Difference between revisions of "Delta integral"

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There are a few equivalent definitions of $\Delta$-integration.
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__NOTOC__
  
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==Cauchy definition 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
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Let $\mathbb{T}$ be a [[time_scale]]. We say that $f$ is regulated if its right-sided limits exist (i.e. are finite) at all right-dense points of $\mathbb{T}$ and its left-sided limits exist (i.e. are finite) at all left-dense points of $\mathbb{T}$. We say that $f$ is pre-differentiable with region of differentiation $D$ if $D \subset \mathbb{T}^{\kappa}$, $\mathbb{T}^{\kappa} \setminus D$ is countable with no right-scattered elements of $\mathbb{T}$, and $f$ is $\Delta$-differentiable at each $t \in D$.
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$$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s).$$
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Now suppose that $f$ is regulated. It is known that there exists a function $F$ which is pre-differentiable with region of differentiation $D$ such that $F^{\Delta}(t)=f(t)$. We define the indefinite integral of a regulated function $f$ by
 
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$$\displaystyle\int f(t) \Delta t = F(t)+C$$
 
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for an arbitrary constant $C$.
 
  
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Now we define the definite integral, i.e. the Cauchy integral, by the formula
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==Properties of $\Delta$-integrals==
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$$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s)$$
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[[Delta integral from t to sigma(t)]]<br />
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for all $s,t \in \mathbb{T}$.
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[[Delta integral is linear]]<br />
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[[Interchanging limits of delta integral]]<br />
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A function $F \colon \mathbb{T}\rightarrow \mathbb{R}$ is called an antiderivative of $f \colon \mathbb{T}\rightarrow \mathbb{R}$ if $F^{\Delta}(t)=f(t)$ for all $t \in \mathbb{T}^{\kappa}$. It is known that all rd-continuous functions possess an antiderivative, in particular if $t_0 \in \mathbb{T}$ then $F$ defined by
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[[Delta integrals are additive over intervals]]<br />
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$$F(t) = \displaystyle\int_{t_0}^t f(\tau) \Delta \tau$$
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[[Integration by parts for delta integrals with sigma in integrand]]<br />
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is an antiderivative of $f$.
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[[Integration by parts for delta integrals with no sigma in integrand]]<br />
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[[Delta integral over degenerate interval]]<br />
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==Riemann definition of $\Delta$-integral==
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[[Modulus of delta integral]]<br />
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[[Delta integral of nonnegative function]]<br />
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[[Delta integral of delta derivative]]<br />
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[[Delta derivative of the delta integral]]<br />
  
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==Related definitions==
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==References==
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If $a \in \mathbb{T}$, $\sup \mathbb{T}=\infty$, and $f$ is rd-continuous on $[a, \infty) \cap \mathbb{T}$ then we define the improper integral by
 
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$$\displaystyle\int_a^{\infty} f(t) \Delta t = \displaystyle\lim_{b \rightarrow \infty} \displaystyle\int_a^b f(t) \Delta t$$
 
  
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==Properties of $\Delta$-integrals==
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[[Category:Definition]]
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*$\displaystyle\int_t^{\sigma(t)} f(\tau) \Delta \tau = \mu(t)f(t)$
 
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*$\displaystyle\int_a^b [f(t)+g(t)]\Delta t = \displaystyle\int_a^b f(t) \Delta t + \displaystyle\int_a^b g(t) \Delta t$
 
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*If $\alpha$ is constant with respect to $t$, then
 
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$$\displaystyle\int_a^b (\alpha f)(t) \Delta t=\alpha \displaystyle\int_a^b f(t) \Delta t$$
 
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*$\displaystyle\int_a^b f(t) \Delta t = -\displaystyle\int_b^a f(t) \Delta t$
 
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*$\displaystyle\int_a^b f(t) \Delta t = \displaystyle\int_a^c f(t) \Delta t + \displaystyle\int_c^b f(t) \Delta t$
 
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*$\displaystyle\int_a^b f(\sigma(t))g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \displaystyle\int_a^b f^{\Delta}(t)g(t) \Delta t$
 
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*$\displaystyle\int_a^b f(t) g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \displaystyle\int_a^b f^{\Delta}(t) g(\sigma(t)) \Delta t$
 
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*$\displaystyle\int_a^a f(t) \Delta t = 0$
 
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*if $|f(t)| \leq g(t)$ on $[a,b)$ then
 
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$$\left| \displaystyle\int_a^b f(t) \Delta t \right| \leq \displaystyle\int_a^b g(t) \Delta t$$
 
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*if $f(t) \geq 0$ for all $a \leq t < b$ then $\displaystyle\int_a^b f(t) \Delta t \geq 0$
 

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

References

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