\begin{definition}[Definition:Beta Function/Definition 3]
The '''beta function''' $\Beta: \C \times \C \to \C$ is defined for $\map \Re x, \map \Re y > 0$ as:
:$\map \Beta {x, y} := \dfrac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$
where $\Gamma$ is the gamma function.
\end{definition}