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 or 'bell curve' is a mathematical function having a characteristic "bell"-shaped curve. These functions are typically continuous or smooth, asymptotically approach zero for large negative/positive x, and have a single, unimodal maximum at small x. Hence, the integral of a bell-shaped function is typically a sigmoid function. Bell curves are also commonly symmetric.

Many common probability distribution functions are bell curves.

Some examples include:

• Gaussian function or normal distribution. This is the prototypical bell curve and is frequently encountered in nature as a consequence of the central limit theorem.

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

• Fuzzy Logic generalized membership bell-shaped function

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

• Hyperbolic secant. This is also the derivative of the Gudermannian function.

$f(x)=\operatorname {sech} (x)={\frac {2}{e^{x}+e^{-x}}}$

• Witch of Agnesi, the probability density function of the Cauchy distribution. This is also a scaled version of the derivative of the arctangent function.

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

• Bump function

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

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

$f(x;\mu ,s)={\begin{cases}{\frac {1}{2s}}\left[1+\cos \left({\frac {x-\mu }{s}}\pi \right)\right]&{\text{for }}\mu -s\leq x\leq \mu +s,\\[3pt]0&{\text{otherwise.}}\end{cases}}$

• Most of the window functions like the Kaiser window

• The derivative of the logistic function. This is a scaled version of the derivative of the hyperbolic tangent function.

$f(x)={\frac {e^{x}}{\left(1+e^{x}\right)^{2}}}$

• Some algebraic functions. For example

$f(x)={\frac {1}{(1+x^{2})^{3/2}}}$