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

References

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. (See §6.2)