Beta Function expressed using Gamma Functions
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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:
\(\ds \map {\Gamma_m} x\) | \(=\) | \(\ds m^x \int_0^1 \paren {1 - t}^m t^{x - 1} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds m^x \, \map \Beta {x, m + 1}\) |
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:
\(\ds \map \Beta {x, y}\) | \(=\) | \(\ds \lim_{m \mathop \to \infty} \dfrac {\map {\Gamma_m} y \, m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{m \mathop \to \infty} \dfrac {\map {\Gamma_m} y \, m^x} {\map {\Gamma_m} {x + y} } \frac {\map \Gamma x} {\paren {m + y}^x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{m \mathop \to \infty} \dfrac {\map {\Gamma_m} y \, \map \Gamma x} {\map {\Gamma_m} {x + y} }\) | as $\displaystyle \lim_{m \mathop \to \infty} \frac {m^x} {\paren {m + y}^x} = 1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\map \Gamma y \, \map \Gamma x} {\map \Gamma {x + y} }\) |
$\blacksquare$
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: Exercise $41 \ \text{(b)}$