Bell-shaped function

Gaussian function

sech(x) (in blue)

Witch of Agnesi

φ_b for b = 1

Raised cosine PDF

Kaiser window

A bell shaped function is a mathematical function having a characteristic "bell"-shaped curve.

Some examples include:

• Gaussian function

$f(x)=ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}$

• Fuzzy Logic generalized membership bell-shaped function

$f(x)={\frac {1}{1+|{\frac {x-c}{a}}|^{2b}}}$

• Hyperbolic secant

$f(x)=sech(x)$

• Witch of Agnesi

$f(x)={\frac {8a^{3}}{x^{2}+4a^{2}}}$

• Bump function

$\phi _{b}(x)={\begin{cases}\exp {\frac {b^{2}}{x^{2}-b^{2}}}&|x|<b\\0&|x|\geq b.\end{cases}}$

• Raised cosines type like the Raised cosine distribution or the Raised-cosine filter

$f(x;\mu ,s)={\frac {1}{2s}}\left[1+\cos \left({\frac {x\!-\!\mu }{s}}\,\pi \right)\right]$for $\mu -s\leq x\leq \mu +s$ and zero otherwise.

• Most of the Window functions like the Kaiser window