Inverse hyperbolic function

Inverse hyperbolic function

The inverses of the hyperbolic functions are the area hyperbolic functions. The names hint at the fact that they compute the area of a sector of the unit hyperbola x^{2} - y^{2} = 1 in the same way that the inverse trigonometric functions compute the arclength of a sector on the unit circle x^{2} + y^{2} = 1.The usual abbreviations for them in mathematics are "arsinh, arcsinh" (in the USA) or "asinh" (in computer science). The notation "sinh-1 (x), cosh-1(x)" etc. are also used, despite the fact that care must be taken to avoid misinterpretations of the superscript "-1" as a power as opposed to a shorthand for inverse. The acronyms "arcsinh", "arccosh" etc. are commonly used, even though they are misnomers, since the prefix "arc" is the abbreviation for "arcus", while the prefix "ar" stands for "area".

Logarithmic representation

The operators are defined in the complex plane by:

:operatorname{arsinh}, x = ln(x + sqrt{x^2 + 1}):operatorname{arcosh}, x = ln(x + sqrt{x-1}sqrt{x+1}):operatorname{artanh}, x = lnleft(frac{sqrt{1 - x^2{1-x} ight) = frac{1}{2} lnleft(frac{1+x}{1-x} ight):operatorname{arcsch}, x = lnleft(sqrt{1+frac{1}{x^2+frac{1}{x} ight):operatorname{arsech}, x = lnleft(sqrt{frac{1}{x}-1}sqrt{frac{1}{x}+1}+frac{1}{x} ight):operatorname{arcoth}, x = frac{1}{2} lnfrac{x+1}{x-1}

The above square roots are principal square roots. For real arguments which return real values, certain simplifications can be made e.g. sqrt{x - 1}sqrt{x+1}=sqrt{x^2-1}, which are not generally true when using principal square roots.

eries expansions

Expansion series can be obtained for the above functions:

:operatorname{arsinh}, x::= x - left( frac {1} {2} ight) frac {x^3} {3} + left( frac {1 cdot 3} {2 cdot 4} ight) frac {x^5} {5} - left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} ight) frac {x^7} {7} +cdots::= sum_{n=0}^infty left( frac {(-1)^n(2n)!} {2^{2n}(n!)^2} ight) frac {x^{2n+1 {(2n+1)} , qquad left| x ight| < 1

:operatorname{arcosh}, x::= ln 2x - left( left( frac {1} {2} ight) frac {x^{-2 {2} + left( frac {1 cdot 3} {2 cdot 4} ight) frac {x^{-4 {4} + left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} ight) frac {x^{-6 {6} +cdots ight)::= ln 2x - sum_{n=1}^infty left( frac {(-1)^n(2n)!} {2^{2n}(n!)^2} ight) frac {x^{-2n {(2n)} , qquad x > 1

:operatorname{artanh}, x = x + frac {x^3} {3} + frac {x^5} {5} + frac {x^7} {7} +cdots = sum_{n=0}^infty frac {x^{2n+1 {(2n+1)} , qquad left| x ight| < 1

:operatorname{arcsch}, x = operatorname{arsinh}, x^{-1}::= x^{-1} - left( frac {1} {2} ight) frac {x^{-3 {3} + left( frac {1 cdot 3} {2 cdot 4} ight) frac {x^{-5 {5} - left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} ight) frac {x^{-7 {7} +cdots::= sum_{n=0}^infty left( frac {(-1)^n(2n)!} {2^{2n}(n!)^2} ight) frac {x^{-(2n+1) {(2n+1)} , qquad left| x ight| < 1

:operatorname{arsech}, x = operatorname{arcosh}, x^{-1}::= ln frac{2}{x} - left( left( frac {1} {2} ight) frac {x^{2 {2} + left( frac {1 cdot 3} {2 cdot 4} ight) frac {x^{4 {4} + left( frac {1 cdot 3 cdot 5} {2 cdot 4 cdot 6} ight) frac {x^{6 {6} +cdots ight)::= ln frac{2}{x} - sum_{n=1}^infty left( frac {(-1)^n(2n)!} {2^{2n}(n!)^2} ight) frac {x^{2n {2n} , qquad 0 < x le 1

:operatorname{arcoth}, x = operatorname{artanh}, x^{-1}::= x^{-1} + frac {x^{-3 {3} + frac {x^{-5 {5} + frac {x^{-7 {7} +cdots::= sum_{n=0}^infty frac {x^{-(2n+1) {(2n+1)} , qquad left| x ight| > 1

Asymptotic expansion for the arsinh "x" is given by

:operatorname{arsinh}, x = ln 2x + sumlimits_{n = 1}^infty {left( { - 1} ight)^{n - 1} fracleft( {2n - 1} ight)!!2nleft( {2n} ight)!!} frac{1}x^{2n}

Derivatives

:egin{align}frac{d}{dx} operatorname{arsinh}, x & {}= frac{1}{sqrt{1+x^2\frac{d}{dx} operatorname{arcosh}, x & {}= frac{1}{sqrt{x^2-1\frac{d}{dx} operatorname{artanh}, x & {}= frac{1}{1-x^2}\frac{d}{dx} operatorname{arcoth}, x & {}= frac{1}{1-x^2}\frac{d}{dx} operatorname{arsech}, x & {}= frac{mp 1}{x,sqrt{1-x^2; qquad Re{x} gtrless 0\frac{d}{dx} operatorname{arcsch}, x & {}= frac{mp 1}{x,sqrt{1+x^2; qquad Re{x} gtrless 0end{align}

For an example derivation: let "&theta;" = arsinh "x", so::frac{d,operatorname{arsinh}, x}{dx} = frac{d heta}{d sinh heta} = frac{1} {cosh heta} = frac{1} {sqrt{1+sinh^2 heta = frac{1}{sqrt{1+x^2

ee also

* List of integrals of arc hyperbolic functions

External links

* [http://mathworld.wolfram.com/InverseHyperbolicFunctions.html Inverse hyperbolic functions] at MathWorld


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