Sigmoidal function is any function with the following properties:

- Monotonically increasing
- Differentiable
- Bounded

Sigmoidal function is useful in fuzzy logic, where it gives a value between 0 and 1 to tell how is the membership of a member to a set, with 1 means the full membership and 0 means not a member.

Usually \(x\) is a measure (e.g. length) between two quantity, \(\ell_1\) and \(\ell_2\), where they are the hard boundaries. In other words, \(x<\ell_1\) implies membership quality \(\mu\)=0 and \(x\ge\ell_2\) implies \(\mu\)=1. Between the boundaries, membership quality of measure \(x\) is determined by a sigmoidal function.

Examples of sigmoidal functions are:

- \(\sin^2\alpha\) where \(\alpha=\frac{\pi}{2}\frac{x-\ell_1}{\ell_2-\ell_1}\)
- \(1/(1+e^{-\lambda x})\) with \(\lambda>0\), here \(\ell_1=-\infty\) and \(\ell_2=\infty\)
- this one is called the “unipolar sigmoidal function”

- \(\dfrac{1-e^{-\lambda x}}{1+e^{-\lambda x}}\) with \(\lambda>0\)
- this one is called the “bipolar sigmoidal function”
- with \(\ell_1=0, \ell_2=\infty\)

- signum function: \((x)=1\) if \(x\ge 0\) and \(f(x)=-1\) if \(x<0\)
- saturation function: \(f(x)=2x\) when \(-0.5<x<0.5\), \(f(x)=-1\) when \(x<-0.5\) and \(f(x)=1\) when \(x>0.5\)
- the op-amp

## Reference

MathWorks’ Documentation on Fuzzy logic toolbox: Membership functions, http://www.mathworks.com/access/helpdesk/help/toolbox/fuzzy/fp608.html