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25.1 Beta

The beta function, \(\text{B}(a, b)\), computes the normalizing constant for the beta distribution, and is defined for \(a > 0\) and \(b > 0\) by \[ \text{B}(a,b) \ = \ \int_0^1 u^{a - 1} (1 - u)^{b - 1} \, du \ = \ \frac{\Gamma(a) \, \Gamma(b)}{\Gamma(a+b)} \, , \] where \(\Gamma(x)\) is the Gamma function.