP and OQ, where O is the origin, P is the point  (x,y)  of the hyperbola and Q is the point  (x,-y)  of the hyperbola.  Thus, conversely, A is the area having hyperbolic cosine equal to x (area cosini hyperbolici x), similarly A is the area having hyperbolic sine equal to y (area sini hyperbolici y).

Note.  In some countries the abbreviation “ar” in the symbols arsinh etc. is replaced by  “a”, “Ar”, “arc” or “arg”.

Title area functions
Canonical name AreaFunctions
Date of creation 2013-03-22 14:21:18
Last modified on 2013-03-22 14:21:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 38
Author pahio (2872)
Entry type Definition
Classification msc 26A09
Synonym inverse hyperbolic functionsDlmfMathworld
Related topic UnitHyperbola
Related topic CyclometricFunctions
Related topic HyperbolicAngle
Related topic IntegralTables
Related topic IntegrationOfSqrtx21
Related topic IntegralRelatedToArcSine
Related topic ArcLengthOfParabola
Related topic ListOfImproperIntegrals
Related topic InverseGudermannianFunction
Related topic EulersSubstitutionsForIntegration
Related topic ArcoshCurve
Related topic EqualArcLength
Defines arsinh
Defines arcosh
Defines artanh
Defines arcoth