nLab Euler beta function

This entry is about the concept in arithmetic. For the beta function related to renormalization group flow see there.

Idea

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

$$B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} d t = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)},\,\,\,\,\,\,Re x\gt 0, \,\,\,\,Re y \gt 0,$$

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

A multidimensional generalization is the Selberg integral.

Related concepts

• Gamma function

• factorial

References

• G. E. Andrews, R. Askey, R. Roy, Special functions, Enc. of Math. and its Appl. 71, Cambridge Univ. Press 1999