# uniform convergence of integral

Let the function $f(x,t)$ be continuous^{} in the domain

$$ |

where $b$ is a real number or $\mathrm{\infty}$, and let the improper integral

$F(t):={\displaystyle {\int}_{a}^{b}}f(x,t)\mathit{d}x=\underset{u\to b-}{lim}{\displaystyle {\int}_{a}^{u}}f(x,t)\mathit{d}x$ | (1) |

be convergent (http://planetmath.org/ImproperIntegral) in every point $t$ of the interval $[c,d]$. We say that the on the interval $[c,d]$, if for each positive number $\epsilon $ there is a value ${x}_{\epsilon}\in [a,b]$ such that

$$ |

when $$.

Title | uniform convergence of integral |
---|---|

Canonical name | UniformConvergenceOfIntegral |

Date of creation | 2013-03-22 14:40:30 |

Last modified on | 2013-03-22 14:40:30 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 26A42 |

Related topic | SumFunctionOfSeries |

Related topic | ConvergenceOfIntegrals |

Defines | integral converging uniformly |

Defines | uniformly converging integral |