BetaFunction 2003-02-09 03:44:12 2007-04-28 08:51:20 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{property} \PMlinkescapeword{side} The \emph{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 \PMlinkname{L.~Euler}{EulerLeonhard} in 1730, and the name was given by J.~Binet.