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.

fn beta betaIncomplete betaIncompleteCompl betaIncompleteInverse digamma erf erfc gamma gammaIncomplete gammaIncompleteCompl gammaIncompleteComplInverse logGamma logmdigamma logmdigammaInverse normalDistribution normalDistributionInverse sgnGamma

Functions 17

fnreal gamma(real x) pure nothrow @safe @nogcThe Gamma function, (x)
fnreal logGamma(real x) pure nothrow @safe @nogcNatural logarithm of the gamma function, (x)
fnreal sgnGamma(real x) pure nothrow @safe @nogcThe sign of (x).
fnreal beta(real x, real y) pure nothrow @safe @nogcBeta function, B(x,y)
fnreal digamma(real x) pure nothrow @safe @nogcDigamma function, (x)
fnreal logmdigamma(real x) pure nothrow @safe @nogcLog Minus Digamma function
fnreal logmdigammaInverse(real x) pure nothrow @safe @nogcInverse of the Log Minus Digamma function
fnreal betaIncomplete(real a, real b, real x ) pure nothrow @safe @nogcRegularized incomplete beta function I(a,b) Mathematically, if a and b are positive real numbers, and 0 x 1, then I(a,b) = 0t(1-t, b-1)dt/B(a,b) where B is the beta function. It is also the c...
fnreal betaIncompleteCompl(real a, real b, real x) pure nothrow @safe @nogcRegularized incomplete beta function complement IC(a,b) Mathematically, if a 0, b 0, and 0 x 1, then IC(a,b) = xt(1-t, b-1)dt/B(a,b) where B is the beta function. It is also the complement ...
fnreal betaIncompleteInverse(real a, real b, real y ) pure nothrow @safe @nogcInverse of incomplete beta integral
fnreal gammaIncomplete(real a, real x ) pure nothrow @safe @nogcRegularized lower incomplete gamma function P(a,x)
fnreal gammaIncompleteCompl(real a, real x ) pure nothrow @safe @nogcRegularized upper incomplete gamma function Q(a,x)
fnreal gammaIncompleteComplInverse(real a, real p) pure nothrow @safe @nogcInverse regularized upper incomplete gamma function Q-1(a,p) with respect to p
fnreal erf(real x) pure nothrow @safe @nogcError function
fnreal erfc(real x) pure nothrow @safe @nogcComplementary error function
fnreal normalDistribution(real x) pure nothrow @safe @nogcStandard normal distribution function.
fnreal normalDistributionInverse(real p) pure nothrow @safe @nogcInverse of Standard normal distribution function