Properties of Beta Function

Theorem

Let $\Beta \left({x, y}\right)$ denote the Beta function:

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

$\Beta \left({x, y}\right)$ has the following properties:

Commutativity of Parameters of Beta Function

$\map \Beta {x, y} = \map \Beta {y, x}$

Beta Function of Real Number with 1

$\map \Beta {x, 1} = \map \Beta {1, x} = \dfrac 1 x$

Beta Function of $x$ with $y+1$ by $\dfrac {x+y} y$

$\map \Beta {x, y} = \dfrac {x + y} y \map \Beta {x, y + 1}$

Beta Function of $x+1$ with $y$ plus Beta Function of $x$ with $y+1$

$\map \Beta {x + 1, y} + \map \Beta {x, y + 1} = \map \Beta {x, y}$