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 beta function with respect to $N \in 2, 3, \ldots$ is the D4881: Complex function $$\begin{equation} Z^N \to \mathbb{C}, \quad (z_1, \dots, z_N) \mapsto \frac{\prod_{n = 1}^N \Gamma(z_n)}{\Gamma \left( \sum_{n = 1}^N z_n \right)} \end{equation}$$