Definition:Beta Function

Definition
The Beta Function $B: \C \times \C \to \C$ is defined by:


 * $\displaystyle B \left({x, y}\right) = \int_0^1 t^{x - 1} \left({1-t}\right)^{y - 1} \ \mathrm d t$

or equivalently:


 * $\displaystyle B \left({x, y}\right) = 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2x - 1} \left({\cos \theta}\right)^{2y - 1} \ \mathrm d \theta$

for $\operatorname{Re} \left({x}\right), \operatorname{Re} \left({y}\right) > 0$.

It is also defined in terms of the Gamma function:


 * $B \left({x, y}\right) = \dfrac {\Gamma \left({x}\right) \Gamma \left({y}\right)} {\Gamma \left({x + y}\right)}$

for $\operatorname{Re} \left({x}\right), \operatorname{Re} \left({y}\right) > 0$.

Also known as
Also called the Eulerian Integral of the First Kind.

Also see

 * Definition:Gamma Function
 * Equivalence of Definitions of Beta Function