The D Programming Language

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.

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.
Boost License 1.0.
Stephen L. Moshier (original C code). Conversion to D by Don Clugston

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

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 +∞
±∞ +∞

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.

real  beta(real x, real y);

Beta function

The  beta function is defined as

 beta(x, y) = (Γ(x) * Γ(y)) / Γ(x + y)

real  digamma(real x);

Digamma function

The  digamma function is the logarithmic derivative of the gamma function.

 digamma(x) = d/dx logGamma(x)

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.

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.

real  gammaIncomplete(real a, real x);
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.

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.

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.

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)).

real  normalDistribution(real x);

Normal distribution function.

The normal (or Gaussian, or bell-shaped) distribution is defined as:

normalDist(x) = 1/√ π -∞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).

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.