std.mathspecialMathematical 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. License:
Boost License 1.0. Authors:
Stephen L. Moshier (original C code). Conversion to D by Don Clugston Source:
- 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 z.re > 0 then
Γ(z) = ∫0∞ tz-1e-t dt
Special Values x Γ(x) NAN NAN ±0.0 ±∞ integer > 0 (x-1)! integer < 0 NAN +∞ +∞ -∞ NAN
- 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) NAN NAN integer <= 0 +∞ ±∞ +∞
- 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.
- Beta function The beta function is defined as beta(x, y) = (Γ(x) * Γ(y)) / Γ(x + y)
- Digamma function The digamma function is the logarithmic derivative of the gamma function. digamma(x) = d/dx logGamma(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.
- 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.
- 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.
- Inverse of complemented incomplete gamma integral Given a and y, the function finds x such that gammaIncompleteCompl( a, x ) = p.
- 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.
- 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)).
- Normal distribution function.
The normal (or Gaussian, or bell-shaped) distribution is
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).
http://www.netlib.org/cephes/ldoubdoc.html, G. Marsaglia, "Evaluating the Normal Distribution", Journal of Statistical Software 11, (July 2004).
- 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.