Hyperbolic Functions

$$\sinh (x)=\frac{e^x-e^{-x}}{2}$$ $$\cosh (x)=\frac{e^x+e^{-x}}{2}$$ $$\tanh (x)=\frac{\sinh (x)}{\cosh (x)}\text{horizontal asymptotes at −1,1}$$

Invers Hyperbolic Functions

These are the original functions reflected about the line $y=x$. $\cosh x$ has a range of $0\le y\le \infty$.

$${\sinh }^{-1}x=\ln (x+\sqrt{x^2+1})$$ $${\cosh }^{-1}x=\ln (x+\sqrt{x^2-1})$$ $${\tanh }^{-1}x=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right)$$