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.