# Commutativity of Parameters of Beta Function

## Theorem

Let $\map \Beta {x, y}$ be the Beta function.

Then:

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

## Proof

 $\ds \map \Beta {x, y}$ $=$ $\ds \frac {\map \Gamma x \map \Gamma y} {\map \Gamma {x + y} }$ Definition 3 of Beta Function $\ds$ $=$ $\ds \frac {\map \Gamma y \map \Gamma x} {\map \Gamma {y + x} }$ Commutative Law of Addition and Commutative Law of Multiplication $\ds$ $=$ $\ds \map \Beta {y, x}$ Definition 3 of Beta Function

$\blacksquare$