Eikonal Blog

2010.01.04

Beta function

Filed under: mathematics — Tags: , — sandokan65 @ 20:19

Definition:

    B(x,y):\equiv \int_0^1t^{x-1} (1-t)^y dt, (\Re x, \Re y > 0).

Properties:

  • B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}.
  • B(x,y) = (1-e^{2\pi i x})^{-1}(1-e^{2\pi i y})^{-1} \int_{1+,0+,1-,0-} \zeta^{x-1} (1-\zeta)^{y-1} d\zeta. where the closed countour goes once around z=1 in positive direction, the around z=0 in positive direction, the around z=1 in negative direction and z=0 in negative direction.

Source: “The Functions of Mathematical Physics” by Harry Hochstadt

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