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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.

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

Source: std/mathspecial.d

pure nothrow @nogc @safe real gamma(real x);
The Gamma function, Γ(x)
Γ(x) is a generalisation of the factorial function to real and complex numbers. Like x!, Γ(x+1) = x * Γ(x).

Mathematically, if > 0 then Γ(z) = 0 tz-1e-t dt

Special Values
x Γ(x)
±0.0 ±∞
integer > 0 (x-1)!
integer < 0 NAN
+∞ +∞
-∞ NAN
pure nothrow @nogc @safe real logGamma(real x);
Natural logarithm of the gamma function, Γ(x)
Returns the base e (2.718...) logarithm of the absolute value of the gamma function of the argument.

For reals, logGamma is equivalent to log(fabs(gamma(x))).

Special Values
x logGamma(x)
integer <= 0 +∞
±∞ +∞
pure nothrow @nogc @safe real sgnGamma(real x);
The sign of Γ(x).
Returns -1 if Γ(x) < 0, +1 if Γ(x) > 0, NAN if sign is indeterminate.

Note that this function can be used in conjunction with logGamma(x) to evaluate gamma for very large values of x.
pure nothrow @nogc @safe real beta(real x, real y);
Beta function
The beta function is defined as

beta(x, y) = (Γ(x) * Γ(y)) / Γ(x + y)
pure nothrow @nogc @safe real digamma(real x);
Digamma function
The digamma function is the logarithmic derivative of the gamma function.

digamma(x) = d/dx logGamma(x)
pure nothrow @nogc @safe real logmdigamma(real x);
Log Minus Digamma function
logmdigamma(x) = log(x) - digamma(x)
pure nothrow @nogc @safe real logmdigammaInverse(real x);
Inverse of the Log Minus Digamma function
Given y, the function finds x such log(x) - digamma(x) = y.
See Also:
pure nothrow @nogc @safe real betaIncomplete(real a, real b, real x);
Incomplete beta integral
Returns incomplete beta integral of the arguments, evaluated from zero to x. The regularized incomplete beta function is defined as

betaIncomplete(a, b, x) = Γ(a + b) / ( Γ(a) Γ(b) ) * 0x ta-1(1-t)b-1 dt

and is the same as the the cumulative distribution function.

The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation

betaIncompleteCompl(a, b, x ) = betaIncomplete( b, a, 1-x )

The integral is evaluated by a continued fraction expansion or, when b * x is small, by a power series.
pure nothrow @nogc @safe real betaIncompleteInverse(real a, real b, real y);
Inverse of incomplete beta integral
Given y, the function finds x such that

betaIncomplete(a, b, x) == y

Newton iterations or interval halving is used.
pure nothrow @nogc @safe real gammaIncomplete(real a, real x);
pure nothrow @nogc @safe real gammaIncompleteCompl(real a, real x);
Incomplete gamma integral and its complement
These functions are defined by

gammaIncomplete = ( 0x e-t ta-1 dt )/ Γ(a)

gammaIncompleteCompl(a,x) = 1 - gammaIncomplete(a,x) = (x e-t ta-1 dt )/ Γ(a)

In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x.
pure nothrow @nogc @safe real gammaIncompleteComplInverse(real a, real p);
Inverse of complemented incomplete gamma integral
Given a and p, the function finds x such that

gammaIncompleteCompl( a, x ) = p.
pure nothrow @nogc @safe real erf(real x);
Error function
The integral is

erf(x) = 2/ √(π) 0x exp( - t2) dt

The magnitude of x is limited to about 106.56 for IEEE 80-bit arithmetic; 1 or -1 is returned outside this range.
pure nothrow @nogc @safe real erfc(real x);
Complementary error function
erfc(x) = 1 - erf(x) = 2/ √(π) x exp( - t2) dt

This function has high relative accuracy for values of x far from zero. (For values near zero, use erf(x)).
pure nothrow @nogc @safe real normalDistribution(real x);
Normal distribution function.
The normal (or Gaussian, or bell-shaped) distribution is defined as:

normalDist(x) = 1/√(2π) -∞x exp( - t2/2) dt = 0.5 + 0.5 * erf(x/sqrt(2)) = 0.5 * erfc(- x/sqrt(2))

To maintain accuracy at values of x near 1.0, use normalDistribution(x) = 1.0 - normalDistribution(-x).

References:, G. Marsaglia, "Evaluating the Normal Distribution", Journal of Statistical Software 11, (July 2004).

pure nothrow @nogc @safe real normalDistributionInverse(real p);
Inverse of Normal distribution function
Returns the argument, x, for which the area under the Normal probability density function (integrated from minus infinity to x) is equal to p.