 Hyperbolic tangent

The hyperbolic tangent isFor real $x$, it is an odd, strictly increasing smooth function mapping $\mathbb{R}$ onto $(-1,1)$. The exponential expression gives its limits $-1$ and $1$ at the two infinities. The quotient rule and the hyperbolic Pythagorean identity giveHence it is nearly linear near zero, with $\tanh x=x+O(x^3)$, and saturates for large input magnitude. Its real inverse function is $\mathrm{artanh}y=\frac{1}{2}\log ((1+y)/(1-y))$ for $|y|<1$. The profile $u(x)=\tanh x$ solves $u^{\prime }=1-u^2$, connecting a simple nonlinear ordinary differential equation with a smooth transition between two limiting states.