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

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $17.3$: The Beta Function:Some Important Results