Definition:Beta Function

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The Beta Function $\Beta: \C \times \C \to \C$ is defined for $\operatorname{Re} \left({x}\right), \operatorname{Re} \left({y}\right) > 0$ as:

Definition 1

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

Definition 2

$\displaystyle \Beta \left({x, y}\right) := 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2 x - 1} \left({\cos \theta}\right)^{2 y - 1} \rd \theta$

Definition 3

$\map \Beta {x, y} := \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$

where $\Gamma$ is the Gamma function.

Also known as

The Beta function can also be referred to as the Eulerian Integral of the First Kind.

Also see

  • Results about the Beta function can be found here.

Historical Note

The beta function $\Beta \left({x, y}\right)$ was discovered by Leonhard Paul Euler.