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| − | There are a few equivalent definitions of $\Delta$-integration.
| + | __NOTOC__ |
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| − | ==Cauchy $\Delta$-integral==
| + | 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 [[time_scale | 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 $t \in D$. | + | $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s).$$ |
| − | 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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| − | $$\displaystyle\int_s^t f(\tau) \Delta \tau = F(t)-F(s)$$ | |
| − | for all $s,t \in \mathbb{T}$.
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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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| − | $$F(t) = \displaystyle\int_{t_0}^t f(\tau) \Delta \tau$$
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| − | is an antiderivative of $f$.
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| − | | |
| − | ==Riemann $\Delta$-integral==
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| − | | |
| − | ==Lebesgue $\Delta$-integral==
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| − | | |
| − | ==Related definitions==
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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== | | ==Properties of $\Delta$-integrals== |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | [[Delta integral from t to sigma(t)]]<br /> |
| − | <strong>Theorem:</strong> The following formula holds:
| + | [[Delta integral is linear]]<br /> |
| − | $$\int_t^{\sigma(t)} f(\tau) \Delta \tau = \mu(t)f(t)$$
| + | [[Interchanging limits of delta integral]]<br /> |
| − | <div class="mw-collapsible-content">
| + | [[Delta integrals are additive over intervals]]<br /> |
| − | <strong>Proof:</strong> █
| + | [[Integration by parts for delta integrals with sigma in integrand]]<br /> |
| − | </div>
| + | [[Integration by parts for delta integrals with no sigma in integrand]]<br /> |
| − | </div>
| + | [[Delta integral over degenerate interval]]<br /> |
| − | | + | [[Modulus of delta integral]]<br /> |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | [[Delta integral of nonnegative function]]<br /> |
| − | <strong>Theorem:</strong> The following formula holds:
| + | [[Delta integral of delta derivative]]<br /> |
| − | $$\int_a^b [f(t)+g(t)]\Delta t = \int_a^b f(t) \Delta t + \int_a^b g(t) \Delta t$$
| + | [[Delta derivative of the delta integral]]<br /> |
| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> If $\alpha$ is constant with respect to $t$, then
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| − | $$\int_a^b (\alpha f)(t) \Delta t=\alpha \int_a^b f(t) \Delta t.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div> | |
| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\int_a^b f(t) \Delta t = -\int_b^a f(t) \Delta t$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\int_a^b f(t) \Delta t = \int_a^c f(t) \Delta t + \int_c^b f(t) \Delta t.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem (Integration by Parts,I):</strong> The following formula holds:
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| − | $$\int_a^b f(\sigma(t))g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int_a^b f^{\Delta}(t)g(t) \Delta t.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem (Integration by Parts,II):</strong> The following formula holds:
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| − | $$\int_a^b f(t) g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int_a^b f^{\Delta}(t) g(\sigma(t)) \Delta t.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div> | |
| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> The following formula holds:
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| − | $$\int_a^a f(t) \Delta t = 0.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> If $|f(t)| \leq g(t)$ on $[a,b)$ then
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| − | $$\left| \int_a^b f(t) \Delta t \right| \leq \int_a^b g(t) \Delta t$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | | |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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| − | <strong>Theorem:</strong> If $f(t) \geq 0$ for all $a \leq t < b$ then
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| − | $$\displaystyle\int_a^b f(t) \Delta t \geq 0.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div> | |
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | ==References== |
| − | <strong>Theorem (Fundamental theorem of calculus,I):</strong> The following formula holds:
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| − | $$\int_a^b f^{\Delta}(t) \Delta t = f(b)-f(a).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
| + | [[Category:Definition]] |
| − | <strong>Theorem (Fundamental theorem of calculus,II):</strong> The following formula holds:
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| − | $$\left( \int_{t_0}^x f(\tau) \Delta \tau) \right)^{\Delta} = f(x).$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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