Difference between revisions of "Delta Taylor's formula"
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| − | Let $\mathbb{T}$ be a [[time scale]] | + | ==Theorem== |
| − | + | Let $\mathbb{T}$ be a [[time scale]] and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then | |
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$$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$ | $$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$ | ||
where $h_k$ denotes the [[Polynomials | $h_k$ polynomials]]. | where $h_k$ denotes the [[Polynomials | $h_k$ polynomials]]. | ||
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| − | + | ==Proof== | |
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| + | ==References== | ||
| + | *{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Theorem 11 | ||
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| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 17:04, 15 January 2023
Theorem
Let $\mathbb{T}$ be a time scale and $n \in \{1,2,\ldots\}$. Suppose $f$ is $n$-times differentiable on $\mathbb{T}^{\kappa^n}$. Let $\alpha \in \mathbb{T}^{\kappa^{n-1}}, t\in\mathbb{T}$ then $$f(t)=\displaystyle\sum_{k=0}^{n-1} h_k(t,\alpha) f^{\Delta^k}(\alpha) + \displaystyle\int_{\alpha}^{\rho^{n-1}(t)} h_{n-1}(t,\sigma(\tau)) f^{\Delta^n}(\tau) \Delta \tau,$$ where $h_k$ denotes the $h_k$ polynomials.
