# beta function

The *beta function ^{}* is defined as

$$B(p,q)={\int}_{0}^{1}{x}^{p-1}{(1-x)}^{q-1}\mathit{d}x$$ |

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

Title | beta function |
---|---|

Canonical name | BetaFunction |

Date of creation | 2013-03-22 13:26:23 |

Last modified on | 2013-03-22 13:26:23 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 21 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 33B15 |