Inverse hyperbolic tangent

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The function artanh.

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 x2 − y2 = 1 in the same way that the inverse trigonometric functions compute the arclength of a sector on the unit circle x2 + y2 = 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.

Contents

Logarithmic representation

The operators are defined in the complex plane by:

\begin{align} \operatorname{arsinh}\, x &= \ln(x + \sqrt{x^2 + 1}), \\[2.5ex] \operatorname{arcosh}\, x &= \ln(x + \sqrt{x-1}\sqrt{x+1}), \\[1.5ex] \operatorname{artanh}\, x &= \ln\left(\frac{\sqrt{1 - x^2}}{1-x}\right) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right), \\ \operatorname{arcsch}\, x &= \ln\left(\sqrt{1+\frac{1}{x^2}}+\frac{1}{x}\right), \\ \operatorname{arsech}\, x &= \ln\left(\sqrt{\frac{1}{x}-1}\sqrt{\frac{1}{x}+1}+\frac{1}{x}\right), \\ \operatorname{arcoth}\, x &= \frac{1}{2} \ln\frac{x+1}{x-1}. \end{align}

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.


Inverse hyperbolic functions in the complex plane
\operatorname{arsinh}(z) \operatorname{arcosh}(z) \operatorname{artanh}(z) \operatorname{arcoth}(z) \operatorname{arsech}(z) \operatorname{arcsch}(z)

Series expansions

Expansion series can be obtained for the above functions:

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

Asymptotic expansion for the arsinh x is given by

\operatorname{arsinh}\, x = \ln 2x + \sum\limits_{n = 1}^\infty {\left( { - 1} \right)^{n - 1} \frac{{\left( {2n - 1} \right)!!}}{{2n\left( {2n} \right)!!}}} \frac{1}{{x^{2n} }}

Derivatives

\begin{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{-1}{x(x+1)\,\sqrt{\frac{1-x}{1+x}}}\\ \frac{d}{dx} \operatorname{arcsch}\, x & {}= \frac{-1}{x^2\,\sqrt{1+\frac{1}{x^2}}}\\ \end{align}

For real x:

\begin{align} \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 0 \end{align}

For an example derivation: let θ = arsinh x, so:

\frac{d\,\operatorname{arsinh}\, x}{dx} = \frac{d \theta}{d \sinh \theta} = \frac{1} {\cosh \theta} = \frac{1} {\sqrt{1+\sinh^2 \theta}} = \frac{1}{\sqrt{1+x^2}}

See also

External links

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  • This page was last modified on 1 December 2008, at 11:08.

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