Definition:Beta Function/Definition 1

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The beta function $\Beta: \C \times \C \to \C$ is defined for $\map \Re x, \map \Re y > 0$ as:

$\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Also rendered as

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

$\ds \map \Beta {x, y} := \int_0^1 t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Also see

  • Results about the beta function can be found here.