Beta Function expressed using Gamma Functions

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

Then:
 * $\Beta \left({x, y}\right) = \dfrac {\Gamma \left({x}\right) \Gamma \left({y}\right)} {\Gamma \left({x + y}\right)}$

where $\Gamma$ is the Gamma function:

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


 * $\Beta \left({x, y}\right) = \dfrac {\Gamma_m \left({y}\right) m^x} {\Gamma_m \left({x + y}\right)} \Beta \left({x, y + m + 1}\right)$

where $\Gamma_m$ is the partial Gamma function:
 * $\displaystyle \Gamma_m \left({y}\right) := \frac {m^y m!} {y \left({y + 1}\right) \left({y + 2}\right) \cdots \left({y + m}\right)}$

From Partial Gamma Function expressed as Integral:

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

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

Thus:


 * $\displaystyle \lim_{m \mathop \to \infty} \left({m + y}\right)^x \, \Beta \left({x, m + y + 1}\right) = \Gamma \left({x}\right)$

Hence: