Definition:Beta Function

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

Definition 1

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

Definition 2

$\ds \map \Beta {x, y} := 2 \int_0^{\pi / 2} \paren {\sin \theta}^{2 x - 1} \paren {\cos \theta}^{2 y - 1} \rd \theta$

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.