nLab
Euler beta function

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

Context

Arithmetic

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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.