# nLab Euler beta function

The Euler beta function has been defined by Euler around 1730 by the so called Euler beta integral

$B\left(x,y\right)={\int }_{0}^{1}{t}^{x-1}\left(1-t{\right)}^{y-1}dt=\frac{\Gamma \left(x\right)\Gamma \left(y\right)}{\Gamma \left(x+y\right)},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{Re}x>0,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{Re}y>0,$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} d t = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},\,\,\,\,\,\,Re x\gt 0, \,\,\,\,Re y \gt 0,

which can be expressed in terms of the gamma function as stated.

A multidimensional generalization is the Selberg integral.

• G. E. Andrews, R. Askey, R. Roy, Special functions, Enc. of Math. and its Appl. 71, Cambridge Univ. Press 1999
Revised on October 10, 2011 20:50:58 by Zoran Škoda (161.53.130.104)