Inverse Hyperbolic Functions

This exploration delves into the intricate connection between inverse hyperbolic functions and natural logarithms. We’ll clarify their domains and ranges, revealing how these functions are fundamentally expressed using logarithmic forms.

Function	Formulas
Sinh^-1x	$ln(x+\sqrt{x^2+1}), \forall \,\,\ x$
Cosh^-1x	$ln(x+\sqrt{x^2-1}), \forall \,\,\ x\geq 1$
Tanh^-1x	$\frac{1}{2}ln(\frac{1+x}{1-x}) , \|x\| < 1$
Cosech^-1x	$ln(\frac{1}{x}+\frac{\sqrt{1+x^2}}{\|x\|}) , x \neq 0$
Sech^-1x	$ln(\frac{1}{x}+\frac{\sqrt{1-x^2}}{\|x\|}) , 0 < x \leq 1$
Coth^-1x	$\frac{1}{2}ln(\frac{x+1}{x-1}) , \|x\| < 1$

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