nLab
Euler beta function

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

Contents

Idea

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

B(x,y)= 0 1t x1(1t) y1dt=Γ(x)Γ(y)Γ(x+y),Rex>0,Rey>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.

References

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

Last revised on January 29, 2018 at 14:50:56. See the history of this page for a list of all contributions to it.