# beta function

The beta function is defined as

 $B(p,q)=\int_{0}^{1}x^{p-1}(1-x)^{q-1}dx$

for any real numbers $p,q>0$. For other complex values of $p$ and $q$, we can define $B(p,q)$ by analytic continuation.

The beta function has the property

 $B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$

for all complex numbers $p$ and $q$ for which the right-hand side is defined. Here $\Gamma$ is the gamma function.

Also,

 $B(p,q)=B(q,p)$

and

 $B({\textstyle\frac{1}{2},\frac{1}{2}})=\pi.$

The beta function was first defined by L. Euler (http://planetmath.org/EulerLeonhard) in 1730, and the name was given by J. Binet.

Title beta function BetaFunction 2013-03-22 13:26:23 2013-03-22 13:26:23 yark (2760) yark (2760) 21 yark (2760) Definition msc 33B15