\begin{definition}[Definition:Beta Function/Definition 1]
The '''beta function''' $\Beta: \C \times \C \to \C$ is defined for $\map \Re x, \map \Re y > 0$ as:
:$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
\end{definition}