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# Module `std.mathspecial`

Mathematical Special Functions

The technical term 'Special Functions' includes several families of transcendental functions, which have important applications in particular branches of mathematics and physics.

The gamma and related functions, and the error function are crucial for mathematical statistics. The Bessel and related functions arise in problems involving wave propagation (especially in optics). Other major categories of special functions include the elliptic integrals (related to the arc length of an ellipse), and the hypergeometric functions.

## Status

Many more functions will be added to this module. The naming convention for the distribution functions (gammaIncomplete, etc) is not yet finalized and will probably change.

## Functions

NameDescription
``` beta(x, y) ``` Beta function
``` betaIncomplete(a, b, x) ``` Incomplete beta integral
``` betaIncompleteInverse(a, b, y) ``` Inverse of incomplete beta integral
``` digamma(x) ``` Digamma function
``` erf(x) ``` Error function
``` erfc(x) ``` Complementary error function
``` gamma(x) ``` The Gamma function, Γ(x)
``` gammaIncomplete(a, x) ``` Incomplete gamma integral and its complement
``` gammaIncompleteCompl(a, x) ``` Incomplete gamma integral and its complement
``` gammaIncompleteComplInverse(a, p) ``` Inverse of complemented incomplete gamma integral
``` logGamma(x) ``` Natural logarithm of the gamma function, Γ(x)
``` logmdigamma(x) ``` Log Minus Digamma function
``` logmdigammaInverse(x) ``` Inverse of the Log Minus Digamma function
``` normalDistribution(x) ``` Standard normal distribution function.
``` normalDistributionInverse(p) ``` Inverse of Standard normal distribution function
``` sgnGamma(x) ``` The sign of Γ(x).

## Authors

Stephen L. Moshier (original C code). Conversion to D by Don Clugston