Special Functions

Factorial

• Factorial: Factorial
• Logarithmic Factorial: FactorialLn
• Binomial Coefficient: Binomial
• Logarithmic Binomial Coefficient: BinomialLn
• Multinomial Coefficient: Multinomial

Code Sample:

double x = SpecialFunctions.Factorial(14);
// x will now have the value 87178291200.0

double y = SpecialFunctions.Factorial(31);
// y will now have the value 8.2228386541779224E+33

Gamma

Commonly used gamma-related functions in literature:

GAMMA(a) = int(exp(-t)t^(a-1), t=0..infinity)
gamma(a,x) = int(exp(-t)t^(a-1),t=0..x)
Gamma(a,x) = int(exp(-t)t^(a-1),t=x..infinity)
P(a,x) = gamma(a,x)/GAMMA(a)
Q(a,x) = Gamma(a,x)/GAMMA(a)

• Gamma: Gamma = a -> GAMMA(a)
• Logarithmic Gamma: GammaLn = a -> ln(abs(GAMMA(a))
• Lower Incomplete Gamma (Unregularized): GammaLowerIncomplete = (a,x) -> gamma(a,x)
• Upper Incomplete Gamma (Unregularized): GammaUpperIncomplete = (a,x) -> Gamma(a,x)
• Lower Incomplete Regularized Gamma: GammaLowerRegularized = (a,x) -> P(a,x)
• Upper Incomplete Regularized Gamma: GammaUpperRegularized = (a,x) -> Q(a,x)

Digamma (Psi)

• Digamma (Psi): DiGamma
• Inverse Digamma: DiGammaInv

Euler Beta

• Beta: Beta
• Logarithmic Beta: BetaLn
• Lower Incomplete Beta (Unregularized): BetaIncomplete
• Lower Incomplete Regularized Beta: BetaRegularized

Error Function

• Error Function: Erf
• Complementary Error Function: Erfc
• Inverse Error Function: ErfInv
• Inverse Complementary Error Function: ErfcInv

Code Sample:

double x = 0.9;
double res = SpecialFunctions.Erf(x);
// res will now have the value 0.7969082124

Logistic (Sigmoid)

• Logistic Function: Logistic
• Logit Function (inverse of Logistic): Logit

Harmonic Numbers

• t'th Harmonic Number: Harmonic
• Generalized Harmonic Number of order n of m: GeneralHarmonic

Various Numerically Stable Functions

• x -> exp(x)-1: ExponentialMinusOne
• (a,b) -> sqrt(a^2 + b^2): Hypotenuse (also for complex numbers)