Complete Beta Function

BetaC(x, y) = The Complete Beta function:

BetaC(x, y) = BetaC(y, x) = Gam(x) * Gam(y) / Gam(x+y).


BetaC(x, y) = 1 / (x * Bino(x+y−1, y−1).

The Gamma function Gam(n) is infinite if n is an integer <= 0, but BetaC(x, y) may have a finite value even when Gam(x), Gam(y) or Gam(x+y) are infinite.

Let gx = Gam(x), gy = Gam(y), gs = Gam(x+y), and B = BetaC(x, y). If gs is the only one of the three that is infinite, B = 0. If gx and gy are both infinite, or one is infinite without gs being infinite, B is infinite. If exactly one, say gy, of gx and gy is infinite and gs is also infinite, B has a finite value B = Gam(x) * Gam(1 − (x+y)) / Gam(1 − y). The sign of B is negative iff exactly one of the integers y and x+y is odd.

If none of gx, gy, gs are infinite, the equation BetaC(x, y) = 1 / (x * Bino(x+y−1, y−1) is used, where Bino(x, y) is the generalized binomial coefficient.

See: Beta Function -- From MathWorld
And: Wolfram Function Evaluation -- Beta

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