Compute the logarithm of the gamma function Γ(x). Uses Algorithm AS 245 by Macleod.

Gives an accuracy of 10–12 significant decimal digits, except for small regions around x = 1 and x = 2, where the function goes to zero. For greater accuracy, use logGammaL.

Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).

Compute the logarithm of the gamma function, Γ(x). Uses a Lanczos approximation.

This function is slower than logGamma, but gives 14 or more significant decimal digits of accuracy, except around x = 1 and x = 2, where the function goes to zero.

Returns ∞ if the input is outside of the range (0 < x ≤ 1e305).

Compute the normalized lower incomplete gamma function γ(s,x). Normalization means that γ(s,∞)=1. Uses Algorithm AS 239 by Shea.

Inverse incomplete gamma function. It's approximately inverse of incompleteGamma for the same s. So following equality approximately holds:

 invIncompleteGamma s . incompleteGamma s = id

For invIncompleteGamma s p s must be positive and p must be in [0,1] range.

Compute the natural logarithm of the beta function.

Regularized incomplete beta function. Uses algorithm AS63 by Majumder and Bhattachrjee.

Regularized incomplete beta function. Same as incompleteBeta but also takes logarithm of beta function as parameter.

Compute inverse of regularized incomplete beta function. Uses initial approximation from AS109 and Halley method to solve equation.

Compute the natural logarithm of 1 + x. This is accurate even for values of x near zero, where use of log(1+x) would lose precision.

O(log n) Compute the logarithm in base 2 of the given value.

Compute the factorial function n!. Returns ∞ if the input is above 170 (above which the result cannot be represented by a 64-bit Double).

Compute the natural logarithm of the factorial function. Gives 16 decimal digits of precision.

Calculate the error term of the Stirling approximation. This is only defined for non-negative values.

 stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(n/e)^n)

Compute the binomial coefficient n `choose` k. For values of k > 30, this uses an approximation for performance reasons. The approximation is accurate to 12 decimal places in the worst case

Example:

 7 `choose` 3 == 35