Editing Sigmoid function (section)

== Examples ==
[[File:Gjl-t(x).svg|thumb|320px|right|Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at the origin is 1.]]

* [[Logistic function]] <math display="block"> f(x) = \frac{1}{1 + e^{-x}} </math>
* [[Hyperbolic tangent]] (shifted and scaled version of the logistic function, above) <math display="block"> f(x) = \tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}} </math>
* [[Arctangent function]] <math display="block"> f(x) = \arctan x </math>
* [[Gudermannian function]] <math display="block"> f(x) = \operatorname{gd}(x) = \int_0^x \frac{dt}{\cosh t} = 2\arctan\left(\tanh\left(\frac{x}{2}\right)\right) </math>
* [[Error function]] <math display="block"> f(x) = \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt </math>
* [[Generalised logistic function]] <math display="block"> f(x) = \left(1 + e^{-x} \right)^{-\alpha}, \quad \alpha > 0 </math>
* [[Smoothstep]] function <math display="block"> 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 </math>
* Some [[algebraic function]]s, for example <math display="block"> f(x) = \frac{x}{\sqrt{1+x^2}} </math>
* and in a more general form<ref name="Dunning-Kensler-Coudeville-Bailleux_2015" /> <math display="block"> f(x) = \frac{x}{\left(1 + |x|^{k}\right)^{1/k}} </math>
* Up to shifts and scaling, many sigmoids are special cases of <math display="block"> f(x) = \varphi(\varphi(x, \beta), \alpha) , </math> where <math display="block"> \varphi(x, \lambda) = \begin{cases} (1 - \lambda x)^{1/\lambda} & \lambda \ne 0 \\e^{-x} & \lambda = 0 \\  \end{cases} </math> is the inverse of the negative [[Box–Cox transformation]], and <math>\alpha < 1</math> and <math>\beta < 1</math> are shape parameters.<ref name="grex" />
* [[Non-analytic_smooth_function#Smooth_transition_functions|Smooth transition function]]<ref>{{Cite web|url=https://www.youtube.com/watch?v=vD5g8aVscUI|title=Smooth Transition Function in One Dimension &#124; Smooth Transition Function Series Part 1|via=www.youtube.com| date =16 August 2022|author=EpsilonDelta|at=13:29/14:04}}</ref> normalized to (−1,1):
<!--
<math display="block"> 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| < 1 \\
\\
\sgn(x) & |x| \ge 1 \\
\end{cases}</math> AManWithNoPlan simplified below -->
<math display="block">\begin{align}f(x) &= \begin{cases}
{\displaystyle
\frac{2}{1+e^{-2m\frac{x}{1-x^2}}} - 1}, & |x| < 1 \\
\\
\sgn(x) & |x| \ge 1 \\
\end{cases} \\
&= \begin{cases}
{\displaystyle
\tanh\left(m\frac{x}{1-x^2}\right)}, & |x| < 1 \\
\\
\sgn(x) & |x| \ge 1 \\
\end{cases}\end{align}</math> using the hyperbolic tangent mentioned above.  Here, <math>m</math> is a free parameter encoding the slope at <math>x=0</math>, which must be greater than or equal to <math>\sqrt{3}</math> 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 <math>x \leq -1</math> and at 1 for all <math>x \geq 1</math>.  Nonetheless, it is [[Smoothness|smooth]] (infinitely differentiable, <math>C^\infty</math>) ''everywhere'', including at <math>x = \pm 1</math>.