Beta Function as Integral of Power of t by Power of 1 minus t over Power of r plus t

Theorem

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

where $\Beta$ denotes the Beta function.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $17.6$: The Beta Function:Some Important Results