\begin{definition}[Definition:Beta Function/Definition 2]
The '''beta function''' $\Beta: \C \times \C \to \C$ is defined for $\map \Re x, \map \Re y > 0$ as:
:$\ds \map \Beta {x, y} := 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta$
\end{definition}