Inverse hyperbolic functions

The inverse hyperbolic functions.

In mathematics, the inverse hyperbolic functions provide a hyperbolic angle corresponding to a given value of a hyperbolic function. The size of the hyperbolic angle is equal to the area of the corresponding hyperbolic sector of the hyperbola xy = 1, or twice the area of the corresponding sector of the unit hyperbola x² − y² = 1, just as a circular angle is twice the area of the circular sector of the unit circle. Some authors have called inverse hyperbolic functions "area functions" to realize the hyperbolic angles.

Notation

The preferred abbreviations are ar- followed by the hyperbolic function (arsinh, arcosh, etc.;)

However, arc- followed by the hyperbolic function (for example arcsinh, arccosh), are also commonly seen by analogy with the nomenclature for inverse trigonometric functions. The latter are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area.

Other authors prefer to use the notation argsinh, argcosh, argtanh, and so on, where the prefix arg is the abbreviation of the Latin argumentum. In computer science this is often shortened to asinh.

The notation sinh⁻¹(x), cosh⁻¹(x), etc., is 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 (e.g., cosh⁻¹(x) versus cosh(x)⁻¹)

Logarithmic representation

The operators are defined in the complex plane by:

${\begin{aligned}\operatorname {arsinh} \,z&=\operatorname {Log} (z+{\sqrt {z^{2}+1}}\,)\\[2.5ex]\operatorname {arcosh} \,z&=\operatorname {Log} (z+{\sqrt {z+1}}{\sqrt {z-1}}\,)\\[1.5ex]\operatorname {artanh} \,z&={\tfrac {1}{2}}\operatorname {Log} \left({1+z}\right)-{\tfrac {1}{2}}\operatorname {Log} \left({1-z}\right)\\\operatorname {arcoth} \,z&={\tfrac {1}{2}}\operatorname {Log} \left({1+{\frac {1}{z}}}\right)-{\tfrac {1}{2}}\operatorname {Log} \left({1-{\frac {1}{z}}}\right)\\\operatorname {arcsch} \,z&=\operatorname {Log} \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z^{2}}}+1}}\,\right)\\\operatorname {arsech} \,z&=\operatorname {Log} \left({\frac {1}{z}}+{\sqrt {{\frac {1}{z}}+1}}\,{\sqrt {{\frac {1}{z}}-1}}\,\right)\end{aligned}}$

The above square roots are principal square roots, and the $\operatorname {Log} {}$ are principal complex logarithms. For real arguments, i.e., z = x, which return real values, certain simplifications can be made e.g. ${\sqrt {x+1}}{\sqrt {x-1}}={\sqrt {x^{2}-1}}$ and $\ln \left({1+x}\right)-\ln \left({1-x}\right)=\ln \left({\frac {1+x}{1-x}}\right)$, which are not generally true for complex arguments.

$\operatorname {arsinh} (z)$

$\operatorname {arcosh} (z)$

$\operatorname {artanh} (z)$

$\operatorname {arcoth} (z)$

$\operatorname {arsech} (z)$

$\operatorname {arcsch} (z)$

Inverse hyperbolic functions in the complex z-plane: the colour at each point in the plane represents the complex value of the respective function at that point

Series expansions

Expansion series can be obtained for the above functions:

${\begin{aligned}\operatorname {arsinh} \,x&=x-\left({\frac {1}{2}}\right){\frac {x^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n+1}}{(2n+1)}},\qquad \left|x\right|<1\end{aligned}}$

${\begin{aligned}\operatorname {arcosh} \,x&=\ln 2x-\left(\left({\frac {1}{2}}\right){\frac {x^{-2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-6}}{6}}+\cdots \right)\\&=\ln 2x-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-2n}}{(2n)}},\qquad x>1\end{aligned}}$

${\begin{aligned}\operatorname {artanh} \,x&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+{\frac {x^{7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)}},\qquad \left|x\right|<1\end{aligned}}$

${\begin{aligned}\operatorname {arcsch} \,x=\operatorname {arsinh} {\frac {1}{x}}&=x^{-1}-\left({\frac {1}{2}}\right){\frac {x^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{-5}}{5}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{-7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }\left({\frac {(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|>1\end{aligned}}$

${\begin{aligned}\operatorname {arsech} \,x=\operatorname {arcosh} {\frac {1}{x}}&=\ln {\frac {2}{x}}-\left(\left({\frac {1}{2}}\right){\frac {x^{2}}{2}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {x^{4}}{4}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {x^{6}}{6}}+\cdots \right)\\&=\ln {\frac {2}{x}}-\sum _{n=1}^{\infty }\left({\frac {(2n)!}{2^{2n}(n!)^{2}}}\right){\frac {x^{2n}}{2n}},\qquad 0<x\leq 1\end{aligned}}$

${\begin{aligned}\operatorname {arcoth} \,x=\operatorname {artanh} {\frac {1}{x}}&=x^{-1}+{\frac {x^{-3}}{3}}+{\frac {x^{-5}}{5}}+{\frac {x^{-7}}{7}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {x^{-(2n+1)}}{(2n+1)}},\qquad \left|x\right|>1\end{aligned}}$

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

${\begin{aligned}{\frac {d}{dx}}\operatorname {arsinh} \,x&{}={\frac {1}{\sqrt {1+x^{2}}}},{\text{ for all real }}x\\{\frac {d}{dx}}\operatorname {arcosh} \,x&{}={\frac {1}{\sqrt {x^{2}-1}}},{\text{ for all real }}x>1\\{\frac {d}{dx}}\operatorname {artanh} \,x&{}={\frac {1}{1-x^{2}}},{\text{ for all real }}|x|<1\\{\frac {d}{dx}}\operatorname {arcoth} \,x&{}={\frac {1}{1-x^{2}}},{\text{ for all real }}|x|>1\\{\frac {d}{dx}}\operatorname {arsech} \,x&{}={\frac {-1}{x{\sqrt {1-x^{2}}}}},{\text{ for all real }}x\in (0,1)\\{\frac {d}{dx}}\operatorname {arcsch} \,x&{}={\frac {-1}{|x|{\sqrt {1+x^{2}}}}},{\text{ for all real }}x{\text{, except }}0\\\end{aligned}}$

For an example differentiation: let θ = arsinh x, so (where sinh² θ = (sinh θ)²):

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

Composition of hyperbolic and inverse hyperbolic functions

${\begin{aligned}&\sinh(\operatorname {arcosh} \,x)={\sqrt {x^{2}-1}}\quad {\text{for}}\quad |x|>1\\&\sinh(\operatorname {artanh} \,x)={\frac {x}{\sqrt {1-x^{2}}}}\quad {\text{for}}\quad -1<x<1\\&\cosh(\operatorname {arsinh} \,x)={\sqrt {1+x^{2}}}\\&\cosh(\operatorname {artanh} \,x)={\frac {1}{\sqrt {1-x^{2}}}}\quad {\text{for}}\quad -1<x<1\\&\tanh(\operatorname {arsinh} \,x)={\frac {x}{\sqrt {1+x^{2}}}}\\&\tanh(\operatorname {arcosh} \,x)={\frac {\sqrt {x^{2}-1}}{x}}\quad {\text{for}}\quad |x|>1\end{aligned}}$

Addition formulae

$\operatorname {arsinh} \;u\pm \operatorname {arsinh} \;v=\operatorname {arsinh} \left(u{\sqrt {1+v^{2}}}\pm v{\sqrt {1+u^{2}}}\right)$

$\operatorname {arcosh} \;u\pm \operatorname {arcosh} \;v=\operatorname {arcosh} \left(uv\pm {\sqrt {(u^{2}-1)(v^{2}-1)}}\right)$

$\operatorname {artanh} \;u\pm \operatorname {artanh} \;v=\operatorname {artanh} \left({\frac {u\pm v}{1\pm uv}}\right)$

${\begin{aligned}\operatorname {arsinh} \;u+\operatorname {arcosh} \;v&=\operatorname {arsinh} \left(uv+{\sqrt {(1+u^{2})(v^{2}-1)}}\right)\\&=\operatorname {arcosh} \left(v{\sqrt {1+u^{2}}}+u{\sqrt {v^{2}-1}}\right)\end{aligned}}$

Other identities

${\begin{aligned}2\operatorname {arcosh} x&=\operatorname {arcosh} (2x^{2}-1)&\quad {\hbox{ for }}x\geq 1\\4\operatorname {arcosh} x&=\operatorname {arcosh} (8x^{4}-8x^{2}+1)&\quad {\hbox{ for }}x\geq 1\\2\operatorname {arsinh} x&=\operatorname {arcosh} (2x^{2}+1)&\quad {\hbox{ for }}x\geq 0\\4\operatorname {arsinh} x&=\operatorname {arcosh} (8x^{4}+8x^{2}+1)&\quad {\hbox{ for }}x\geq 0\end{aligned}}$

See also

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

References

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

External links

• "Inverse hyperbolic functions", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Inverse hyperbolic functions at MathWorld