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

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

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

Also,

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

and

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

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