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[edit] Inverse hyperbolic functions

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 <math>x^{2} - y^{2} = 1</math> in the same way the inverse trigonometric functions compute the arclength of a sector on the unit circle <math>x^{2} + y^{2} = 1.</math> The usual abbreviations for them in mathematics are arsinh, arcsinh (US) 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 in the US, even though they are – technically speaking – misnomers. The prefix arc is the abbreviation for arcus in contrast to the prefix ar which means area.

<math>\operatorname{arsinh}\, x = \ln(x + \sqrt{x^2 + 1})</math>

<math>\operatorname{arcosh}\, x = \ln(x \pm \sqrt{x^2 - 1})</math>

<math>\operatorname{artanh}\, x = \ln\left(\frac{\sqrt{1 - x^2}}{1-x}\right) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)</math>

<math>\operatorname{arcoth}\, x = \ln\frac{\sqrt{x^2 - 1}}{x-1} = \frac{1}{2} \ln\frac{x+1}{x-1}</math>

<math>\operatorname{arsech}\, x = \pm \ln\frac{1 + \sqrt{1 - x^2}}{x}</math>

<math>\operatorname{arcsch}\, x =

\begin{cases}

 \ln\frac{1 - \sqrt{1 + x^2}}{x},  & \mbox{for }x < 0\!\, \\
 \ln\frac{1 + \sqrt{1 + x^2}}{x},  & \mbox{for }x > 0\!\, 

\end{cases}</math>

Expansion series can be obtained for the above functions:

<math>\operatorname{arsinh}\, x</math>

<math>= 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</math>

<math>= \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n+1}} {(2n+1)} , \left| x \right| < 1</math>

<math>\operatorname{arcosh}\, x</math>

<math>= \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)</math>

<math>= \ln 2x - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-2n}} {(2n)} , x > 1</math>

<math>\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)} , \left| x \right| < 1 </math>

<math>\operatorname{arcsch}\, x = \operatorname{arsinh}\, x^{-1}</math>

<math>= 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</math>

<math>= \sum_{n=0}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{-(2n+1)}} {(2n+1)} , \left| x \right| < 1</math>

<math>\operatorname{arsech}\, x = \operatorname{arcosh}\, x^{-1}</math>

<math>= \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)</math>

<math>= \ln \frac{2}{x} - \sum_{n=1}^\infty \left( \frac {(-1)^n(2n)!} {2^{2n}(n!)^2} \right) \frac {x^{2n}} {2n} , 0 < x \le 1 </math>

<math>\operatorname{arcoth}\, x = \operatorname{artanh}\, x^{-1}</math>

<math>= x^{-1} + \frac {x^{-3}} {3} + \frac {x^{-5}} {5} + \frac {x^{-7}} {7} +\cdots</math>

<math>= \sum_{n=0}^\infty \frac {x^{-(2n+1)}} {(2n+1)} , \left| x \right| > 1 </math>

Asymptotic expansion for the arsinh x is given by

<math>\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} }}</math>

[edit] Applications of inverse trigonometric functions and inverse hyperbolic functions to integrals

<math>\int \frac {dx} {\sqrt{1 - x^2}} = \operatorname{arcsin}\, x + {C} = - \operatorname{arccos}\, x + \frac {\pi}{2}+{C}</math>

<math>\int \frac {dx} {\sqrt{x^2 + 1}} = \operatorname{arsinh}\, x + {C} = \ln (x + \sqrt{x^2 + 1}) + {C}</math>

<math>\int \frac {dx} {\sqrt{x^2 - 1}} = \operatorname{arcosh}\, x + {C} = \ln (x + \sqrt{x^2 - 1}) + {C}</math>

<math>\int \sqrt{1 - x^2}\; dx = \frac{\operatorname{arcsin}\, x + x\sqrt{1 - x^2}}{2} + {C}</math>

<math>\int \sqrt{x^2 + 1}\; dx = \frac{\operatorname{arsinh}\, x + x\sqrt{x^2 + 1}}{2} + {C} = \frac{\ln(x + \sqrt{x^2 + 1}) + x\sqrt{x^2 + 1}}{2} + {C}</math>

<math>\int \sqrt{x^2 - 1}\; dx = \frac{- \operatorname{arcosh}\, x + x\sqrt{x^2 - 1}}{2} + {C}

= \frac{- \ln(x + \sqrt{x^2 - 1}) + x\sqrt{x^2 - 1}}{2} + {C}</math>

<math>\int \frac {dx} {1 + x^2} = \operatorname{arctan}\, x + {C}</math>

<math>\int \frac {dx} {1 - x^2} = \operatorname{artanh}\, x + {C} = \frac{1}{2} \ln\frac{1+x}{1-x} + {C}</math>

[edit] Hyperbolic functions for complex numbers

<math>\operatorname{arsinh}\, x = i \arcsin(-i x)</math>

<math>\operatorname{arsinh}\, ix = i \arcsin x</math>

<math>\operatorname{arcosh}\, x = i \arccos x</math>

<math>\operatorname{artanh}\, x = i \arctan(-i x)</math>

<math>\operatorname{artanh}\, ix = i \arctan x</math>

<math>\ 2\sum_{j=n}^{kn-1} \operatorname{artanh}\, \frac{1}{1 + 2\,j} = \ln k</math>