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