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