Gamma Function¶
It extends the factorial function beyond the realm of nonnegative integers.
\(\Gamma(a) = \int_{0}^{\infty} x^a e^{-x}\frac{dx}{x}\)
Properties of the Gamma function:
\(\Gamma(a+1) = a\Gamma (a)​\)
\(\Gamma(n) = (n-1)! ​\) if n is a positive integer.
Multivariate:¶
\[\Gamma_D(x) = \pi^{(D)(D-1)/4} \prod_{i=1}^D \Gamma(x + (1 - i)/2)\]
\(\Gamma_1(a) = \Gamma(a)\) is the single variable gamma function.