•
Logistic function f(x) = \frac{1}{1 + e^{-x}} •
Hyperbolic tangent (shifted and scaled version of the logistic function, above) f(x) = \tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}} •
Arctangent function f(x) = \arctan x •
Gudermannian function f(x) = \operatorname{gd}(x) = \int_0^x \frac{dt}{\cosh t} = 2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) •
Error function f(x) = \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt •
Generalised logistic function f(x) = \left(1 + e^{-x} \right)^{-\alpha}, \quad \alpha > 0 •
Smoothstep function f(x) = \begin{cases} {\displaystyle \left( \int_0^1 \left(1 - u^2\right)^N du \right)^{-1} \int_0^x \left( 1 - u^2 \right)^N \ du}, & |x| \le 1 \\ \\ \sgn(x) & |x| \ge 1 \\ \end{cases} \quad N \in \mathbb{Z} \ge 1 • Some
algebraic functions, for example f(x) = \frac{x}{\sqrt{1+x^2}} • and in a more general form normalized to (−1,1): f(x) = \begin{cases} {\displaystyle 2\frac{e^{\frac{1}{u}}}{e^{\frac{1}{u}}+e^{\frac{-1}{1+u}}} - 1}, u=\frac{x+1}{-2}, & |x| AManWithNoPlan simplified below --> \begin{align}f(x) &= \begin{cases} {\displaystyle \frac{2}{1+e^{-2m\frac{x}{1-x^2}}} - 1}, & |x| using the hyperbolic tangent mentioned above. Here, m is a free parameter encoding the slope at x=0, which must be greater than or equal to \sqrt{3} because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid. This function is unusual because it actually attains the limiting values of −1 and 1 within a finite range, meaning that its value is constant at −1 for all x \leq -1 and at 1 for all x \geq 1. Nonetheless, it is
smooth (infinitely differentiable, C^\infty)
everywhere, including at x = \pm 1. == Applications ==