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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