# Difference between revisions of "Delta integral"

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− | + | Let $\mathbb{T}$ be a [[time scale]]. Delta integration is defined as the inverse operation of [[delta derivative|delta differentiation]] in the sense that if $F^{\Delta}(t)=f(t)$, then | |

− | Let $\mathbb{T}$ be a [[ | + | $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s).$$ |

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− | $$\displaystyle\ | ||

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− | + | ==Properties of $\Delta$-integrals== | |

− | + | [[Delta integral from t to sigma(t)]]<br /> | |

− | + | [[Delta integral is linear]]<br /> | |

− | + | [[Interchanging limits of delta integral]]<br /> | |

− | + | [[Delta integrals are additive over intervals]]<br /> | |

− | + | [[Integration by parts for delta integrals with sigma in integrand]]<br /> | |

− | + | [[Integration by parts for delta integrals with no sigma in integrand]]<br /> | |

− | + | [[Delta integral over degenerate interval]]<br /> | |

− | + | [[Modulus of delta integral]]<br /> | |

+ | [[Delta integral of nonnegative function]]<br /> | ||

+ | [[Delta integral of delta derivative]]<br /> | ||

+ | [[Delta derivative of the delta integral]]<br /> | ||

− | == | + | ==References== |

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− | + | [[Category:Definition]] | |

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## Latest revision as of 23:46, 22 August 2016

Let $\mathbb{T}$ be a time scale. Delta integration is defined as the inverse operation of delta differentiation in the sense that if $F^{\Delta}(t)=f(t)$, then
$$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s).$$

## Properties of $\Delta$-integrals

Delta integral from t to sigma(t)

Delta integral is linear

Interchanging limits of delta integral

Delta integrals are additive over intervals

Integration by parts for delta integrals with sigma in integrand

Integration by parts for delta integrals with no sigma in integrand

Delta integral over degenerate interval

Modulus of delta integral

Delta integral of nonnegative function

Delta integral of delta derivative

Delta derivative of the delta integral