Inverse Hyperbolic Functions

We explore the intimate connection between inverse hyperbolic functions and natural logarithms, revealing how the former can be elegantly expressed through the latter.

FUNCTIONS FORMULAS
Sinh-1 x ln(x+\sqrt{x^2+1})  For all x
Cosh-1 x ln(x+\sqrt{x^2-1}), x\geq 1
Tanh-1 x \frac{1}{2}ln(\frac{1+x}{1-x}), |x|<1
Cosech-1 x ln(\frac{1}{x}+\frac{\sqrt{1+x^2}}{|x|}), x \neq 0
Sech-1 x ln(\frac{1}{x}+\frac{\sqrt{1-x^2}}{x}), 0<x \leq 1
Coth-1 x \frac{1}{2}ln(\frac{1-x}{1+x}), |x|<1

 

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