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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.
License:
Authors:
Stephen L. Moshier (original C code). Conversion to D by Don Clugston
Source std/mathspecial.d
- pure nothrow @nogc @safe real
gamma
(realx
); - 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 - pure nothrow @nogc @safe real
logGamma
(realx
); - 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 +∞ ±∞ +∞ - pure nothrow @nogc @safe real
sgnGamma
(realx
); - 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
(realx
, realy
); - Beta functionThe beta function is defined as beta(x, y) = (Γ(x) * Γ(y)) / Γ(x + y)
- pure nothrow @nogc @safe real
digamma
(realx
); - Digamma functionThe digamma function is the logarithmic derivative of the gamma function. digamma(x) = d/dx logGamma(x)See Also:
- pure nothrow @nogc @safe real
logmdigamma
(realx
); - Log Minus Digamma functionlogmdigamma(x) = log(x) - digamma(x)See Also:
- pure nothrow @nogc @safe real
logmdigammaInverse
(realx
); - Inverse of the Log Minus Digamma functionGiven y, the function finds x such log(x) - digamma(x) = y.See Also:
- pure nothrow @nogc @safe real
betaIncomplete
(reala
, realb
, realx
); - Incomplete beta integralReturns regularized 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 cumulative distribution function of the Beta distribution. 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
(reala
, realb
, realy
); - Inverse of incomplete beta integralGiven 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
(reala
, realx
);
pure nothrow @nogc @safe realgammaIncompleteCompl
(reala
, realx
); - 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.
- pure nothrow @nogc @safe real
gammaIncompleteComplInverse
(reala
, realp
); - Inverse of complemented incomplete gamma integralGiven a and p, the function finds x such that gammaIncompleteCompl( a, x ) = p.
- pure nothrow @nogc @safe real
erf
(realx
); - Error functionThe 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
(realx
); - Complementary error functionerfc(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
(realx
); - Standard 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).
- pure nothrow @nogc @safe real
normalDistributionInverse
(realp
); - Inverse of Standard 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.
Note This function is only implemented to 80 bit precision.
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