Beta function

A separate article treats the beta-function (written with a hyphen) of physics.

In mathematics, the beta function (occasionally written as Beta function), also called the Euler integral of the first kind, is a special function defined by

$\mathrm{\Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,dt \!$

for Re(x), Re(y) > 0. The beta function is symmetric, meaning that

$\mathrm{\Beta}(x,y) = \mathrm{\Beta}(y,x). \!$

It has many other forms, including:

$\mathrm{\Beta}(x,y)=\frac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \!$

$\mathrm{\Beta}(x,y) = 2\int_0^{\pi/2}\sin^{2x-1}\theta\cos^{2y-1}\theta\,d\theta, \qquad \Re(x)>0,\ \Re(y)>0 \!$

$\mathrm{\Beta}(x,y) = \int_0^\infty\frac{t^{x-1}}{(1+t)^{x+y}}\,dt, \qquad \Re(x)>0,\ \Re(y)>0 \!$

$\mathrm{\Beta}(x,y) = \frac{1}{y}\sum_{n=0}^\infty(-1)^n\frac{(y)_{n+1}}{n!(x+n)} \!$

where (x)_n is the falling factorial. Euler's beta function was the first known scattering amplitude in string theory, first conjectured by Gabriele Veneziano .

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