Let $Z : = \{ (x, y) \in \mathbb{C} : x > 0 \}$ be the D3740: Complex open right half-plane such that

(i)	$\Gamma : Z \to \mathbb{C}$ is the D258: Gamma function

The standard beta function is the D4881: Complex function \begin{equation} Z \times Z \to \mathbb{C}, \quad (z, w) \mapsto \frac{\Gamma(z) \Gamma(w)}{\Gamma(z + w)} \end{equation}