Gamma function

Γ (α) = \int_{0}^{\infty} y^{α - 1} e^{- y} d y, for y > 0

Pi function brings it back to the right factorial

Γ (x) = Π (x - 1) = \int_{0}^{\infty} t^{x} e^{- t} d t = \int_{0}^{\infty} (- \ln (t))^{x} d t = lim_{N \to \infty} N^{x} \prod_{k = 1}^{N} \frac{k}{x + k}

Properties

\begin{aligned} Γ (α) & = \int_{0}^{\infty} y^{α - 1} e^{- y} d y (integration by parts) \\ = {[y^{α - 1} (- e^{- y}) |}_{0}^{\infty} - \int_{0}^{\infty} - e^{- y} (α - 1) y^{α - 2} d y \\ = 0 - 0 + (α - 1) \int_{0}^{\infty} y^{α - 1 - 1} e^{- y} d y \\ = (α - 1) Γ (α - 1) \end{aligned}