# alternate integral representation of beta function

By making the change of variable ${x}^{p}=y$, we see that

$${\int}_{0}^{1}{x}^{p-1}{(1-x)}^{q-1}\mathit{d}x=\frac{1}{p}{\int}_{0}^{1}{(1-{y}^{\frac{1}{p}})}^{q-1}\mathit{d}y.$$ |

Hence, we have

$${\int}_{0}^{1}{(1-{y}^{\frac{1}{p}})}^{q-1}\mathit{d}y=p\frac{\mathrm{\Gamma}(p)\mathrm{\Gamma}(q)}{\mathrm{\Gamma}(p+q)}.$$ |

Title | alternate integral representation of beta function |
---|---|

Canonical name | AlternateIntegralRepresentationOfBetaFunction |

Date of creation | 2013-03-22 17:10:08 |

Last modified on | 2013-03-22 17:10:08 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 4 |

Author | rspuzio (6075) |

Entry type | Result |

Classification | msc 33B15 |