Beta Function expressed using Gamma Functions

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

Then:
 * $\map \Beta {x, y} = \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$

where $\Gamma$ is the Gamma function:

Proof
From Beta Function of x with y+m+1:


 * $\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y \, m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$

where $\Gamma_m$ is the partial Gamma function:
 * $\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \dotsm \paren {y + m} }$

From Partial Gamma Function expressed as Integral:

Thus:
 * $\displaystyle \lim_{m \mathop \to \infty} m^x \, \map \Beta {x, m + 1} = \map \Gamma x$

As $m^x$ is monotone, it does not matter if $m$ is integer or real.

Thus:


 * $\displaystyle \lim_{m \mathop \to \infty} \paren {m + y}^x \, \map \Beta {x, m + y + 1} = \map \Gamma x$

Hence: