The beta function B(p,q), or the beta integral (also called the Eulerian integral of the first kind) is defined by B(p, q) = (Γ(p) * Γ(q)) / Γ(p+q)

Special mathematical functions

Error function (erf) and related functions. the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: erf (z) = 2/√π * \int e^(-t²) dt from 0 to z This integral is a special (non-elementary) sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of x, the error function has the following interpretation: for a random variable Y that is normally distributed with mean 0 and standard deviation 1/√2 , erf x is the probability that Y falls in the range [−x, x].

Functions for computing the factorial of a number. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n: n! = n * (n-1) * (n-2) * ... * 2 * 1 For example, 5! = 5 * 4 * 3 * 2 * 1 = 120 The value of 0! is 1.

Approximations for the gamma function and related functions. The gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers: Γ(x) = (x-1)! The gamma function is defined for all complex numbers except the non-positive integers.

Logistic (Sigmoid) Functions