Hyperbolic functions: Difference between revisions

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

</math>

:<math>\operatorname{arctanh} (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{arccotharccsch} (x) = \operatorname{arcsinh} (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)} , \left| x \right| >< 1 </math>

:<math>\operatorname{arcsech} (x) = \operatorname{arccosh} (x^{-1}) = \ln 2 - (\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 ) = \ln 2 - \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{arccoth} (x) = \operatorname{arctanh} (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)} , \left| x \right| > 1 </math>

==Applications of inverse trigonometric functions and inverse hyperbolic functions to integrals==