Definition:Beta Function/Definition 1

Definition
The Beta Function $\Beta: \C \times \C \to \C$ is defined for $\operatorname{Re} \left({x}\right), \operatorname{Re} \left({y}\right) > 0$ as:
 * $\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \rd t$

Also rendered as
In a frequently-seen abuse of notation, the improper nature of the integral is often ignored, and the expression is rendered:


 * $\displaystyle \Beta \left({x, y}\right) := \int_0^1 t^{x - 1} \left({1 - t}\right)^{y - 1} \rd t$

Also see

 * Equivalence of Definitions of Beta Function