# Gamma and Beta function

The Gamma and Beta function as described in 'Numerical Recipes in C++', the approximation is taken from [Lanczos, C. 1964 SIAM Journal on Numerical Analysis, ser. B, vol. 1, pp. 86-96]

cof :: [Double]
cof = [76.18009172947146,-86.50532032941677,24.01409824083091,-1.231739572450155,0.001208650973866179,-0.000005395239384953]

ser :: Double
ser = 1.000000000190015

gammaln :: Double -> Double
gammaln xx = let tmp' = (xx+5.5) - (xx+0.5)*log(xx+5.5)
ser' = foldl (+) ser $map (\(y,c) -> c/(xx+y))$ zip [1..] cof
in -tmp' + log(2.5066282746310005 * ser' / xx) where


the beta function relates to the gamma function by $\Beta(z,w)= \frac{\Gamma(z)\cdot\Gamma(w)}{\Gamma(z+w)}$, so we can compute the Beta function using gammaln like this:

beta z w = exp ((gammaln z) + (gammaln w) - (gammaln (z+w))