Beta-function

$B$-function, Euler $B$-function, Euler integral of the first kind

2020 Mathematics Subject Classification: Primary: 33B15 [MSN][ZBL] $\newcommand{\Re}{\mathop{\mathrm{Re}}}$

A function of two variables $p$ and $q$ which, for $p,\,q > 0$, is defined by the equation \begin{equation} \label{eq1} B(p,q) = \int_0^1 x^{p-1}(1-x)^{q-1} \rd x. \end{equation} The values of the beta-function for various values of the parameters $p$ and $q$ are connected by the following relationships: $$ B(p,q) = B(q,p), $$ $$ B(p,q) = \frac{q-1}{p+q-1}B(p,q-1), \quad q > 1. $$ The following formula is valid: $$ B(p,1-p) = \frac{\pi}{\sin p\pi}, \quad 0 < p < 1. $$ If $p$ and $q$ are complex, the integral \ref{eq1} converges if $\Re p > 0$ and $\Re q > 0$. The beta-function can be expressed by the gamma-function: $$ B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}. $$