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.
- 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)
- 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
- Digamma functionThe digamma function is the logarithmic derivative of the gamma function. digamma(x) = d/dx logGamma(x)
- Log Minus Digamma function
- Inverse of the Log Minus Digamma functionGiven y, the function finds x such log(x) - digamma(x) = y.
- Incomplete beta integralReturns 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 integralGiven y, the function finds x such that betaIncomplete(a, b, x) == y Newton iterations or interval halving is used.
- Incomplete gamma integral and its complementThese 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 integralGiven a and p, the function finds x such that gammaIncompleteCompl( a, x ) = p.
- Error function
- Complementary error function
- 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: http://www.netlib.org/cephes/ldoubdoc.html, G. Marsaglia, "Evaluating the Normal Distribution", Journal of Statistical Software 11, (July 2004).
- Inverse of Normal distribution functionReturns the argument, x, for which the area under the Normal probability density function (integrated from minus infinity to x) is equal to p.