Logistic sigmoid function, varying smoothly from 0 to 1 along the real axis.

Solution to the logistic equation \(y'(x) = -\frac1\beta y(x) \left(1-y(x)\right)\) shifted by \(\mu\).

\[ f_{\mu,\beta}(x) = \frac{1}{1+\exp\left(-\frac{x-\mu}\beta\right)} \]

Smoothstep function, varying symmetrically from 0 to 1 between its parameters. It should be called smoothstepRamp to follow this module’s nomenclature, but the name smoothstep is very much standard.

https://en.wikipedia.org/wiki/Smoothstep

Gaussian falloff. Equivalent to the Gaussian distribution, but normalized to \(f(\mu)=1\), \(\int = \sigma\sqrt{2\pi}\).

\[ f_{\mu,\sigma}(x) = \exp\left(-\frac12\left(\frac{x-\mu}\sigma\right)^2\right) \]

Logistic distribution, normalized to normalized to \(f(\mu)=1\), (int = beta). Compared to a Gaussian falloff, this has a heavier tail (higher kurtosis).

\[ f_{\mu,\beta}(x) = \frac{\exp\left(-\frac{x-\mu}\beta\right)}{\left(1+\exp\left(-\frac{x-\mu}\beta\right)\right)^2} \]

Cauchy falloff. This is a much gentler (quadratic) falloff than Gaussian (exponential). Normalized to \(f(\mu)=1\), \(\int = \pi\gamma\). Its falloff is in fact so slow that it does not have a standard deviation (or any higher moments, for that matter).

\[ f_{\mu,\gamma}(x) = \frac1{1 + \left(\frac{x-\mu}\gamma\right)^2} \]

Smooth bump function: nonzero between -1 and 1, smooth over all of \(\mathbb R\).

\[ f(x) = \begin{cases} \exp \left(-{\frac1{1-x^2}}\right) & x \in (-1,1) \\ 0 & \text{otherwise} \end{cases} \]