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→Applications of inverse hyperbolic functions to integrals: add trig functions to heading - since they're mentioned |
→Inverse hyperbolic functions: add series for some inverse hyperbolic functions (more to follow) |
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:<math>\operatorname{arcsech}(x) = \ln\left(\frac{1 \pm \sqrt{1 - x^2}}{x}\right)</math>
:<math>\operatorname{arccsch}(x) = \ln\left(\frac{1 \pm \sqrt{1 + x^2}}{x}\right)</math>
Expansion series can be obtained for the above functions:
:<math>\operatorname{arcsinh} (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)} , \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
==Applications of inverse trigonometric functions and inverse hyperbolic functions to integrals==
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